10, Semi-Rigorously Proved Theorems about the global limits of certain Third-\
    order rational recurreces with arbitrary posiitive initial conditions



                              By Shalosh B. Ekhad





                             Theorem number, 1, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                            1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
                x[n + 1] = -------------------------------------
                           1 + 2 x[n - 2] + 3 x[n - 1] + 35 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], -----------------------------]
                                            1 + 2 x[1] + 3 x[2] + 35 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], -----------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 3 x[2] + 35 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+3*a[2]+35*a
[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+3*a[2]+35*a[3]), (2*a[1]*a[2]+4
*a[1]*a[3]+3*a[2]^2+41*a[2]*a[3]+70*a[3]^2+39*a[1]+78*a[2]+1406*a[3]+38)/(4*a[1
]*a[2]+6*a[1]*a[3]+6*a[2]^2+79*a[2]*a[3]+105*a[3]^2+37*a[1]+75*a[2]+1333*a[3]+
36)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+3*a[2]+35*a[3]), (2*a[1]*a[2]+4*a[1]*a
[3]+3*a[2]^2+41*a[2]*a[3]+70*a[3]^2+39*a[1]+78*a[2]+1406*a[3]+38)/(4*a[1]*a[2]+
6*a[1]*a[3]+6*a[2]^2+79*a[2]*a[3]+105*a[3]^2+37*a[1]+75*a[2]+1333*a[3]+36), (8*
a[1]^2*a[2]*a[3]+12*a[1]^2*a[3]^2+24*a[1]*a[2]^2*a[3]+316*a[1]*a[2]*a[3]^2+420*
a[1]*a[3]^3+18*a[2]^3*a[3]+447*a[2]^2*a[3]^2+3080*a[2]*a[3]^3+3675*a[3]^4+164*a
[1]^2*a[2]+394*a[1]^2*a[3]+496*a[1]*a[2]^2+7127*a[1]*a[2]*a[3]+15401*a[1]*a[3]^
2+375*a[2]^3+9874*a[2]^2*a[3]+76914*a[2]*a[3]^2+148855*a[3]^3+3034*a[1]^2+10746
*a[1]*a[2]+164189*a[1]*a[3]+9312*a[2]^2+276519*a[2]*a[3]+1971565*a[3]^2+4510*a[
1]+7581*a[2]+109191*a[3]+1514)/(16*a[1]^2*a[2]*a[3]+24*a[1]^2*a[3]^2+48*a[1]*a[
2]^2*a[3]+632*a[1]*a[2]*a[3]^2+840*a[1]*a[3]^3+36*a[2]^3*a[3]+894*a[2]^2*a[3]^2
+6160*a[2]*a[3]^3+7350*a[3]^4+160*a[1]^2*a[2]+458*a[1]^2*a[3]+486*a[1]*a[2]^2+
7303*a[1]*a[2]*a[3]+19135*a[1]*a[3]^2+369*a[2]^3+10029*a[2]^2*a[3]+83460*a[2]*a
[3]^2+194600*a[3]^3+2915*a[1]^2+10349*a[1]*a[2]+158644*a[1]*a[3]+8994*a[2]^2+
268244*a[2]*a[3]+1925024*a[3]^2+4353*a[1]+7344*a[2]+106420*a[3]+1474)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
12*a[1]*a[2]^3+140*a[1]*a[2]^2*a[3]+12*a[1]*a[2]*a[3]^2+140*a[1]*a[3]^3+9*a[2]^
4+210*a[2]^3*a[3]+1234*a[2]^2*a[3]^2+210*a[2]*a[3]^3+1225*a[3]^4-8*a[1]^2*a[2]-\
8*a[1]^2*a[3]-20*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-276*a[1]*a[3]^2-12*a[2]^3-368*a
[2]^2*a[3]-2864*a[2]*a[3]^2-2380*a[3]^3+9*a[1]^2+18*a[1]*a[2]+268*a[1]*a[3]+8*a
[2]^2+264*a[2]*a[3]+2315*a[3]^2+8*a[1]+10*a[2]+138*a[3]+2)/(1+2*a[1]+3*a[2]+35*
a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+192*a[1]^4*a[2]*a[3]^3+144*a[1]^4*a[3]^4+384*
a[1]^3*a[2]^3*a[3]^2+5632*a[1]^3*a[2]^2*a[3]^3+14304*a[1]^3*a[2]*a[3]^4+10080*a
[1]^3*a[3]^5+864*a[1]^2*a[2]^4*a[3]^2+22752*a[1]^2*a[2]^3*a[3]^3+180024*a[1]^2*
a[2]^2*a[3]^4+398160*a[1]^2*a[2]*a[3]^5+264600*a[1]^2*a[3]^6+864*a[1]*a[2]^5*a[
3]^2+32832*a[1]*a[2]^4*a[3]^3+445464*a[1]*a[2]^3*a[3]^4+2498440*a[1]*a[2]^2*a[3
]^5+4909800*a[1]*a[2]*a[3]^6+3087000*a[1]*a[3]^7+324*a[2]^6*a[3]^2+16092*a[2]^5
*a[3]^3+310689*a[2]^4*a[3]^4+2885820*a[2]^3*a[3]^5+12771850*a[2]^2*a[3]^6+
22638000*a[2]*a[3]^7+13505625*a[3]^8-128*a[1]^4*a[2]^2*a[3]+800*a[1]^4*a[2]*a[3
]^2+1488*a[1]^4*a[3]^3-768*a[1]^3*a[2]^3*a[3]-3472*a[1]^3*a[2]^2*a[3]^2+87992*a
[1]^3*a[2]*a[3]^3+137208*a[1]^3*a[3]^4-1728*a[1]^2*a[2]^4*a[3]-26424*a[1]^2*a[2
]^3*a[3]^2+176796*a[1]^2*a[2]^2*a[3]^3+3298296*a[1]^2*a[2]*a[3]^4+4469220*a[1]^
2*a[3]^5-1728*a[1]*a[2]^5*a[3]-45036*a[1]*a[2]^4*a[3]^2-43470*a[1]*a[2]^3*a[3]^
3+6856834*a[1]*a[2]^2*a[3]^4+51747710*a[1]*a[2]*a[3]^5+62261850*a[1]*a[3]^6-648
*a[2]^6*a[3]-23868*a[2]^5*a[3]^2-173556*a[2]^4*a[3]^3+3327162*a[2]^3*a[3]^4+
59248070*a[2]^2*a[3]^5+292032650*a[2]*a[3]^6+316674750*a[3]^7+96*a[1]^4*a[2]^2-\
2096*a[1]^4*a[2]*a[3]+2120*a[1]^4*a[3]^2+560*a[1]^3*a[2]^3-7904*a[1]^3*a[2]^2*a
[3]-173928*a[1]^3*a[2]*a[3]^2+287312*a[1]^3*a[3]^3+1228*a[1]^2*a[2]^4-7788*a[1]
^2*a[2]^3*a[3]-733226*a[1]^2*a[2]^2*a[3]^2-4505904*a[1]^2*a[2]*a[3]^3+13114650*
a[1]^2*a[3]^4+1200*a[1]*a[2]^5+1832*a[1]*a[2]^4*a[3]-1000404*a[1]*a[2]^3*a[3]^2
-14959018*a[1]*a[2]^2*a[3]^3-37064480*a[1]*a[2]*a[3]^4+240492350*a[1]*a[3]^5+
441*a[2]^6+4206*a[2]^5*a[3]-447362*a[2]^4*a[3]^2-11317090*a[2]^3*a[3]^3-\
84262495*a[2]^2*a[3]^4-4258800*a[2]*a[3]^5+1533234500*a[3]^6+1448*a[1]^4*a[2]-\
8764*a[1]^4*a[3]+9252*a[1]^3*a[2]^2+48778*a[1]^3*a[2]*a[3]-986806*a[1]^3*a[3]^2
+22004*a[1]^2*a[2]^3+357184*a[1]^2*a[2]^2*a[3]-2136718*a[1]^2*a[2]*a[3]^2-\
39278106*a[1]^2*a[3]^3+23148*a[1]*a[2]^4+625588*a[1]*a[2]^3*a[3]+585080*a[1]*a[
2]^2*a[3]^2-96835008*a[1]*a[2]*a[3]^3-650968144*a[1]*a[3]^4+9108*a[2]^5+344394*
a[2]^4*a[3]+2407904*a[2]^3*a[3]^2-53935706*a[2]^2*a[3]^3-907106016*a[2]*a[3]^4-\
3829718620*a[3]^5+6861*a[1]^4+49572*a[1]^3*a[2]+652730*a[1]^3*a[3]+131607*a[1]^
2*a[2]^2+3416948*a[1]^2*a[2]*a[3]+21786463*a[1]^2*a[3]^2+152886*a[1]*a[2]^3+
5887020*a[1]*a[2]^2*a[3]+74349874*a[1]*a[2]*a[3]^2+307512764*a[1]*a[3]^3+65880*
a[2]^4+3360378*a[2]^3*a[3]+63671673*a[2]^2*a[3]^2+536112524*a[2]*a[3]^3+
1743260445*a[3]^4+18844*a[1]^3+96334*a[1]^2*a[2]+1304032*a[1]^2*a[3]+162744*a[1
]*a[2]^2+4361502*a[1]*a[2]*a[3]+28820470*a[1]*a[3]^2+91314*a[2]^3+3663570*a[2]^
2*a[3]+49006658*a[2]*a[3]^2+223867642*a[3]^3+18557*a[1]^2+61714*a[1]*a[2]+
832590*a[1]*a[3]+51522*a[2]^2+1403836*a[2]*a[3]+9753986*a[3]^2+7872*a[1]+13212*
a[2]+184676*a[3]+1300)/(4*a[1]*a[2]+6*a[1]*a[3]+6*a[2]^2+79*a[2]*a[3]+105*a[3]^
2+37*a[1]+75*a[2]+1333*a[3]+36)^2/(1+2*a[1]+3*a[2]+35*a[3])^2, (12288*a[1]^8*a[
2]^4*a[3]^2+69632*a[1]^8*a[2]^3*a[3]^3+148480*a[1]^8*a[2]^2*a[3]^4+141312*a[1]^
8*a[2]*a[3]^5+50688*a[1]^8*a[3]^6+143360*a[1]^7*a[2]^5*a[3]^2+2371584*a[1]^7*a[
2]^4*a[3]^3+10516480*a[1]^7*a[2]^3*a[3]^4+20271104*a[1]^7*a[2]^2*a[3]^5+
18214656*a[1]^7*a[2]*a[3]^6+6287616*a[1]^7*a[3]^7+732160*a[1]^6*a[2]^6*a[3]^2+
20099072*a[1]^6*a[2]^5*a[3]^3+185909760*a[1]^6*a[2]^4*a[3]^4+687289856*a[1]^6*a
[2]^3*a[3]^5+1212435904*a[1]^6*a[2]^2*a[3]^6+1032484224*a[1]^6*a[2]*a[3]^7+
343468224*a[1]^6*a[3]^8+2138112*a[1]^5*a[2]^7*a[3]^2+82102272*a[1]^5*a[2]^6*a[3
]^3+1187036160*a[1]^5*a[2]^5*a[3]^4+7972343808*a[1]^5*a[2]^4*a[3]^5+25535318272
*a[1]^5*a[2]^3*a[3]^6+41654456960*a[1]^5*a[2]^2*a[3]^7+33750064896*a[1]^5*a[2]*
a[3]^8+10839669120*a[1]^5*a[3]^9+3905280*a[1]^4*a[2]^8*a[3]^2+192890880*a[1]^4*
a[2]^7*a[3]^3+3822662400*a[1]^4*a[2]^6*a[3]^4+38567808640*a[1]^4*a[2]^5*a[3]^5+
208247778960*a[1]^4*a[2]^4*a[3]^6+593241529280*a[1]^4*a[2]^3*a[3]^7+
903013344160*a[1]^4*a[2]^2*a[3]^8+699080894400*a[1]^4*a[2]*a[3]^9+217304514000*
a[1]^4*a[3]^10+4568832*a[1]^3*a[2]^9*a[3]^2+276127488*a[1]^3*a[2]^8*a[3]^3+
6976321920*a[1]^3*a[2]^7*a[3]^4+94999248128*a[1]^3*a[2]^6*a[3]^5+749349821872*a
[1]^3*a[2]^5*a[3]^6+3434975292240*a[1]^3*a[2]^4*a[3]^7+8872936109920*a[1]^3*a[2
]^3*a[3]^8+12707142711200*a[1]^3*a[2]^2*a[3]^9+9442198374000*a[1]^3*a[2]*a[3]^
10+2849393610000*a[1]^3*a[3]^11+3343680*a[1]^2*a[2]^10*a[3]^2+239210496*a[1]^2*
a[2]^9*a[3]^3+7362148320*a[1]^2*a[2]^8*a[3]^4+126894267936*a[1]^2*a[2]^7*a[3]^5
+1334805924804*a[1]^2*a[2]^6*a[3]^6+8756896099704*a[1]^2*a[2]^5*a[3]^7+
35264786420460*a[1]^2*a[2]^4*a[3]^8+83852875458800*a[1]^2*a[2]^3*a[3]^9+
113838269681500*a[1]^2*a[2]^2*a[3]^10+81575266395000*a[1]^2*a[2]*a[3]^11+
23984099212500*a[1]^2*a[3]^12+1399680*a[1]*a[2]^11*a[3]^2+115769088*a[1]*a[2]^
10*a[3]^3+4207645440*a[1]*a[2]^9*a[3]^4+88020998784*a[1]*a[2]^8*a[3]^5+
1165084954056*a[1]*a[2]^7*a[3]^6+10095612703296*a[1]*a[2]^6*a[3]^7+
57277601218856*a[1]*a[2]^5*a[3]^8+207495381525600*a[1]*a[2]^4*a[3]^9+
459674516959000*a[1]*a[2]^3*a[3]^10+595581481880000*a[1]*a[2]^2*a[3]^11+
413478310875000*a[1]*a[2]*a[3]^12+118877591700000*a[1]*a[3]^13+256608*a[2]^12*a
[3]^2+24109488*a[2]^11*a[3]^3+1011798540*a[2]^10*a[3]^4+24946380432*a[2]^9*a[3]
^5+399523703913*a[2]^8*a[3]^6+4334819521680*a[2]^7*a[3]^7+32227076185984*a[2]^6
*a[3]^8+162415697365880*a[2]^5*a[3]^9+538482445940250*a[2]^4*a[3]^10+
1122171809990000*a[2]^3*a[3]^11+1395614353787500*a[2]^2*a[3]^12+942490760925000
*a[2]*a[3]^13+265851812953125*a[3]^14+159744*a[1]^8*a[2]^4*a[3]+1314816*a[1]^8*
a[2]^3*a[3]^2+3673088*a[1]^8*a[2]^2*a[3]^3+4328704*a[1]^8*a[2]*a[3]^4+1857408*a
[1]^8*a[3]^5+1843200*a[1]^7*a[2]^5*a[3]+34853888*a[1]^7*a[2]^4*a[3]^2+209456384
*a[1]^7*a[2]^3*a[3]^3+522724864*a[1]^7*a[2]^2*a[3]^4+581955776*a[1]^7*a[2]*a[3]
^5+241267008*a[1]^7*a[3]^6+9307136*a[1]^6*a[2]^6*a[3]+275525888*a[1]^6*a[2]^5*a
[3]^2+2967162368*a[1]^6*a[2]^4*a[3]^3+14328059840*a[1]^6*a[2]^3*a[3]^4+
32353500224*a[1]^6*a[2]^2*a[3]^5+34070167936*a[1]^6*a[2]*a[3]^6+13632079488*a[1
]^6*a[3]^7+26865408*a[1]^5*a[2]^7*a[3]+1083330304*a[1]^5*a[2]^6*a[3]^2+
17183827776*a[1]^5*a[2]^5*a[3]^3+135421084992*a[1]^5*a[2]^4*a[3]^4+553623337152
*a[1]^5*a[2]^3*a[3]^5+1143780278848*a[1]^5*a[2]^2*a[3]^6+1142002387136*a[1]^5*a
[2]*a[3]^7+441016377024*a[1]^5*a[3]^8+48493440*a[1]^4*a[2]^8*a[3]+2477223360*a[
1]^4*a[2]^7*a[3]^2+52267433600*a[1]^4*a[2]^6*a[3]^3+585583846320*a[1]^4*a[2]^5*
a[3]^4+3722964723160*a[1]^4*a[2]^4*a[3]^5+13326447979840*a[1]^4*a[2]^3*a[3]^6+
25455200771600*a[1]^4*a[2]^2*a[3]^7+24187406163280*a[1]^4*a[2]*a[3]^8+
9028661976600*a[1]^4*a[3]^9+56059776*a[1]^3*a[2]^9*a[3]+3469857984*a[1]^3*a[2]^
8*a[3]^2+91703216112*a[1]^3*a[2]^7*a[3]^3+1345112678336*a[1]^3*a[2]^6*a[3]^4+
11893409995852*a[1]^3*a[2]^5*a[3]^5+64333583842660*a[1]^3*a[2]^4*a[3]^6+
206636344669480*a[1]^3*a[2]^3*a[3]^7+368714145549640*a[1]^3*a[2]^2*a[3]^8+
335159602003100*a[1]^3*a[2]*a[3]^9+121315440508500*a[1]^3*a[3]^10+40538880*a[1]
^2*a[2]^10*a[3]+2950258032*a[1]^2*a[2]^9*a[3]^2+93919676256*a[1]^2*a[2]^8*a[3]^
3+1712068556196*a[1]^2*a[2]^7*a[3]^4+19611154207384*a[1]^2*a[2]^6*a[3]^5+
145459453215792*a[1]^2*a[2]^5*a[3]^6+693106909050780*a[1]^2*a[2]^4*a[3]^7+
2036715436818660*a[1]^2*a[2]^3*a[3]^8+3429929023463600*a[1]^2*a[2]^2*a[3]^9+
3001801262887000*a[1]^2*a[2]*a[3]^10+1058578201117500*a[1]^2*a[3]^11+16768944*a
[1]*a[2]^11*a[3]+1404263952*a[1]*a[2]^10*a[3]^2+52399559772*a[1]*a[2]^9*a[3]^3+
1145965567140*a[1]*a[2]^8*a[3]^4+16227696699486*a[1]*a[2]^7*a[3]^5+
154840436831414*a[1]*a[2]^6*a[3]^6+1002138786022242*a[1]*a[2]^5*a[3]^7+
4313136970656770*a[1]*a[2]^4*a[3]^8+11778946909600850*a[1]*a[2]^3*a[3]^9+
18900352878594250*a[1]*a[2]^2*a[3]^10+16032375951121250*a[1]*a[2]*a[3]^11+
5537552367506250*a[1]*a[3]^12+3038472*a[2]^12*a[3]+288084276*a[2]^11*a[3]^2+
12351019644*a[2]^10*a[3]^3+315999778608*a[2]^9*a[3]^4+5354644192308*a[2]^8*a[3]
^5+62940471866502*a[2]^7*a[3]^6+521454652919274*a[2]^6*a[3]^7+3026932058709626*
a[2]^5*a[3]^8+11986780262268350*a[2]^4*a[3]^9+30795148949447750*a[2]^3*a[3]^10+
47446546841986250*a[2]^2*a[3]^11+39224628453193750*a[2]*a[3]^12+
13328459516793750*a[3]^13+820224*a[1]^8*a[2]^4+10847232*a[1]^8*a[2]^3*a[3]+
41022336*a[1]^8*a[2]^2*a[3]^2+60924608*a[1]^8*a[2]*a[3]^3+31733776*a[1]^8*a[3]^
4+9493504*a[1]^7*a[2]^5+228616960*a[1]^7*a[2]^4*a[3]+1876060352*a[1]^7*a[2]^3*a
[3]^2+6093028512*a[1]^7*a[2]^2*a[3]^3+8460420032*a[1]^7*a[2]*a[3]^4+4255507232*
a[1]^7*a[3]^5+48080768*a[1]^6*a[2]^6+1679477312*a[1]^6*a[2]^5*a[3]+22213311008*
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912677035904399429*a[3]^6+128254818398*a[1]^5+960124435326*a[1]^4*a[2]+
10709734984254*a[1]^4*a[3]+2816264605258*a[1]^3*a[2]^2+59928255756408*a[1]^3*a[
2]*a[3]+283366401464154*a[1]^3*a[3]^2+4040654510150*a[1]^2*a[2]^3+
122150263621156*a[1]^2*a[2]^2*a[3]+1059335511167074*a[1]^2*a[2]*a[3]^2+
2212923512144100*a[1]^2*a[3]^3+2833303090518*a[1]*a[2]^4+107413845338344*a[1]*a
[2]^3*a[3]+1265192044799900*a[1]*a[2]^2*a[3]^2+4240346416897000*a[1]*a[2]*a[3]^
3-2163902218020866*a[1]*a[3]^4+777187046754*a[2]^5+34520326337826*a[2]^4*a[3]+
493246990445360*a[2]^3*a[3]^2+2230613124756032*a[2]^2*a[3]^3+2749828913221230*a
[2]*a[3]^4+45780020440787454*a[3]^5+76691997405*a[1]^4+431811029380*a[1]^3*a[2]
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]*a[3]+53444831414639*a[1]^2*a[3]^2+789351237180*a[1]*a[2]^3+19381502445232*a[1
]*a[2]^2*a[3]+109665551341068*a[1]*a[2]*a[3]^2+4196943445000*a[1]*a[3]^3+
257145177573*a[2]^4+7751916731628*a[2]^3*a[3]+60526064335850*a[2]^2*a[3]^2+
134099323779124*a[2]*a[3]^3+1452703496003233*a[3]^4+24511762968*a[1]^3+
94550660676*a[1]^2*a[2]+676307828420*a[1]^2*a[3]+117653733276*a[1]*a[2]^2+
1456525298344*a[1]*a[2]*a[3]+1190910541612*a[1]*a[3]^2+48008500344*a[2]^3+
829452266040*a[2]^2*a[3]+3289201391896*a[2]*a[3]^2+30042179066136*a[3]^3+
3517307124*a[1]^2+7858746704*a[1]*a[2]+21920001248*a[1]*a[3]+4580893524*a[2]^2+
41219830520*a[2]*a[3]+397352880212*a[3]^2+132309360*a[1]+210470544*a[2]+
3083686192*a[3]+10764304)/(16*a[1]^2*a[2]*a[3]+24*a[1]^2*a[3]^2+48*a[1]*a[2]^2*
a[3]+632*a[1]*a[2]*a[3]^2+840*a[1]*a[3]^3+36*a[2]^3*a[3]+894*a[2]^2*a[3]^2+6160
*a[2]*a[3]^3+7350*a[3]^4+160*a[1]^2*a[2]+458*a[1]^2*a[3]+486*a[1]*a[2]^2+7303*a
[1]*a[2]*a[3]+19135*a[1]*a[3]^2+369*a[2]^3+10029*a[2]^2*a[3]+83460*a[2]*a[3]^2+
194600*a[3]^3+2915*a[1]^2+10349*a[1]*a[2]+158644*a[1]*a[3]+8994*a[2]^2+268244*a
[2]*a[3]+1925024*a[3]^2+4353*a[1]+7344*a[2]+106420*a[3]+1474)^2/(4*a[1]*a[2]+6*
a[1]*a[3]+6*a[2]^2+79*a[2]*a[3]+105*a[3]^2+37*a[1]+75*a[2]+1333*a[3]+36)^2/(1+2
*a[1]+3*a[2]+35*a[3])^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+9830400*a[1]^10*a[2]^3*a[3]^3+22118400*a[1
]^10*a[2]^2*a[3]^4+22118400*a[1]^10*a[2]*a[3]^5+8294400*a[1]^10*a[3]^6+19660800
*a[1]^9*a[2]^5*a[3]^2+347340800*a[1]^9*a[2]^4*a[3]^3+1641676800*a[1]^9*a[2]^3*a
[3]^4+3361996800*a[1]^9*a[2]^2*a[3]^5+3196108800*a[1]^9*a[2]*a[3]^6+1161216000*
a[1]^9*a[3]^7+104857600*a[1]^8*a[2]^6*a[3]^2+3037593600*a[1]^8*a[2]^5*a[3]^3+
29917184000*a[1]^8*a[2]^4*a[3]^4+118235136000*a[1]^8*a[2]^3*a[3]^5+222732288000
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[1]^7*a[2]^5*a[3]^4+1404588851200*a[1]^7*a[2]^4*a[3]^5+4810054963200*a[1]^7*a[2
]^3*a[3]^6+8404662988800*a[1]^7*a[2]^2*a[3]^7+7281601228800*a[1]^7*a[2]*a[3]^8+
2490518016000*a[1]^7*a[3]^9+683827200*a[1]^6*a[2]^8*a[3]^2+33606451200*a[1]^6*a
[2]^7*a[3]^3+674568294400*a[1]^6*a[2]^6*a[3]^4+7027767398400*a[1]^6*a[2]^5*a[3]
^5+39906616870400*a[1]^6*a[2]^4*a[3]^6+121184449075200*a[1]^6*a[2]^3*a[3]^7+
197693893555200*a[1]^6*a[2]^2*a[3]^8+164084514048000*a[1]^6*a[2]*a[3]^9+
54525804480000*a[1]^6*a[3]^10+1006387200*a[1]^5*a[2]^9*a[3]^2+57518899200*a[1]^
5*a[2]^8*a[3]^3+1397549260800*a[1]^5*a[2]^7*a[3]^4+18725422284800*a[1]^5*a[2]^6
*a[3]^5+149277205324800*a[1]^5*a[2]^5*a[3]^6+709994586649600*a[1]^5*a[2]^4*a[3]
^7+1941065132697600*a[1]^5*a[2]^3*a[3]^8+2971149624576000*a[1]^5*a[2]^2*a[3]^9+
2366981205120000*a[1]^5*a[2]*a[3]^10+764854876800000*a[1]^5*a[3]^11+1103155200*
a[1]^4*a[2]^10*a[3]^2+70292275200*a[1]^4*a[2]^9*a[3]^3+1938543667200*a[1]^4*a[2
]^8*a[3]^4+30434891827200*a[1]^4*a[2]^7*a[3]^5+299984504915200*a[1]^4*a[2]^6*a[
3]^6+1915334041152000*a[1]^4*a[2]^5*a[3]^7+7810483935968000*a[1]^4*a[2]^4*a[3]^
8+19383530355840000*a[1]^4*a[2]^3*a[3]^9+27923907967200000*a[1]^4*a[2]^2*a[3]^
10+21383056296000000*a[1]^4*a[2]*a[3]^11+6725152980000000*a[1]^4*a[3]^12+
920678400*a[1]^3*a[2]^11*a[3]^2+64455782400*a[1]^3*a[2]^10*a[3]^3+1955162880000
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a[2]^5*a[3]^8+49712795779360000*a[1]^3*a[2]^4*a[3]^9+111642052250400000*a[1]^3*
a[2]^3*a[3]^10+151176823308000000*a[1]^3*a[2]^2*a[3]^11+111314441700000000*a[1]
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223948800*a[1]*a[2]^13*a[3]^2+19632844800*a[1]*a[2]^12*a[3]^3+753102489600*a[1]
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a[3]^6+2081065469932800*a[1]*a[2]^8*a[3]^7+12412354078732800*a[1]*a[2]^7*a[3]^8
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33353491500000000*a[1]*a[3]^15+41990400*a[2]^14*a[3]^2+4171046400*a[2]^13*a[3]^
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36173937694944000*a[2]^7*a[3]^9+127668829484980000*a[2]^6*a[3]^10+
297769530643200000*a[2]^5*a[3]^11+463853325722500000*a[2]^4*a[3]^12+
507075425430000000*a[2]^3*a[3]^13+415945833581250000*a[2]^2*a[3]^14+
244592271000000000*a[2]*a[3]^15+72960762656250000*a[3]^16+32768000*a[1]^10*a[2]
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948326400*a[1]^10*a[2]*a[3]^4+418867200*a[1]^10*a[3]^5+394444800*a[1]^9*a[2]^5*
a[3]+7933542400*a[1]^9*a[2]^4*a[3]^2+48946176000*a[1]^9*a[2]^3*a[3]^3+
126467481600*a[1]^9*a[2]^2*a[3]^4+146134195200*a[1]^9*a[2]*a[3]^5+62674560000*a
[1]^9*a[3]^6+2110054400*a[1]^8*a[2]^6*a[3]+66607513600*a[1]^8*a[2]^5*a[3]^2+
757191168000*a[1]^8*a[2]^4*a[3]^3+3790027776000*a[1]^8*a[2]^3*a[3]^4+
8920815552000*a[1]^8*a[2]^2*a[3]^5+9811976601600*a[1]^8*a[2]*a[3]^6+
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a[3]^7+141576637110652800*a[1]^5*a[2]^2*a[3]^8+135778566683592000*a[1]^5*a[2]*a
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[2]^2*a[3]+83460*a[2]*a[3]^2+194600*a[3]^3+2915*a[1]^2+10349*a[1]*a[2]+158644*a
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                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



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a[3]^5+5028311575211616*a[1]*a[2]^6+492659319008945796*a[1]*a[2]^5*a[3]+
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2]+14686142254244134*a[1]^5*a[3]+3298410067135446*a[1]^4*a[2]^2+
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]^3+940505642242063192015*a[1]^2*a[3]^4+10351091397247026*a[1]*a[2]^5+
831066623510353666*a[1]*a[2]^4*a[3]+26709456962737418138*a[1]*a[2]^3*a[3]^2+
428180954374320707062*a[1]*a[2]^2*a[3]^3+3415482487768802529860*a[1]*a[2]*a[3]^
4+10825067911706081613816*a[1]*a[3]^5+3697144106204619*a[2]^6+
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3]+5053853078726142*a[1]^3*a[2]^2+166161209326144992*a[1]^3*a[2]*a[3]+
1353427139472378646*a[1]^3*a[3]^2+9361243775512250*a[1]^2*a[2]^3+
451822924190556244*a[1]^2*a[2]^2*a[3]+7221021525558777726*a[1]^2*a[2]*a[3]^2+
38170862427792163300*a[1]^2*a[3]^3+8866850477560482*a[1]*a[2]^4+
558052858142212456*a[1]*a[2]^3*a[3]+13109886808918643300*a[1]*a[2]^2*a[3]^2+
136074640830751763400*a[1]*a[2]*a[3]^3+525997696956225086666*a[1]*a[3]^4+
3407405787647046*a[2]^5+262420602804487974*a[2]^4*a[3]+8057751949269662240*a[2]
^3*a[3]^2+123155316933640817968*a[2]^2*a[3]^3+935984377834513599770*a[2]*a[3]^4
+2827408850795857750346*a[3]^5+179567342557595*a[1]^4+1269560978927620*a[1]^3*a
[2]+20225811428646300*a[1]^3*a[3]+3415194157533785*a[1]^2*a[2]^2+
107289387910296926*a[1]^2*a[2]*a[3]+836384802948543361*a[1]^2*a[3]^2+
4130634863902020*a[1]*a[2]^3+191815633836661168*a[1]*a[2]^2*a[3]+
2949796720066826932*a[1]*a[2]*a[3]^2+15011714275607863000*a[1]*a[3]^3+
1889332118361927*a[2]^4+115290161853891972*a[2]^3*a[3]+2622974496880173950*a[2]
^2*a[3]^2+26350540368637660476*a[2]*a[3]^3+98568135472476725867*a[3]^4+
117611500069432*a[1]^3+621040515284524*a[1]^2*a[2]+9589077514753580*a[1]^2*a[3]
+1099891809134724*a[1]*a[2]^2+33584406984088056*a[1]*a[2]*a[3]+
254456366180874388*a[1]*a[3]^2+652407687991656*a[2]^3+29545085532578760*a[2]^2*
a[3]+442862651463852104*a[2]*a[3]^2+2196141434935732264*a[3]^3+45328269509676*a
[1]^2+157990479786896*a[1]*a[2]+2382656635655552*a[1]*a[3]+138007768932876*a[2]
^2+4123094141739080*a[2]*a[3]+30555734415675788*a[3]^2+9528159497040*a[1]+
16421302094256*a[2]+242791096632208*a[3]+843649234096


We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 2, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                            1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
                x[n + 1] = -------------------------------------
                           1 + 2 x[n - 2] + 4 x[n - 1] + 34 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], -----------------------------]
                                            1 + 2 x[1] + 4 x[2] + 34 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], -----------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 4 x[2] + 34 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+4*a[2]+34*a
[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+4*a[2]+34*a[3]), (2*a[1]*a[2]+4
*a[1]*a[3]+4*a[2]^2+42*a[2]*a[3]+68*a[3]^2+39*a[1]+79*a[2]+1405*a[3]+38)/(4*a[1
]*a[2]+8*a[1]*a[3]+8*a[2]^2+84*a[2]*a[3]+136*a[3]^2+36*a[1]+74*a[2]+1296*a[3]+
35)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+4*a[2]+34*a[3]), (2*a[1]*a[2]+4*a[1]*a
[3]+4*a[2]^2+42*a[2]*a[3]+68*a[3]^2+39*a[1]+79*a[2]+1405*a[3]+38)/(4*a[1]*a[2]+
8*a[1]*a[3]+8*a[2]^2+84*a[2]*a[3]+136*a[3]^2+36*a[1]+74*a[2]+1296*a[3]+35), (8*
a[1]^2*a[2]*a[3]+16*a[1]^2*a[3]^2+32*a[1]*a[2]^2*a[3]+336*a[1]*a[2]*a[3]^2+544*
a[1]*a[3]^3+32*a[2]^3*a[3]+608*a[2]^2*a[3]^2+3400*a[2]*a[3]^3+4624*a[3]^4+164*a
[1]^2*a[2]+400*a[1]^2*a[3]+656*a[1]*a[2]^2+7344*a[1]*a[2]*a[3]+15296*a[1]*a[3]^
2+656*a[2]^3+13088*a[2]^2*a[3]+80844*a[2]*a[3]^2+144432*a[3]^3+3030*a[1]^2+
12288*a[1]*a[2]+162382*a[1]*a[3]+12456*a[2]^2+327702*a[2]*a[3]+1912868*a[3]^2+
4503*a[1]+9049*a[2]+107492*a[3]+1511)/(16*a[1]^2*a[2]*a[3]+32*a[1]^2*a[3]^2+64*
a[1]*a[2]^2*a[3]+672*a[1]*a[2]*a[3]^2+1088*a[1]*a[3]^3+64*a[2]^3*a[3]+1216*a[2]
^2*a[3]^2+6800*a[2]*a[3]^3+9248*a[3]^4+160*a[1]^2*a[2]+464*a[1]^2*a[3]+640*a[1]
*a[2]^2+7632*a[1]*a[2]*a[3]+19168*a[1]*a[3]^2+640*a[2]^3+13408*a[2]^2*a[3]+
90288*a[2]*a[3]^2+191760*a[3]^3+2868*a[1]^2+11640*a[1]*a[2]+155340*a[1]*a[3]+
11808*a[2]^2+313700*a[2]*a[3]+1868016*a[3]^2+4300*a[1]+8644*a[2]+104618*a[3]+
1467)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
16*a[1]*a[2]^3+136*a[1]*a[2]^2*a[3]+16*a[1]*a[2]*a[3]^2+136*a[1]*a[3]^3+16*a[2]
^4+272*a[2]^3*a[3]+1172*a[2]^2*a[3]^2+272*a[2]*a[3]^3+1156*a[3]^4-8*a[1]^2*a[2]
-8*a[1]^2*a[3]-28*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-268*a[1]*a[3]^2-24*a[2]^3-508*
a[2]^2*a[3]-2848*a[2]*a[3]^2-2244*a[3]^3+9*a[1]^2+28*a[1]*a[2]+258*a[1]*a[3]+21
*a[2]^2+380*a[2]*a[3]+2186*a[3]^2+8*a[1]+14*a[2]+134*a[3]+2)/(1+2*a[1]+4*a[2]+
34*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+256*a[1]^4*a[2]*a[3]^3+256*a[1]^4*a[3]^4+
512*a[1]^3*a[2]^3*a[3]^2+6400*a[1]^3*a[2]^2*a[3]^3+19456*a[1]^3*a[2]*a[3]^4+
17408*a[1]^3*a[3]^5+1536*a[1]^2*a[2]^4*a[3]^2+32256*a[1]^2*a[2]^3*a[3]^3+221568
*a[1]^2*a[2]^2*a[3]^4+548352*a[1]^2*a[2]*a[3]^5+443904*a[1]^2*a[3]^6+2048*a[1]*
a[2]^5*a[3]^2+60416*a[1]*a[2]^4*a[3]^3+660992*a[1]*a[2]^3*a[3]^4+3242240*a[1]*a
[2]^2*a[3]^5+6806528*a[1]*a[2]*a[3]^6+5030912*a[1]*a[3]^7+1024*a[2]^6*a[3]^2+
38912*a[2]^5*a[3]^3+587264*a[2]^4*a[3]^4+4430336*a[2]^3*a[3]^5+17182784*a[2]^2*
a[3]^6+31443200*a[2]*a[3]^7+21381376*a[3]^8-128*a[1]^4*a[2]^2*a[3]+640*a[1]^4*a
[2]*a[3]^2+1792*a[1]^4*a[3]^3-1024*a[1]^3*a[2]^3*a[3]-3456*a[1]^3*a[2]^2*a[3]^2
+80128*a[1]^3*a[2]*a[3]^3+165888*a[1]^3*a[3]^4-3072*a[1]^2*a[2]^4*a[3]-36096*a[
1]^2*a[2]^3*a[3]^2+222336*a[1]^2*a[2]^2*a[3]^3+3240960*a[1]^2*a[2]*a[3]^4+
5352960*a[1]^2*a[3]^5-4096*a[1]*a[2]^5*a[3]-82432*a[1]*a[2]^4*a[3]^2-14848*a[1]
*a[2]^3*a[3]^3+8568704*a[1]*a[2]^2*a[3]^4+53298944*a[1]*a[2]*a[3]^5+73392128*a[
1]*a[3]^6-2048*a[2]^6*a[3]-58368*a[2]^5*a[3]^2-306688*a[2]^4*a[3]^3+5500672*a[2
]^3*a[3]^4+75124224*a[2]^2*a[3]^5+309660032*a[2]*a[3]^6+365998848*a[3]^7+96*a[1
]^4*a[2]^2-1920*a[1]^4*a[2]*a[3]+960*a[1]^4*a[3]^2+768*a[1]^3*a[2]^3-9728*a[1]^
3*a[2]^2*a[3]-167136*a[1]^3*a[2]*a[3]^2+176192*a[1]^3*a[3]^3+2304*a[1]^2*a[2]^4
-12288*a[1]^2*a[2]^3*a[3]-898592*a[1]^2*a[2]^2*a[3]^2-4470560*a[1]^2*a[2]*a[3]^
3+9262016*a[1]^2*a[3]^4+3072*a[1]*a[2]^5+6144*a[1]*a[2]^4*a[3]-1558016*a[1]*a[2
]^3*a[3]^2-18665920*a[1]*a[2]^2*a[3]^3-37060512*a[1]*a[2]*a[3]^4+181016000*a[1]
*a[3]^5+1536*a[2]^6+14336*a[2]^5*a[3]-874112*a[2]^4*a[3]^2-18040064*a[2]^3*a[3]
^3-105675776*a[2]^2*a[3]^4+1184288*a[2]*a[3]^5+1186000512*a[3]^6+1392*a[1]^4*a[
2]-7584*a[1]^4*a[3]+11312*a[1]^3*a[2]^2+43712*a[1]^3*a[2]*a[3]-873376*a[1]^3*a[
3]^2+34464*a[1]^2*a[2]^3+452432*a[1]^2*a[2]^2*a[3]-2498864*a[1]^2*a[2]*a[3]^2-\
35038080*a[1]^2*a[3]^3+46656*a[1]*a[2]^4+1042496*a[1]*a[2]^3*a[3]+615792*a[1]*a
[2]^2*a[3]^2-107872288*a[1]*a[2]*a[3]^3-576858528*a[1]*a[3]^4+23680*a[2]^5+
746304*a[2]^4*a[3]+4240032*a[2]^3*a[3]^2-74858672*a[2]^2*a[3]^3-1000641440*a[2]
*a[3]^4-3338062880*a[3]^5+6516*a[1]^4+55280*a[1]^3*a[2]+613264*a[1]^3*a[3]+
175420*a[1]^2*a[2]^2+3852704*a[1]^2*a[2]*a[3]+20132156*a[1]^2*a[3]^2+246832*a[1
]*a[2]^3+8055640*a[1]*a[2]^2*a[3]+83710168*a[1]*a[2]*a[3]^2+278531400*a[1]*a[3]
^3+129968*a[2]^4+5606576*a[2]^3*a[3]+86925344*a[2]^2*a[3]^2+577700280*a[2]*a[3]
^3+1573194168*a[3]^4+17892*a[1]^3+109928*a[1]^2*a[2]+1221012*a[1]^2*a[3]+225048
*a[1]*a[2]^2+4979824*a[1]*a[2]*a[3]+26514968*a[1]*a[3]^2+153520*a[2]^3+5076272*
a[2]^2*a[3]+53964332*a[2]*a[3]^2+204049552*a[3]^3+17618*a[1]^2+71414*a[1]*a[2]+
778216*a[1]*a[3]+72361*a[2]^2+1574786*a[2]*a[3]+9004755*a[3]^2+7474*a[1]+15082*
a[2]+172836*a[3]+1234)/(4*a[1]*a[2]+8*a[1]*a[3]+8*a[2]^2+84*a[2]*a[3]+136*a[3]^
2+36*a[1]+74*a[2]+1296*a[3]+35)^2/(1+2*a[1]+4*a[2]+34*a[3])^2, (12288*a[1]^8*a[
2]^4*a[3]^2+98304*a[1]^8*a[2]^3*a[3]^3+294912*a[1]^8*a[2]^2*a[3]^4+393216*a[1]^
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*a[2]^2*a[3]^2+6800*a[2]*a[3]^3+9248*a[3]^4+160*a[1]^2*a[2]+464*a[1]^2*a[3]+640
*a[1]*a[2]^2+7632*a[1]*a[2]*a[3]+19168*a[1]*a[3]^2+640*a[2]^3+13408*a[2]^2*a[3]
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11808*a[2]^2+313700*a[2]*a[3]+1868016*a[3]^2+4300*a[1]+8644*a[2]+104618*a[3]+
1467)^2/(4*a[1]*a[2]+8*a[1]*a[3]+8*a[2]^2+84*a[2]*a[3]+136*a[3]^2+36*a[1]+74*a[
2]+1296*a[3]+35)^2/(1+2*a[1]+4*a[2]+34*a[3])^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+13107200*a[1]^10*a[2]^3*a[3]^3+39321600*a[
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a[2]^3*a[3]^4+6186598400*a[1]^9*a[2]^2*a[3]^5+7549747200*a[1]^9*a[2]*a[3]^6+
3565158400*a[1]^9*a[3]^7+185139200*a[1]^8*a[2]^6*a[3]^2+4600627200*a[1]^8*a[2]^
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11900619664101376*a[1]^3*a[2]^3+544535846640636000*a[1]^3*a[2]^2*a[3]+
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26454255378632814400*a[1]^2*a[2]^2*a[3]^2+246008190624315014304*a[1]^2*a[2]*a[3
]^3+834195966937778023120*a[1]^2*a[3]^4+20898246036793856*a[1]*a[2]^5+
1468117738454336128*a[1]*a[2]^4*a[3]+40638285721201839552*a[1]*a[2]^3*a[3]^2+
551670521825180047648*a[1]*a[2]^2*a[3]^3+3661021653129754964064*a[1]*a[2]*a[3]^
4+9481673867852362193568*a[1]*a[3]^5+8857959735447296*a[2]^6+720546762211485952
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3]^3+4144340502916345441904*a[2]^2*a[3]^4+21070320271228907298848*a[2]*a[3]^5+
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16904221669384317008*a[1]*a[2]^2*a[3]^2+146720109562150985664*a[1]*a[2]*a[3]^3+
466256910812675241168*a[1]*a[3]^4+7124209013829824*a[2]^5+475188392532400752*a[
2]^4*a[3]+12459454835451087216*a[2]^3*a[3]^2+160120578801704566560*a[2]^2*a[3]^
3+1006597708022999115888*a[2]*a[3]^4+2472883428756278239904*a[3]^5+
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758388489497634872*a[1]^2*a[3]^2+6283024215464896*a[1]*a[2]^3+
248597807164157464*a[1]*a[2]^2*a[3]+3207367450907151392*a[1]*a[2]*a[3]^2+
13471351300680638576*a[1]*a[3]^3+3424949107991548*a[2]^4+178863635119889200*a[2
]^3*a[3]+3431411312002050000*a[2]^2*a[3]^2+28611396753542509048*a[2]*a[3]^3+
87370316812660970596*a[3]^4+108701726060692*a[1]^3+680723637145528*a[1]^2*a[2]+
8794586962140148*a[1]^2*a[3]+1435296043635004*a[1]*a[2]^2+36861660317546384*a[1
]*a[2]*a[3]+231190243729779660*a[1]*a[3]^2+1017547099765764*a[2]^3+
38948510110159748*a[2]^2*a[3]+485882316703688696*a[2]*a[3]^2+
1972958388920241292*a[3]^3+42049928680513*a[1]^2+175125674748324*a[1]*a[2]+
2191709387352044*a[1]*a[3]+183470251767710*a[2]^2+4572597586442194*a[2]*a[3]+
27820091754729214*a[3]^2+8872875089274*a[1]+18412012831978*a[2]+224009017381298
*a[3]+788696926999)/(16*a[1]^2*a[2]*a[3]+32*a[1]^2*a[3]^2+64*a[1]*a[2]^2*a[3]+
672*a[1]*a[2]*a[3]^2+1088*a[1]*a[3]^3+64*a[2]^3*a[3]+1216*a[2]^2*a[3]^2+6800*a[
2]*a[3]^3+9248*a[3]^4+160*a[1]^2*a[2]+464*a[1]^2*a[3]+640*a[1]*a[2]^2+7632*a[1]
*a[2]*a[3]+19168*a[1]*a[3]^2+640*a[2]^3+13408*a[2]^2*a[3]+90288*a[2]*a[3]^2+
191760*a[3]^3+2868*a[1]^2+11640*a[1]*a[2]+155340*a[1]*a[3]+11808*a[2]^2+313700*
a[2]*a[3]+1868016*a[3]^2+4300*a[1]+8644*a[2]+104618*a[3]+1467)^2/(4*a[1]*a[2]+8
*a[1]*a[3]+8*a[2]^2+84*a[2]*a[3]+136*a[3]^2+36*a[1]+74*a[2]+1296*a[3]+35)^2/(1+
2*a[1]+4*a[2]+34*a[3])^2


                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



1638400*a[1]^10*a[2]^4*a[3]^2+13107200*a[1]^10*a[2]^3*a[3]^3+39321600*a[1]^10*a
[2]^2*a[3]^4+52428800*a[1]^10*a[2]*a[3]^5+26214400*a[1]^10*a[3]^6+26214400*a[1]
^9*a[2]^5*a[3]^2+432537600*a[1]^9*a[2]^4*a[3]^3+2411724800*a[1]^9*a[2]^3*a[3]^4
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9*a[3]^7+185139200*a[1]^8*a[2]^6*a[3]^2+4600627200*a[1]^8*a[2]^5*a[3]^3+
42659020800*a[1]^8*a[2]^4*a[3]^4+186869350400*a[1]^8*a[2]^3*a[3]^5+421016371200
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8+760217600*a[1]^7*a[2]^7*a[3]^2+25021644800*a[1]^7*a[2]^6*a[3]^3+328885862400*
a[1]^7*a[2]^5*a[3]^4+2202848460800*a[1]^7*a[2]^4*a[3]^5+8045094502400*a[1]^7*a[
2]^3*a[3]^6+16218744422400*a[1]^7*a[2]^2*a[3]^7+16977703731200*a[1]^7*a[2]*a[3]
^8+7215880601600*a[1]^7*a[3]^9+2018508800*a[1]^6*a[2]^8*a[3]^2+81657856000*a[1]
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5*a[3]^5+67221274624000*a[1]^6*a[2]^4*a[3]^6+211930028441600*a[1]^6*a[2]^3*a[3]
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153473830092800*a[1]^6*a[3]^10+3670016000*a[1]^5*a[2]^9*a[3]^2+172857753600*a[1
]^5*a[2]^8*a[3]^3+3517159833600*a[1]^5*a[2]^7*a[3]^4+40310446489600*a[1]^5*a[2]
^6*a[3]^5+284189156966400*a[1]^5*a[2]^5*a[3]^6+1264340513587200*a[1]^5*a[2]^4*a
[3]^7+3517250849996800*a[1]^5*a[2]^3*a[3]^8+5890639921152000*a[1]^5*a[2]^2*a[3]
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4771020800*a[1]^4*a[2]^10*a[3]^2+250295091200*a[1]^4*a[2]^9*a[3]^3+
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55806298344652800*a[1]^4*a[2]^2*a[3]^10+48318907311718400*a[1]^4*a[2]*a[3]^11+
17870314589388800*a[1]^4*a[3]^12+4613734400*a[1]^3*a[2]^11*a[3]^2+261514854400*
a[1]^3*a[2]^10*a[3]^3+6577297817600*a[1]^3*a[2]^9*a[3]^4+97113132236800*a[1]^3*
a[2]^8*a[3]^5+938347357798400*a[1]^3*a[2]^7*a[3]^6+6248013325926400*a[1]^3*a[2]
^6*a[3]^7+29254751931596800*a[1]^3*a[2]^5*a[3]^8+95829849905561600*a[1]^3*a[2]^
4*a[3]^9+213145497685196800*a[1]^3*a[2]^3*a[3]^10+303781696136806400*a[1]^3*a[2
]^2*a[3]^11+248371537248256000*a[1]^3*a[2]*a[3]^12+88138616039014400*a[1]^3*a[3
]^13+3355443200*a[1]^2*a[2]^12*a[3]^2+205101465600*a[1]^2*a[2]^11*a[3]^3+
5539941580800*a[1]^2*a[2]^10*a[3]^4+87288289689600*a[1]^2*a[2]^9*a[3]^5+
893982002380800*a[1]^2*a[2]^8*a[3]^6+6297747062784000*a[1]^2*a[2]^7*a[3]^7+
31602430017536000*a[1]^2*a[2]^6*a[3]^8+115480142020608000*a[1]^2*a[2]^5*a[3]^9+
308936454748569600*a[1]^2*a[2]^4*a[3]^10+591498281602252800*a[1]^2*a[2]^3*a[3]^
11+762366405889228800*a[1]^2*a[2]^2*a[3]^12+585779366225510400*a[1]^2*a[2]*a[3]
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58988481740800*a[1]*a[2]^10*a[3]^5+662714187776000*a[1]*a[2]^9*a[3]^6+
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381884375236608000*a[1]*a[2]^2*a[3]^13+242976725296742400*a[1]*a[2]*a[3]^14+
86054256875929600*a[1]*a[3]^15+419430400*a[2]^14*a[3]^2+31876710400*a[2]^13*a[3
]^3+1087163596800*a[2]^12*a[3]^4+21942501376000*a[2]^11*a[3]^5+291029070643200*
a[2]^10*a[3]^6+2663983559475200*a[2]^9*a[3]^7+17210087387955200*a[2]^8*a[3]^8+
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1756364800*a[1]^10*a[2]*a[3]^4+996147200*a[1]^10*a[3]^5+524288000*a[1]^9*a[2]^5
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191522406400*a[1]^9*a[2]^2*a[3]^4+270375321600*a[1]^9*a[2]*a[3]^5+145489920000*
a[1]^9*a[3]^6+3702784000*a[1]^8*a[2]^6*a[3]+99631104000*a[1]^8*a[2]^5*a[3]^2+
1036828672000*a[1]^8*a[2]^4*a[3]^3+5282614476800*a[1]^8*a[2]^3*a[3]^4+
13893618892800*a[1]^8*a[2]^2*a[3]^5+18102065561600*a[1]^8*a[2]*a[3]^6+
9252529766400*a[1]^8*a[3]^7+15204352000*a[1]^7*a[2]^7*a[3]+533508915200*a[1]^7*
a[2]^6*a[3]^2+7660332646400*a[1]^7*a[2]^5*a[3]^3+57681195827200*a[1]^7*a[2]^4*a
[3]^4+243130197606400*a[1]^7*a[2]^3*a[3]^5+569434282393600*a[1]^7*a[2]^2*a[3]^6
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[1]^6*a[2]^8*a[3]+1725182771200*a[1]^6*a[2]^7*a[3]^2+31384829952000*a[1]^6*a[2]
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[3]^5+6827800618598400*a[1]^6*a[2]^3*a[3]^6+14441972996505600*a[1]^6*a[2]^2*a[3
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7578129334272000*a[1]^5*a[2]^5*a[3]^5+37881552234086400*a[1]^5*a[2]^4*a[3]^6+
120464514901606400*a[1]^5*a[2]^3*a[3]^7+232452680476262400*a[1]^5*a[2]^2*a[3]^8
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5457313792000*a[1]^3*a[2]^10*a[3]^2+144703488000000*a[1]^3*a[2]^9*a[3]^3+
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4271269478400*a[1]^2*a[2]^11*a[3]^2+121322654924800*a[1]^2*a[2]^10*a[3]^3+
2031852873318400*a[1]^2*a[2]^9*a[3]^4+22359952116940800*a[1]^2*a[2]^8*a[3]^5+
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3]^9+23764225014058188800*a[1]^2*a[2]^3*a[3]^10+34620920232817459200*a[1]^2*a[2
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73788293120000*a[1]*a[2]^11*a[3]^3+1362022105088000*a[1]*a[2]^10*a[3]^4+
16400854391193600*a[1]*a[2]^9*a[3]^5+134865814736076800*a[1]*a[2]^8*a[3]^6+
770550840662425600*a[1]*a[2]^7*a[3]^7+3055200038138675200*a[1]*a[2]^6*a[3]^8+
8283225829448089600*a[1]*a[2]^5*a[3]^9+15035879689476505600*a[1]*a[2]^4*a[3]^10
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We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 3, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                            1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
                x[n + 1] = -------------------------------------
                           1 + 2 x[n - 2] + 5 x[n - 1] + 33 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], -----------------------------]
                                            1 + 2 x[1] + 5 x[2] + 33 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], -----------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 5 x[2] + 33 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+5*a[2]+33*a
[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+5*a[2]+33*a[3]), (2*a[1]*a[2]+4
*a[1]*a[3]+5*a[2]^2+43*a[2]*a[3]+66*a[3]^2+39*a[1]+80*a[2]+1404*a[3]+38)/(4*a[1
]*a[2]+10*a[1]*a[3]+10*a[2]^2+91*a[2]*a[3]+165*a[3]^2+35*a[1]+73*a[2]+1259*a[3]
+34)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+5*a[2]+33*a[3]), (2*a[1]*a[2]+4*a[1]*
a[3]+5*a[2]^2+43*a[2]*a[3]+66*a[3]^2+39*a[1]+80*a[2]+1404*a[3]+38)/(4*a[1]*a[2]
+10*a[1]*a[3]+10*a[2]^2+91*a[2]*a[3]+165*a[3]^2+35*a[1]+73*a[2]+1259*a[3]+34),
(8*a[1]^2*a[2]*a[3]+20*a[1]^2*a[3]^2+40*a[1]*a[2]^2*a[3]+364*a[1]*a[2]*a[3]^2+
660*a[1]*a[3]^3+50*a[2]^3*a[3]+785*a[2]^2*a[3]^2+3828*a[2]*a[3]^3+5445*a[3]^4+
164*a[1]^2*a[2]+406*a[1]^2*a[3]+816*a[1]*a[2]^2+7571*a[1]*a[2]*a[3]+15181*a[1]*
a[3]^2+1015*a[2]^3+16324*a[2]^2*a[3]+84730*a[2]*a[3]^2+139953*a[3]^3+3026*a[1]^
2+13828*a[1]*a[2]+160577*a[1]*a[3]+15672*a[2]^2+378669*a[2]*a[3]+1854315*a[3]^2
+4496*a[1]+10515*a[2]+105795*a[3]+1508)/(16*a[1]^2*a[2]*a[3]+40*a[1]^2*a[3]^2+
80*a[1]*a[2]^2*a[3]+728*a[1]*a[2]*a[3]^2+1320*a[1]*a[3]^3+100*a[2]^3*a[3]+1570*
a[2]^2*a[3]^2+7656*a[2]*a[3]^3+10890*a[3]^4+160*a[1]^2*a[2]+474*a[1]^2*a[3]+790
*a[1]*a[2]^2+7985*a[1]*a[2]*a[3]+19413*a[1]*a[3]^2+975*a[2]^3+16835*a[2]^2*a[3]
+97620*a[2]*a[3]^2+191268*a[3]^3+2819*a[1]^2+12841*a[1]*a[2]+151976*a[1]*a[3]+
14520*a[2]^2+356030*a[2]*a[3]+1811348*a[3]^2+4243*a[1]+9858*a[2]+102748*a[3]+
1458)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
20*a[1]*a[2]^3+132*a[1]*a[2]^2*a[3]+20*a[1]*a[2]*a[3]^2+132*a[1]*a[3]^3+25*a[2]
^4+330*a[2]^3*a[3]+1114*a[2]^2*a[3]^2+330*a[2]*a[3]^3+1089*a[3]^4-8*a[1]^2*a[2]
-8*a[1]^2*a[3]-36*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-260*a[1]*a[3]^2-40*a[2]^3-644*
a[2]^2*a[3]-2828*a[2]*a[3]^2-2112*a[3]^3+9*a[1]^2+38*a[1]*a[2]+248*a[1]*a[3]+40
*a[2]^2+484*a[2]*a[3]+2063*a[3]^2+8*a[1]+18*a[2]+130*a[3]+2)/(1+2*a[1]+5*a[2]+
33*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+320*a[1]^4*a[2]*a[3]^3+400*a[1]^4*a[3]^4+
640*a[1]^3*a[2]^3*a[3]^2+7424*a[1]^3*a[2]^2*a[3]^3+25120*a[1]^3*a[2]*a[3]^4+
26400*a[1]^3*a[3]^5+2400*a[1]^2*a[2]^4*a[3]^2+43680*a[1]^2*a[2]^3*a[3]^3+277944
*a[1]^2*a[2]^2*a[3]^4+720720*a[1]^2*a[2]*a[3]^5+653400*a[1]^2*a[3]^6+4000*a[1]*
a[2]^5*a[3]^2+99200*a[1]*a[2]^4*a[3]^3+943720*a[1]*a[2]^3*a[3]^4+4258584*a[1]*a
[2]^2*a[3]^5+9016920*a[1]*a[2]*a[3]^6+7187400*a[1]*a[3]^7+2500*a[2]^6*a[3]^2+
78500*a[2]^5*a[3]^3+999025*a[2]^4*a[3]^4+6554460*a[2]^3*a[3]^5+23202234*a[2]^2*
a[3]^6+41686920*a[2]*a[3]^7+29648025*a[3]^8-128*a[1]^4*a[2]^2*a[3]+480*a[1]^4*a
[2]*a[3]^2+2000*a[1]^4*a[3]^3-1280*a[1]^3*a[2]^3*a[3]-4048*a[1]^3*a[2]^2*a[3]^2
+72648*a[1]^3*a[2]*a[3]^3+186920*a[1]^3*a[3]^4-4800*a[1]^2*a[2]^4*a[3]-48360*a[
1]^2*a[2]^3*a[3]^2+240972*a[1]^2*a[2]^2*a[3]^3+3223896*a[1]^2*a[2]*a[3]^4+
5985540*a[1]^2*a[3]^5-8000*a[1]*a[2]^5*a[3]-135900*a[1]*a[2]^4*a[3]^2-32290*a[1
]*a[2]^3*a[3]^3+9988062*a[1]*a[2]^2*a[3]^4+55678194*a[1]*a[2]*a[3]^5+80792910*a
[1]*a[3]^6-5000*a[2]^6*a[3]-119500*a[2]^5*a[3]^2-529800*a[2]^4*a[3]^3+7741430*a
[2]^3*a[3]^4+90501642*a[2]^2*a[3]^5+331750782*a[2]*a[3]^6+394947630*a[3]^7+96*a
[1]^4*a[2]^2-1744*a[1]^4*a[2]*a[3]-56*a[1]^4*a[3]^2+976*a[1]^3*a[2]^3-10880*a[1
]^3*a[2]^2*a[3]-162632*a[1]^3*a[2]*a[3]^2+78704*a[1]^3*a[3]^3+3724*a[1]^2*a[2]^
4-15716*a[1]^2*a[2]^3*a[3]-1044234*a[1]^2*a[2]^2*a[3]^2-4610216*a[1]^2*a[2]*a[3
]^3+5941738*a[1]^2*a[3]^4+6320*a[1]*a[2]^5+16552*a[1]*a[2]^4*a[3]-2172416*a[1]*
a[2]^3*a[3]^2-22437286*a[1]*a[2]^2*a[3]^3-40872436*a[1]*a[2]*a[3]^4+130789626*a
[1]*a[3]^5+4025*a[2]^6+37730*a[2]^5*a[3]-1442426*a[2]^4*a[3]^2-26061594*a[2]^3*
a[3]^3-129697271*a[2]^2*a[3]^4-20561772*a[2]*a[3]^5+898089588*a[3]^6+1336*a[1]^
4*a[2]-6492*a[1]^4*a[3]+13188*a[1]^3*a[2]^2+41286*a[1]^3*a[2]*a[3]-770530*a[1]^
3*a[3]^2+48764*a[1]^2*a[2]^3+548148*a[1]^2*a[2]^2*a[3]-2688810*a[1]^2*a[2]*a[3]
^2-31246378*a[1]^2*a[3]^3+80016*a[1]*a[2]^4+1563028*a[1]*a[2]^3*a[3]+830180*a[1
]*a[2]^2*a[3]^2-115178256*a[1]*a[2]*a[3]^3-510852192*a[1]*a[3]^4+49140*a[2]^5+
1379854*a[2]^4*a[3]+6888988*a[2]^3*a[3]^2-94939526*a[2]^2*a[3]^3-1062976716*a[2
]*a[3]^4-2902364388*a[3]^5+6189*a[1]^4+60296*a[1]^3*a[2]+576278*a[1]^3*a[3]+
220387*a[1]^2*a[2]^2+4229108*a[1]^2*a[2]*a[3]+18601243*a[1]^2*a[3]^2+358222*a[1
]*a[2]^3+10346220*a[1]*a[2]^2*a[3]+91353578*a[1]*a[2]*a[3]^2+252155628*a[1]*a[3
]^3+218556*a[2]^4+8429122*a[2]^3*a[3]+112763945*a[2]^2*a[3]^2+603426660*a[2]*a[
3]^3+1422009497*a[3]^4+16992*a[1]^3+121926*a[1]^2*a[2]+1143552*a[1]^2*a[3]+
291412*a[1]*a[2]^2+5507082*a[1]*a[2]*a[3]+24394086*a[1]*a[3]^2+231746*a[2]^3+
6658186*a[2]^2*a[3]+57684378*a[2]*a[3]^2+186093258*a[3]^3+16733*a[1]^2+79998*a[
1]*a[2]+727650*a[1]*a[3]+95922*a[2]^2+1716656*a[2]*a[3]+8315630*a[3]^2+7100*a[1
]+16740*a[2]+161812*a[3]+1172)/(4*a[1]*a[2]+10*a[1]*a[3]+10*a[2]^2+91*a[2]*a[3]
+165*a[3]^2+35*a[1]+73*a[2]+1259*a[3]+34)^2/(1+2*a[1]+5*a[2]+33*a[3])^2, (12288
*a[1]^8*a[2]^4*a[3]^2+126976*a[1]^8*a[2]^3*a[3]^3+492544*a[1]^8*a[2]^2*a[3]^4+
849920*a[1]^8*a[2]*a[3]^5+550400*a[1]^8*a[3]^6+249856*a[1]^7*a[2]^5*a[3]^2+
4034560*a[1]^7*a[2]^4*a[3]^3+25085952*a[1]^7*a[2]^3*a[3]^4+75972608*a[1]^7*a[2]
^2*a[3]^5+112861440*a[1]^7*a[2]*a[3]^6+66092800*a[1]^7*a[3]^7+2223104*a[1]^6*a[
2]^6*a[3]^2+48783360*a[1]^6*a[2]^5*a[3]^3+432734720*a[1]^6*a[2]^4*a[3]^4+
1986626048*a[1]^6*a[2]^3*a[3]^5+4994955712*a[1]^6*a[2]^2*a[3]^6+6546060160*a[1]
^6*a[2]*a[3]^7+3505259200*a[1]^6*a[3]^8+11304960*a[1]^5*a[2]^7*a[3]^2+313419776
*a[1]^5*a[2]^6*a[3]^3+3645584384*a[1]^5*a[2]^5*a[3]^4+23027955712*a[1]^5*a[2]^4
*a[3]^5+85210411264*a[1]^5*a[2]^3*a[3]^6+184753658496*a[1]^5*a[2]^2*a[3]^7+
217616513280*a[1]^5*a[2]*a[3]^8+107609673600*a[1]^5*a[3]^9+35936000*a[1]^4*a[2]
^8*a[3]^2+1203507200*a[1]^4*a[2]^7*a[3]^3+17377186560*a[1]^4*a[2]^6*a[3]^4+
141169678720*a[1]^4*a[2]^5*a[3]^5+704617844880*a[1]^4*a[2]^4*a[3]^6+
2208486096320*a[1]^4*a[2]^3*a[3]^7+4239558552480*a[1]^4*a[2]^2*a[3]^8+
4555729886400*a[1]^4*a[2]*a[3]^9+2098897218000*a[1]^4*a[3]^10+73120000*a[1]^3*a
[2]^9*a[3]^2+2869536000*a[1]^3*a[2]^8*a[3]^3+49548841600*a[1]^3*a[2]^7*a[3]^4+
494141393920*a[1]^3*a[2]^6*a[3]^5+3134160599120*a[1]^3*a[2]^5*a[3]^6+
13087866781232*a[1]^3*a[2]^4*a[3]^7+35897366762144*a[1]^3*a[2]^3*a[3]^8+
62222862167136*a[1]^3*a[2]^2*a[3]^9+61754648622480*a[1]^3*a[2]*a[3]^10+
26718253887600*a[1]^3*a[3]^11+93000000*a[1]^2*a[2]^10*a[3]^2+4183840000*a[1]^2*
a[2]^9*a[3]^3+84117676000*a[1]^2*a[2]^8*a[3]^4+996270082400*a[1]^2*a[2]^7*a[3]^
5+7699260317060*a[1]^2*a[2]^6*a[3]^6+40531716214104*a[1]^2*a[2]^5*a[3]^7+
146900308176780*a[1]^2*a[2]^4*a[3]^8+360911984521968*a[1]^2*a[2]^3*a[3]^9+
573559998484572*a[1]^2*a[2]^2*a[3]^10+531116268015960*a[1]^2*a[2]*a[3]^11+
217265115107700*a[1]^2*a[3]^12+67600000*a[1]*a[2]^11*a[3]^2+3428800000*a[1]*a[2
]^10*a[3]^3+78682560000*a[1]*a[2]^9*a[3]^4+1080009296000*a[1]*a[2]^8*a[3]^5+
9863288131800*a[1]*a[2]^7*a[3]^6+62936905575376*a[1]*a[2]^6*a[3]^7+
285986531163480*a[1]*a[2]^5*a[3]^8+923143242471120*a[1]*a[2]^4*a[3]^9+
2067455547979272*a[1]*a[2]^3*a[3]^10+3048625971669168*a[1]*a[2]^2*a[3]^11+
2655583608423240*a[1]*a[2]*a[3]^12+1032845637898800*a[1]*a[3]^13+21500000*a[2]^
12*a[3]^2+1213650000*a[2]^11*a[3]^3+31300887500*a[2]^10*a[3]^4+488734730000*a[2
]^9*a[3]^5+5154871071625*a[2]^8*a[3]^6+38732498007040*a[2]^7*a[3]^7+
212544501446704*a[2]^6*a[3]^8+856817001851976*a[2]^5*a[3]^9+2510613353332458*a[
2]^4*a[3]^10+5194232274265056*a[2]^3*a[3]^11+7172873112172284*a[2]^2*a[3]^12+
5914721359392120*a[2]*a[3]^13+2196431861577525*a[3]^14+159744*a[1]^8*a[2]^4*a[3
]+1961984*a[1]^8*a[2]^3*a[3]^2+8774656*a[1]^8*a[2]^2*a[3]^3+17079552*a[1]^8*a[2
]*a[3]^4+12273280*a[1]^8*a[3]^5+3260416*a[1]^7*a[2]^5*a[3]+58283008*a[1]^7*a[2]
^4*a[3]^2+412385536*a[1]^7*a[2]^3*a[3]^3+1422365696*a[1]^7*a[2]^2*a[3]^4+
2386631872*a[1]^7*a[2]*a[3]^5+1562038080*a[1]^7*a[3]^6+29117440*a[1]^6*a[2]^6*a
[3]+683041024*a[1]^6*a[2]^5*a[3]^2+6679372288*a[1]^6*a[2]^4*a[3]^3+34471102144*
a[1]^6*a[2]^3*a[3]^4+97984961600*a[1]^6*a[2]^2*a[3]^5+144761192320*a[1]^6*a[2]*
a[3]^6+86789436800*a[1]^6*a[3]^7+148606720*a[1]^5*a[2]^7*a[3]+4314358016*a[1]^5
*a[2]^6*a[3]^2+54002045504*a[1]^5*a[2]^5*a[3]^3+375418362944*a[1]^5*a[2]^4*a[3]
^4+1551999904192*a[1]^5*a[2]^3*a[3]^5+3783248693568*a[1]^5*a[2]^2*a[3]^6+
5007978033856*a[1]^5*a[2]*a[3]^7+2770888987840*a[1]^5*a[3]^8+474064000*a[1]^4*a
[2]^8*a[3]+16402993600*a[1]^4*a[2]^7*a[3]^2+250438075520*a[1]^4*a[2]^6*a[3]^3+
2197627790960*a[1]^4*a[2]^5*a[3]^4+12057087251400*a[1]^4*a[2]^4*a[3]^5+
42052436576640*a[1]^4*a[2]^3*a[3]^6+90395732440880*a[1]^4*a[2]^2*a[3]^7+
108883787191760*a[1]^4*a[2]*a[3]^8+56083700171400*a[1]^4*a[3]^9+967920000*a[1]^
3*a[2]^9*a[3]+38878680000*a[1]^3*a[2]^8*a[3]^2+700731316400*a[1]^3*a[2]^7*a[3]^
3+7435348972960*a[1]^3*a[2]^6*a[3]^4+51054765821580*a[1]^3*a[2]^5*a[3]^5+
234186439730756*a[1]^3*a[2]^4*a[3]^6+713278855932776*a[1]^3*a[2]^3*a[3]^7+
1381847986793704*a[1]^3*a[2]^2*a[3]^8+1536137770015068*a[1]^3*a[2]*a[3]^9+
743475676876020*a[1]^3*a[3]^10+1235200000*a[1]^2*a[2]^10*a[3]+56490630000*a[1]^
2*a[2]^9*a[3]^2+1173911476000*a[1]^2*a[2]^8*a[3]^3+14615431584500*a[1]^2*a[2]^7
*a[3]^4+120690407526240*a[1]^2*a[2]^6*a[3]^5+689225797152864*a[1]^2*a[2]^5*a[3]
^6+2745743132729572*a[1]^2*a[2]^4*a[3]^7+7493411864143444*a[1]^2*a[2]^3*a[3]^8+
13320754308343944*a[1]^2*a[2]^2*a[3]^9+13840831554421608*a[1]^2*a[2]*a[3]^10+
6351128553961620*a[1]^2*a[3]^11+900750000*a[1]*a[2]^11*a[3]+46211550000*a[1]*a[
2]^10*a[3]^2+1087729667500*a[1]*a[2]^9*a[3]^3+15542709514500*a[1]*a[2]^8*a[3]^4
+150011951017550*a[1]*a[2]^7*a[3]^5+1026807469225590*a[1]*a[2]^6*a[3]^6+
5076954934962866*a[1]*a[2]^5*a[3]^7+18064874735729282*a[1]*a[2]^4*a[3]^8+
45079477335061026*a[1]*a[2]^3*a[3]^9+74622967865704170*a[1]*a[2]^2*a[3]^10+
73240135445805570*a[1]*a[2]*a[3]^11+32099279887219050*a[1]*a[3]^12+287375000*a[
2]^12*a[3]+16345212500*a[2]^11*a[3]^2+429809162500*a[2]^10*a[3]^3+6931053382500
*a[2]^9*a[3]^4+76543913540500*a[2]^8*a[3]^5+610914877106230*a[2]^7*a[3]^6+
3613733048649234*a[2]^6*a[3]^7+15932088664624786*a[2]^5*a[3]^8+
51737911678206006*a[2]^4*a[3]^9+119933940347044758*a[2]^3*a[3]^10+
186964646691984282*a[2]^2*a[3]^11+174658016074628394*a[2]*a[3]^12+
73470198794660910*a[3]^13+820224*a[1]^8*a[2]^4+13038592*a[1]^8*a[2]^3*a[3]+
71909248*a[1]^8*a[2]^2*a[3]^2+166804032*a[1]^8*a[2]*a[3]^3+139287184*a[1]^8*a[3
]^4+16714752*a[1]^7*a[2]^5+360368896*a[1]^7*a[2]^4*a[3]+3035333440*a[1]^7*a[2]^
3*a[3]^2+12408713248*a[1]^7*a[2]^2*a[3]^3+24442363456*a[1]^7*a[2]*a[3]^4+
18537421600*a[1]^7*a[3]^5+149025664*a[1]^6*a[2]^6+4055831744*a[1]^6*a[2]^5*a[3]
+45890938272*a[1]^6*a[2]^4*a[3]^2+272875923360*a[1]^6*a[2]^3*a[3]^3+
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95074233520726756*a[2]*a[3]^5+2714888968064515721*a[3]^6+120042384054*a[1]^5+
1507704188210*a[1]^4*a[2]+9953415021702*a[1]^4*a[3]+7549615180930*a[1]^3*a[2]^2
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a[1]^2*a[2]^3+395695509559628*a[1]^2*a[2]^2*a[3]+2117544967822538*a[1]^2*a[2]*a
[3]^2+2377328228199484*a[1]^2*a[3]^3+23413460920110*a[1]*a[2]^4+675411414778900
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2]*a[3]^3+8469869587574942*a[1]*a[3]^4+11599731060570*a[2]^5+430443575814714*a[
2]^4*a[3]+5011960734885848*a[2]^3*a[3]^2+17570245233634600*a[2]^2*a[3]^3+
6239965680439342*a[2]*a[3]^4+138549659740732798*a[3]^5+75216701697*a[1]^4+
771375309412*a[1]^3*a[2]+4205480650540*a[1]^3*a[3]+2956663084807*a[1]^2*a[2]^2+
33362392206138*a[1]^2*a[2]*a[3]+63338548565239*a[1]^2*a[3]^2+5016480263844*a[1]
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333226218146628*a[1]*a[3]^3+3177495144081*a[2]^4+77637996492172*a[2]^3*a[3]+
464434353662174*a[2]^2*a[3]^2+494590129307468*a[2]*a[3]^3+4482668535366441*a[3]
^4+26383129160*a[1]^3+207932424896*a[1]^2*a[2]+867102187264*a[1]^2*a[3]+
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94786042768232*a[3]^3+4820429896*a[1]^2+25592682512*a[1]*a[2]+89628671824*a[1]*
a[3]+34546918908*a[2]^2+193008056744*a[2]*a[3]+1285780815276*a[3]^2+448791552*a
[1]+1036359072*a[2]+10261506912*a[3]+36902224)/(16*a[1]^2*a[2]*a[3]+40*a[1]^2*a
[3]^2+80*a[1]*a[2]^2*a[3]+728*a[1]*a[2]*a[3]^2+1320*a[1]*a[3]^3+100*a[2]^3*a[3]
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3]+790*a[1]*a[2]^2+7985*a[1]*a[2]*a[3]+19413*a[1]*a[3]^2+975*a[2]^3+16835*a[2]^
2*a[3]+97620*a[2]*a[3]^2+191268*a[3]^3+2819*a[1]^2+12841*a[1]*a[2]+151976*a[1]*
a[3]+14520*a[2]^2+356030*a[2]*a[3]+1811348*a[3]^2+4243*a[1]+9858*a[2]+102748*a[
3]+1458)^2/(4*a[1]*a[2]+10*a[1]*a[3]+10*a[2]^2+91*a[2]*a[3]+165*a[3]^2+35*a[1]+
73*a[2]+1259*a[3]+34)^2/(1+2*a[1]+5*a[2]+33*a[3])^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+16384000*a[1]^10*a[2]^3*a[3]^3+61440000*a[
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32768000*a[1]^9*a[2]^5*a[3]^2+543948800*a[1]^9*a[2]^4*a[3]^3+3391488000*a[1]^9*
a[2]^3*a[3]^4+10158080000*a[1]^9*a[2]^2*a[3]^5+14796800000*a[1]^9*a[2]*a[3]^6+
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                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



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a[2]*a[3]^5+16772445423684500075847*a[1]^2*a[3]^6-4855498516443750*a[1]*a[2]^7-\
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2]+12096952571547430*a[1]^5*a[3]+4879823331011342*a[1]^4*a[2]^2+
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]^3+736384935029939341659*a[1]^2*a[3]^4+35896812798432198*a[1]*a[2]^5+
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665416434574623553154*a[1]*a[2]^2*a[3]^3+3790149263938905038016*a[1]*a[2]*a[3]^
4+8263788473475267655436*a[1]*a[3]^5+17308387730558355*a[2]^6+
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3]+7594418528656870*a[1]^3*a[2]^2+187109372644576032*a[1]^3*a[2]*a[3]+
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30538652460269175916*a[1]^2*a[3]^3+24125566151650890*a[1]*a[2]^4+
1167199222368969500*a[1]*a[2]^3*a[3]+20439748911663721148*a[1]*a[2]^2*a[3]^2+
152921945222140509428*a[1]*a[2]*a[3]^3+411284742101478909258*a[1]*a[3]^4+
12569966141838030*a[2]^5+751374468037535886*a[2]^4*a[3]+17405478943973702552*a[
2]^3*a[3]^2+194439222832058368200*a[2]^2*a[3]^3+1044128081636445968058*a[2]*a[3
]^4+2151493362567617072202*a[3]^5+151518652999303*a[1]^4+1449188427559188*a[1]^
3*a[2]+16834340255363260*a[1]^3*a[3]+5284743505737193*a[1]^2*a[2]^2+
122760517600972662*a[1]^2*a[2]*a[3]+684376661790254761*a[1]^2*a[3]^2+
8691407459626956*a[1]*a[2]^3+302289297552908844*a[1]*a[2]^2*a[3]+
3371961197977389308*a[1]*a[2]*a[3]^2+12030501921139540972*a[1]*a[3]^3+
5425807747236219*a[2]^4+250867814282641828*a[2]^3*a[3]+4194742932656980426*a[2]
^2*a[3]^2+29972616491204007332*a[2]*a[3]^3+77050135100472718659*a[3]^4+
99964756636440*a[1]^3+722859986794704*a[1]^2*a[2]+8026645270199536*a[1]^2*a[3]+
1757371943615260*a[1]*a[2]^2+39118376224428120*a[1]*a[2]*a[3]+
209029943057536140*a[1]*a[3]^2+1434931193068304*a[2]^3+47981771769354216*a[2]^2
*a[3]+514334759138187600*a[2]*a[3]^2+1763561582721682968*a[3]^3+38813328462904*
a[1]^2+187654396348688*a[1]*a[2]+2006099445192976*a[1]*a[3]+227889600351492*a[2
]^2+4892171066306456*a[2]*a[3]+25202440989032724*a[3]^2+8220664251648*a[1]+
19910776228128*a[2]+205631417141088*a[3]+733487830576


We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 4, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                            1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
                x[n + 1] = -------------------------------------
                           1 + 2 x[n - 2] + 6 x[n - 1] + 32 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], -----------------------------]
                                            1 + 2 x[1] + 6 x[2] + 32 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], -----------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 6 x[2] + 32 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+6*a[2]+32*a
[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+6*a[2]+32*a[3]), (2*a[1]*a[2]+4
*a[1]*a[3]+6*a[2]^2+44*a[2]*a[3]+64*a[3]^2+39*a[1]+81*a[2]+1403*a[3]+38)/(4*a[1
]*a[2]+12*a[1]*a[3]+12*a[2]^2+100*a[2]*a[3]+192*a[3]^2+34*a[1]+72*a[2]+1222*a[3
]+33)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+6*a[2]+32*a[3]), (2*a[1]*a[2]+4*a[1]
*a[3]+6*a[2]^2+44*a[2]*a[3]+64*a[3]^2+39*a[1]+81*a[2]+1403*a[3]+38)/(4*a[1]*a[2
]+12*a[1]*a[3]+12*a[2]^2+100*a[2]*a[3]+192*a[3]^2+34*a[1]+72*a[2]+1222*a[3]+33)
, (8*a[1]^2*a[2]*a[3]+24*a[1]^2*a[3]^2+48*a[1]*a[2]^2*a[3]+400*a[1]*a[2]*a[3]^2
+768*a[1]*a[3]^3+72*a[2]^3*a[3]+984*a[2]^2*a[3]^2+4352*a[2]*a[3]^3+6144*a[3]^4+
164*a[1]^2*a[2]+412*a[1]^2*a[3]+976*a[1]*a[2]^2+7808*a[1]*a[2]*a[3]+15056*a[1]*
a[3]^2+1452*a[2]^3+19588*a[2]^2*a[3]+88560*a[2]*a[3]^2+135424*a[3]^3+3022*a[1]^
2+15366*a[1]*a[2]+158774*a[1]*a[3]+18960*a[2]^2+429420*a[2]*a[3]+1795906*a[3]^2
+4489*a[1]+11979*a[2]+104100*a[3]+1505)/(16*a[1]^2*a[2]*a[3]+48*a[1]^2*a[3]^2+
96*a[1]*a[2]^2*a[3]+800*a[1]*a[2]*a[3]^2+1536*a[1]*a[3]^3+144*a[2]^3*a[3]+1968*
a[2]^2*a[3]^2+8704*a[2]*a[3]^3+12288*a[3]^4+160*a[1]^2*a[2]+488*a[1]^2*a[3]+936
*a[1]*a[2]^2+8368*a[1]*a[2]*a[3]+19864*a[1]*a[3]^2+1368*a[2]^3+20328*a[2]^2*a[3
]+105672*a[2]*a[3]^2+192896*a[3]^3+2768*a[1]^2+13952*a[1]*a[2]+148552*a[1]*a[3]
+17124*a[2]^2+395252*a[2]*a[3]+1755008*a[3]^2+4182*a[1]+10986*a[2]+100810*a[3]+
1447)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
24*a[1]*a[2]^3+128*a[1]*a[2]^2*a[3]+24*a[1]*a[2]*a[3]^2+128*a[1]*a[3]^3+36*a[2]
^4+384*a[2]^3*a[3]+1060*a[2]^2*a[3]^2+384*a[2]*a[3]^3+1024*a[3]^4-8*a[1]^2*a[2]
-8*a[1]^2*a[3]-44*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-252*a[1]*a[3]^2-60*a[2]^3-776*
a[2]^2*a[3]-2804*a[2]*a[3]^2-1984*a[3]^3+9*a[1]^2+48*a[1]*a[2]+238*a[1]*a[3]+65
*a[2]^2+576*a[2]*a[3]+1946*a[3]^2+8*a[1]+22*a[2]+126*a[3]+2)/(1+2*a[1]+6*a[2]+
32*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+384*a[1]^4*a[2]*a[3]^3+576*a[1]^4*a[3]^4+
768*a[1]^3*a[2]^3*a[3]^2+8704*a[1]^3*a[2]^2*a[3]^3+31488*a[1]^3*a[2]*a[3]^4+
36864*a[1]^3*a[3]^5+3456*a[1]^2*a[2]^4*a[3]^2+57600*a[1]^2*a[2]^3*a[3]^3+350592
*a[1]^2*a[2]^2*a[3]^4+921600*a[1]^2*a[2]*a[3]^5+884736*a[1]^2*a[3]^6+6912*a[1]*
a[2]^5*a[3]^2+152064*a[1]*a[2]^4*a[3]^3+1315584*a[1]*a[2]^3*a[3]^4+5582848*a[1]
*a[2]^2*a[3]^5+11599872*a[1]*a[2]*a[3]^6+9437184*a[1]*a[3]^7+5184*a[2]^6*a[3]^2
+141696*a[2]^5*a[3]^3+1594944*a[2]^4*a[3]^4+9449472*a[2]^3*a[3]^5+31031296*a[2]
^2*a[3]^6+53477376*a[2]*a[3]^7+37748736*a[3]^8-128*a[1]^4*a[2]^2*a[3]+320*a[1]^
4*a[2]*a[3]^2+2112*a[1]^4*a[3]^3-1536*a[1]^3*a[2]^3*a[3]-5248*a[1]^3*a[2]^2*a[3
]^2+65024*a[1]^3*a[2]*a[3]^3+200832*a[1]^3*a[3]^4-6912*a[1]^2*a[2]^4*a[3]-64512
*a[1]^2*a[2]^3*a[3]^2+231552*a[1]^2*a[2]^2*a[3]^3+3220608*a[1]^2*a[2]*a[3]^4+
6395904*a[1]^2*a[3]^5-13824*a[1]*a[2]^5*a[3]-210816*a[1]*a[2]^4*a[3]^2-138240*a
[1]*a[2]^3*a[3]^3+11115904*a[1]*a[2]^2*a[3]^4+58363904*a[1]*a[2]*a[3]^5+
85032960*a[1]*a[3]^6-10368*a[2]^6*a[3]-219456*a[2]^5*a[3]^2-914112*a[2]^4*a[3]^
3+9784704*a[2]^3*a[3]^4+105469952*a[2]^2*a[3]^5+354713600*a[2]*a[3]^6+407371776
*a[3]^7+96*a[1]^4*a[2]^2-1568*a[1]^4*a[2]*a[3]-928*a[1]^4*a[3]^2+1184*a[1]^3*a[
2]^3-11360*a[1]^3*a[2]^2*a[3]-159648*a[1]^3*a[2]*a[3]^2-5920*a[1]^3*a[3]^3+5488
*a[1]^2*a[2]^4-16608*a[1]^2*a[2]^3*a[3]-1170464*a[1]^2*a[2]^2*a[3]^2-4880928*a[
1]^2*a[2]*a[3]^3+3108720*a[1]^2*a[3]^4+11328*a[1]*a[2]^5+38240*a[1]*a[2]^4*a[3]
-2806368*a[1]*a[2]^3*a[3]^2-26358880*a[1]*a[2]^2*a[3]^3-47465312*a[1]*a[2]*a[3]
^4+88821248*a[1]*a[3]^5+8784*a[2]^6+84480*a[2]^5*a[3]-2116160*a[2]^4*a[3]^2-\
35324512*a[2]^3*a[3]^3-157218064*a[2]^2*a[3]^4-61085184*a[2]*a[3]^5+661876736*a
[3]^6+1280*a[1]^4*a[2]-5488*a[1]^4*a[3]+14880*a[1]^3*a[2]^2+41104*a[1]^3*a[2]*a
[3]-677872*a[1]^3*a[3]^2+64592*a[1]^2*a[2]^3+646816*a[1]^2*a[2]^2*a[3]-2739328*
a[1]^2*a[2]*a[3]^2-27872400*a[1]^2*a[3]^3+123792*a[1]*a[2]^4+2186512*a[1]*a[2]^
3*a[3]+1328336*a[1]*a[2]^2*a[3]^2-119606496*a[1]*a[2]*a[3]^3-452195536*a[1]*a[3
]^4+88128*a[2]^5+2288688*a[2]^4*a[3]+10519712*a[2]^3*a[3]^2-113560976*a[2]^2*a[
3]^3-1100867184*a[2]*a[3]^4-2516696320*a[3]^5+5880*a[1]^4+64680*a[1]^3*a[2]+
541712*a[1]^3*a[3]+266004*a[1]^2*a[2]^2+4551968*a[1]^2*a[2]*a[3]+17188420*a[1]^
2*a[3]^2+484728*a[1]*a[2]^3+12726120*a[1]*a[2]^2*a[3]+97477696*a[1]*a[2]*a[3]^2
+228222824*a[1]*a[3]^3+331128*a[2]^4+11758776*a[2]^3*a[3]+140592504*a[2]^2*a[3]
^2+615652160*a[2]*a[3]^3+1288010664*a[3]^4+16144*a[1]^3+132472*a[1]^2*a[2]+
1071508*a[1]^2*a[3]+360756*a[1]*a[2]^2+5953308*a[1]*a[2]*a[3]+22448872*a[1]*a[3
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15354868822*a[3]+56008621)/(16*a[1]^2*a[2]*a[3]+48*a[1]^2*a[3]^2+96*a[1]*a[2]^2
*a[3]+800*a[1]*a[2]*a[3]^2+1536*a[1]*a[3]^3+144*a[2]^3*a[3]+1968*a[2]^2*a[3]^2+
8704*a[2]*a[3]^3+12288*a[3]^4+160*a[1]^2*a[2]+488*a[1]^2*a[3]+936*a[1]*a[2]^2+
8368*a[1]*a[2]*a[3]+19864*a[1]*a[3]^2+1368*a[2]^3+20328*a[2]^2*a[3]+105672*a[2]
*a[3]^2+192896*a[3]^3+2768*a[1]^2+13952*a[1]*a[2]+148552*a[1]*a[3]+17124*a[2]^2
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3]+33)^2/(1+2*a[1]+6*a[2]+32*a[3])^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+19660800*a[1]^10*a[2]^3*a[3]^3+88473600*a[
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39321600*a[1]^9*a[2]^5*a[3]^2+681574400*a[1]^9*a[2]^4*a[3]^3+4639948800*a[1]^9*
a[2]^3*a[3]^4+15571353600*a[1]^9*a[2]^2*a[3]^5+25834291200*a[1]^9*a[2]*a[3]^6+
16986931200*a[1]^9*a[3]^7+414515200*a[1]^8*a[2]^6*a[3]^2+9378201600*a[1]^8*a[2]
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1143477043200*a[1]^8*a[2]^2*a[3]^6+1625260032000*a[1]^8*a[2]*a[3]^7+
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*a[2]*a[3]^5+14879344438152036529856*a[1]^2*a[3]^6-9666265776871680*a[1]*a[2]^7
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40258899404626569344*a[1]*a[2]^4*a[3]^3+2269301668063616834432*a[1]*a[2]^3*a[3]
^4+30335553122255558867328*a[1]*a[2]^2*a[3]^5+147158766852295296236416*a[1]*a[2
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1112315201982682075615616*a[2]*a[3]^7+1324243213179219704104960*a[3]^8+
28428465793088*a[1]^7+285104593144128*a[1]^6*a[2]+3981321880960320*a[1]^6*a[3]+
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12711955091483449536*a[1]^4*a[3]^3+4941696464652416*a[1]^3*a[2]^4+
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[3]^3+3450815763459051404160*a[1]^2*a[2]*a[3]^4+7338738305591163272192*a[1]^2*a
[3]^5+36113433703263360*a[1]*a[2]^6+2758214837394884352*a[1]*a[2]^5*a[3]+
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^3+11193802547704944098944*a[1]*a[2]^2*a[3]^4+46428658670904987881152*a[1]*a[2]
*a[3]^5+74780727767709338783872*a[1]*a[3]^6+24025739769377280*a[2]^7+
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1199775129259653053952*a[2]^4*a[3]^3+12723176446830092557248*a[2]^3*a[3]^4+
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312824243205298075342848*a[3]^7+65225715378160*a[1]^6+894247763674944*a[1]^5*a[
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136990019823656960*a[1]^4*a[2]*a[3]+783894457993745760*a[1]^4*a[3]^2+
20007073791686528*a[1]^3*a[2]^3+739752731543487136*a[1]^3*a[2]^2*a[3]+
8498916259739599136*a[1]^3*a[2]*a[3]^2+30267481726747813184*a[1]^3*a[3]^3+
44326784551726688*a[1]^2*a[2]^4+2133336330983597440*a[1]^2*a[2]^3*a[3]+
36396317577138453008*a[1]^2*a[2]^2*a[3]^2+259075941661434686656*a[1]^2*a[2]*a[3
]^3+646705451179998440128*a[1]^2*a[3]^4+55414834020768000*a[1]*a[2]^5+
3232350815904859264*a[1]*a[2]^4*a[3]+72223917542128795520*a[1]*a[2]^3*a[3]^2+
765289808718675483136*a[1]*a[2]^2*a[3]^3+3820560131792974596704*a[1]*a[2]*a[3]^
4+7163831486661573228192*a[1]*a[3]^5+29656547185144320*a[2]^6+
2017535779740395712*a[2]^5*a[3]+55266582927350578016*a[2]^4*a[3]^2+
772470900728894431968*a[2]^3*a[3]^3+5764813204884727736352*a[2]^2*a[3]^4+
21674070278589470674048*a[2]*a[3]^5+31978155485577162930592*a[3]^6+
123417022522304*a[1]^5+1609979634335488*a[1]^4*a[2]+17599070628019232*a[1]^4*a[
3]+8672632760288064*a[1]^3*a[2]^2+190955698321513872*a[1]^3*a[2]*a[3]+
988775918434962928*a[1]^3*a[3]^2+24064455470417312*a[1]^2*a[2]^3+
795827182534005952*a[1]^2*a[2]^2*a[3]+8293845188919857712*a[1]^2*a[2]*a[3]^2+
27111228180592301200*a[1]^2*a[3]^3+34225478052824448*a[1]*a[2]^4+
1504795523380435216*a[1]*a[2]^3*a[3]+23585442742775548352*a[1]*a[2]^2*a[3]^2+
155266072236701280480*a[1]*a[2]*a[3]^3+360892320652493834576*a[1]*a[3]^4+
19831302170774208*a[2]^5+1084293336647574432*a[2]^4*a[3]+22661752952725767824*a
[2]^3*a[3]^2+224793429801180785488*a[2]^2*a[3]^3+1053651690459769762352*a[2]*a[
3]^4+1861344820561062844688*a[3]^5+138061242694916*a[1]^4+1489448856312640*a[1]
^3*a[2]+15240081520257216*a[1]^3*a[3]+6103841455926896*a[1]^2*a[2]^2+
126073794412354784*a[1]^2*a[2]*a[3]+614411773293704308*a[1]^2*a[3]^2+
11246359158973656*a[1]*a[2]^3+350851251171407512*a[1]*a[2]^2*a[3]+
3453773143579975528*a[1]*a[2]*a[3]^2+10688005306396078144*a[1]*a[3]^3+
7846199133212088*a[2]^4+328000813236158544*a[2]^3*a[3]+4882950075540371444*a[2]
^2*a[3]^2+30551059891284797232*a[2]*a[3]^3+67578429981711869168*a[3]^4+
91434907568452*a[1]^3+748654076093716*a[1]^2*a[2]+7287485911573364*a[1]^2*a[3]+
2053636618049892*a[1]*a[2]^2+40442340807695880*a[1]*a[2]*a[3]+
188011148617595740*a[1]*a[3]^2+1886112844219296*a[2]^3+56282933925188868*a[2]^2
*a[3]+529752752715213464*a[2]*a[3]^2+1567948451122445932*a[3]^3+35633832035685*
a[1]^2+195857128223300*a[1]*a[2]+1826528454197300*a[1]*a[3]+269464044091974*a[2
]^2+5092110121323338*a[2]*a[3]+22709396409367634*a[3]^2+7575225008682*a[1]+
20942924845578*a[2]+187745935846378*a[3]+678390529579)/(16*a[1]^2*a[2]*a[3]+48*
a[1]^2*a[3]^2+96*a[1]*a[2]^2*a[3]+800*a[1]*a[2]*a[3]^2+1536*a[1]*a[3]^3+144*a[2
]^3*a[3]+1968*a[2]^2*a[3]^2+8704*a[2]*a[3]^3+12288*a[3]^4+160*a[1]^2*a[2]+488*a
[1]^2*a[3]+936*a[1]*a[2]^2+8368*a[1]*a[2]*a[3]+19864*a[1]*a[3]^2+1368*a[2]^3+
20328*a[2]^2*a[3]+105672*a[2]*a[3]^2+192896*a[3]^3+2768*a[1]^2+13952*a[1]*a[2]+
148552*a[1]*a[3]+17124*a[2]^2+395252*a[2]*a[3]+1755008*a[3]^2+4182*a[1]+10986*a
[2]+100810*a[3]+1447)^2/(4*a[1]*a[2]+12*a[1]*a[3]+12*a[2]^2+100*a[2]*a[3]+192*a
[3]^2+34*a[1]+72*a[2]+1222*a[3]+33)^2/(1+2*a[1]+6*a[2]+32*a[3])^2


                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



1638400*a[1]^10*a[2]^4*a[3]^2+19660800*a[1]^10*a[2]^3*a[3]^3+88473600*a[1]^10*a
[2]^2*a[3]^4+176947200*a[1]^10*a[2]*a[3]^5+132710400*a[1]^10*a[3]^6+39321600*a[
1]^9*a[2]^5*a[3]^2+681574400*a[1]^9*a[2]^4*a[3]^3+4639948800*a[1]^9*a[2]^3*a[3]
^4+15571353600*a[1]^9*a[2]^2*a[3]^5+25834291200*a[1]^9*a[2]*a[3]^6+16986931200*
a[1]^9*a[3]^7+414515200*a[1]^8*a[2]^6*a[3]^2+9378201600*a[1]^8*a[2]^5*a[3]^3+
86977740800*a[1]^8*a[2]^4*a[3]^4+423532953600*a[1]^8*a[2]^3*a[3]^5+
1143477043200*a[1]^8*a[2]^2*a[3]^6+1625260032000*a[1]^8*a[2]*a[3]^7+
951400857600*a[1]^8*a[3]^8+2516582400*a[1]^7*a[2]^7*a[3]^2+70044876800*a[1]^7*a
[2]^6*a[3]^3+825478348800*a[1]^7*a[2]^5*a[3]^4+5336675123200*a[1]^7*a[2]^4*a[3]
^5+20436772454400*a[1]^7*a[2]^3*a[3]^6+46367244288000*a[1]^7*a[2]^2*a[3]^7+
57736101888000*a[1]^7*a[2]*a[3]^8+30457567641600*a[1]^7*a[3]^9+9702604800*a[1]^
6*a[2]^8*a[3]^2+319016140800*a[1]^6*a[2]^7*a[3]^3+4552563097600*a[1]^6*a[2]^6*a
[3]^4+36800259686400*a[1]^6*a[2]^5*a[3]^5+184162205696000*a[1]^6*a[2]^4*a[3]^6+
583929456230400*a[1]^6*a[2]^3*a[3]^7+1145201747558400*a[1]^6*a[2]^2*a[3]^8+
1269982612684800*a[1]^6*a[2]*a[3]^9+609762882355200*a[1]^6*a[3]^10+24772608000*
a[1]^5*a[2]^9*a[3]^2+931450060800*a[1]^5*a[2]^8*a[3]^3+15503523840000*a[1]^5*a[
2]^7*a[3]^4+149827302195200*a[1]^5*a[2]^6*a[3]^5+925768876032000*a[1]^5*a[2]^5*
a[3]^6+3789699992780800*a[1]^5*a[2]^4*a[3]^7+10269651173376000*a[1]^5*a[2]^3*a[
3]^8+17752605799219200*a[1]^5*a[2]^2*a[3]^9+17753834520576000*a[1]^5*a[2]*a[3]^
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1781150515200*a[1]^4*a[2]^9*a[3]^3+33391617638400*a[1]^4*a[2]^8*a[3]^4+
370775477452800*a[1]^4*a[2]^7*a[3]^5+2699710126489600*a[1]^4*a[2]^6*a[3]^6+
13461650158387200*a[1]^4*a[2]^5*a[3]^7+46517700273766400*a[1]^4*a[2]^4*a[3]^8+
109896261225676800*a[1]^4*a[2]^3*a[3]^9+169702749477273600*a[1]^4*a[2]^2*a[3]^
10+154526534546227200*a[1]^4*a[2]*a[3]^11+62951120909107200*a[1]^4*a[3]^12+
50960793600*a[1]^3*a[2]^11*a[3]^2+2276248780800*a[1]^3*a[2]^10*a[3]^3+
46215423590400*a[1]^3*a[2]^9*a[3]^4+563861834956800*a[1]^3*a[2]^8*a[3]^5+
4599939307929600*a[1]^3*a[2]^7*a[3]^6+26375970829107200*a[1]^3*a[2]^6*a[3]^7+
108544478635622400*a[1]^3*a[2]^5*a[3]^8+320615202213068800*a[1]^3*a[2]^4*a[3]^9
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^11+771741198581760000*a[1]^3*a[2]*a[3]^12+292786202581401600*a[1]^3*a[3]^13+
44192563200*a[1]^2*a[2]^12*a[3]^2+2059134566400*a[1]^2*a[2]^11*a[3]^3+
43596162662400*a[1]^2*a[2]^10*a[3]^4+555556346265600*a[1]^2*a[2]^9*a[3]^5+
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3]^11+2311338797826048000*a[1]^2*a[2]^2*a[3]^12+1768220855894016000*a[1]^2*a[2]
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1414162022400*a[1]*a[2]^12*a[3]^3+31541546188800*a[1]*a[2]^11*a[3]^4+
419976393523200*a[1]*a[2]^10*a[3]^5+3712230162432000*a[1]*a[2]^9*a[3]^6+
22912050659328000*a[1]*a[2]^8*a[3]^7+101129119531008000*a[1]*a[2]^7*a[3]^8+
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2232039097958400*a[2]^10*a[3]^6+16016479322112000*a[2]^9*a[3]^7+
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919138729984000000*a[2]^6*a[3]^10+1920039628308480000*a[2]^5*a[3]^11+
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2651032485730713600*a[2]^2*a[3]^14+1614962678877388800*a[2]*a[3]^15+
569986827839078400*a[3]^16+32768000*a[1]^10*a[2]^4*a[3]+422707200*a[1]^10*a[2]^
3*a[3]^2+2034892800*a[1]^10*a[2]^2*a[3]^3+4335206400*a[1]^10*a[2]*a[3]^4+
3450470400*a[1]^10*a[3]^5+781516800*a[1]^9*a[2]^5*a[3]+14334361600*a[1]^9*a[2]^
4*a[3]^2+104585625600*a[1]^9*a[2]^3*a[3]^3+378224640000*a[1]^9*a[2]^2*a[3]^4+
677132697600*a[1]^9*a[2]*a[3]^5+480013516800*a[1]^9*a[3]^6+8187084800*a[1]^8*a[
2]^6*a[3]+194358476800*a[1]^8*a[2]^5*a[3]^2+1919690342400*a[1]^8*a[2]^4*a[3]^3+
10067460096000*a[1]^8*a[2]^3*a[3]^4+29473343078400*a[1]^8*a[2]^2*a[3]^5+
45566779392000*a[1]^8*a[2]*a[3]^6+29027480371200*a[1]^8*a[3]^7+49397760000*a[1]
^7*a[2]^7*a[3]+1435071283200*a[1]^7*a[2]^6*a[3]^2+17903978086400*a[1]^7*a[2]^5*
a[3]^3+124052208025600*a[1]^7*a[2]^4*a[3]^4+514054658457600*a[1]^7*a[2]^3*a[3]^
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997613140377600*a[1]^7*a[3]^8+189304012800*a[1]^6*a[2]^8*a[3]+6472738406400*a[1
]^6*a[2]^7*a[3]^2+97313765785600*a[1]^6*a[2]^6*a[3]^3+838647891558400*a[1]^6*a[
2]^5*a[3]^4+4520464816537600*a[1]^6*a[2]^4*a[3]^5+15563620260249600*a[1]^6*a[2]
^3*a[3]^6+33332498119065600*a[1]^6*a[2]^2*a[3]^7+40500195228057600*a[1]^6*a[2]*
a[3]^8+21331757157580800*a[1]^6*a[3]^9+480588595200*a[1]^5*a[2]^9*a[3]+
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3356406125363200*a[1]^5*a[2]^6*a[3]^4+22225835697766400*a[1]^5*a[2]^5*a[3]^5+
98336517931008000*a[1]^5*a[2]^4*a[3]^6+289971178119168000*a[1]^5*a[2]^3*a[3]^7+
548112656248012800*a[1]^5*a[2]^2*a[3]^8+601188716091801600*a[1]^5*a[2]*a[3]^9+
290915160057446400*a[1]^5*a[3]^10+824927846400*a[1]^4*a[2]^10*a[3]+
35575942348800*a[1]^4*a[2]^9*a[3]^2+697606859980800*a[1]^4*a[2]^8*a[3]^3+
8185223208960000*a[1]^4*a[2]^7*a[3]^4+63565334121676800*a[1]^4*a[2]^6*a[3]^5+
340843188384563200*a[1]^4*a[2]^5*a[3]^6+1275464176187801600*a[1]^4*a[2]^4*a[3]^
7+3281597479811481600*a[1]^4*a[2]^3*a[3]^8+5542143790546944000*a[1]^4*a[2]^2*a[
3]^9+5534409985464729600*a[1]^4*a[2]*a[3]^10+2475850386427084800*a[1]^4*a[3]^11
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957700576051200*a[1]^3*a[2]^9*a[3]^3+12302743412736000*a[1]^3*a[2]^8*a[3]^4+
106565994165043200*a[1]^3*a[2]^7*a[3]^5+653667645902028800*a[1]^3*a[2]^6*a[3]^6
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]^8+20938894144438272000*a[1]^3*a[2]^3*a[3]^9+31769709076964966400*a[1]^3*a[2]^
2*a[3]^10+29052667800059904000*a[1]^3*a[2]*a[3]^11+12096156162274099200*a[1]^3*
a[3]^12+848019456000*a[1]^2*a[2]^12*a[3]+40771819929600*a[1]^2*a[2]^11*a[3]^2+
899080180531200*a[1]^2*a[2]^10*a[3]^3+12034959453388800*a[1]^2*a[2]^9*a[3]^4+
109081111825612800*a[1]^2*a[2]^8*a[3]^5+707030367731712000*a[1]^2*a[2]^7*a[3]^6
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3453773143579975528*a[1]*a[2]*a[3]^2+10688005306396078144*a[1]*a[3]^3+
7846199133212088*a[2]^4+328000813236158544*a[2]^3*a[3]+4882950075540371444*a[2]
^2*a[3]^2+30551059891284797232*a[2]*a[3]^3+67578429981711869168*a[3]^4+
91434907568452*a[1]^3+748654076093716*a[1]^2*a[2]+7287485911573364*a[1]^2*a[3]+
2053636618049892*a[1]*a[2]^2+40442340807695880*a[1]*a[2]*a[3]+
188011148617595740*a[1]*a[3]^2+1886112844219296*a[2]^3+56282933925188868*a[2]^2
*a[3]+529752752715213464*a[2]*a[3]^2+1567948451122445932*a[3]^3+35633832035685*
a[1]^2+195857128223300*a[1]*a[2]+1826528454197300*a[1]*a[3]+269464044091974*a[2
]^2+5092110121323338*a[2]*a[3]+22709396409367634*a[3]^2+7575225008682*a[1]+
20942924845578*a[2]+187745935846378*a[3]+678390529579


We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 5, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                            1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
                x[n + 1] = -------------------------------------
                           1 + 2 x[n - 2] + 7 x[n - 1] + 31 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], -----------------------------]
                                            1 + 2 x[1] + 7 x[2] + 31 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], -----------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 7 x[2] + 31 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+7*a[2]+31*a
[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+7*a[2]+31*a[3]), (2*a[1]*a[2]+4
*a[1]*a[3]+7*a[2]^2+45*a[2]*a[3]+62*a[3]^2+39*a[1]+82*a[2]+1402*a[3]+38)/(4*a[1
]*a[2]+14*a[1]*a[3]+14*a[2]^2+111*a[2]*a[3]+217*a[3]^2+33*a[1]+71*a[2]+1185*a[3
]+32)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+7*a[2]+31*a[3]), (2*a[1]*a[2]+4*a[1]
*a[3]+7*a[2]^2+45*a[2]*a[3]+62*a[3]^2+39*a[1]+82*a[2]+1402*a[3]+38)/(4*a[1]*a[2
]+14*a[1]*a[3]+14*a[2]^2+111*a[2]*a[3]+217*a[3]^2+33*a[1]+71*a[2]+1185*a[3]+32)
, (8*a[1]^2*a[2]*a[3]+28*a[1]^2*a[3]^2+56*a[1]*a[2]^2*a[3]+444*a[1]*a[2]*a[3]^2
+868*a[1]*a[3]^3+98*a[2]^3*a[3]+1211*a[2]^2*a[3]^2+4960*a[2]*a[3]^3+6727*a[3]^4
+164*a[1]^2*a[2]+418*a[1]^2*a[3]+1136*a[1]*a[2]^2+8055*a[1]*a[2]*a[3]+14921*a[1
]*a[3]^2+1967*a[2]^3+22886*a[2]^2*a[3]+92322*a[2]*a[3]^2+130851*a[3]^3+3018*a[1
]^2+16902*a[1]*a[2]+156973*a[1]*a[3]+22320*a[2]^2+479955*a[2]*a[3]+1737641*a[3]
^2+4482*a[1]+13441*a[2]+102407*a[3]+1502)/(16*a[1]^2*a[2]*a[3]+56*a[1]^2*a[3]^2
+112*a[1]*a[2]^2*a[3]+888*a[1]*a[2]*a[3]^2+1736*a[1]*a[3]^3+196*a[2]^3*a[3]+
2422*a[2]^2*a[3]^2+9920*a[2]*a[3]^3+13454*a[3]^4+160*a[1]^2*a[2]+506*a[1]^2*a[3
]+1078*a[1]*a[2]^2+8787*a[1]*a[2]*a[3]+20515*a[1]*a[3]^2+1813*a[2]^3+23905*a[2]
^2*a[3]+114660*a[2]*a[3]^2+196416*a[3]^3+2715*a[1]^2+14973*a[1]*a[2]+145068*a[1
]*a[3]+19614*a[2]^2+431384*a[2]*a[3]+1698984*a[3]^2+4117*a[1]+12028*a[2]+98804*
a[3]+1434)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
28*a[1]*a[2]^3+124*a[1]*a[2]^2*a[3]+28*a[1]*a[2]*a[3]^2+124*a[1]*a[3]^3+49*a[2]
^4+434*a[2]^3*a[3]+1010*a[2]^2*a[3]^2+434*a[2]*a[3]^3+961*a[3]^4-8*a[1]^2*a[2]-\
8*a[1]^2*a[3]-52*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-244*a[1]*a[3]^2-84*a[2]^3-904*a
[2]^2*a[3]-2776*a[2]*a[3]^2-1860*a[3]^3+9*a[1]^2+58*a[1]*a[2]+228*a[1]*a[3]+96*
a[2]^2+656*a[2]*a[3]+1835*a[3]^2+8*a[1]+26*a[2]+122*a[3]+2)/(1+2*a[1]+7*a[2]+31
*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+448*a[1]^4*a[2]*a[3]^3+784*a[1]^4*a[3]^4+896
*a[1]^3*a[2]^3*a[3]^2+10240*a[1]^3*a[2]^2*a[3]^3+38752*a[1]^3*a[2]*a[3]^4+48608
*a[1]^3*a[3]^5+4704*a[1]^2*a[2]^4*a[3]^2+74592*a[1]^2*a[2]^3*a[3]^3+441528*a[1]
^2*a[2]^2*a[3]^4+1156176*a[1]^2*a[2]*a[3]^5+1130136*a[1]^2*a[3]^6+10976*a[1]*a[
2]^5*a[3]^2+222656*a[1]*a[2]^4*a[3]^3+1801016*a[1]*a[2]^3*a[3]^4+7260200*a[1]*a
[2]^2*a[3]^5+14584136*a[1]*a[2]*a[3]^6+11678072*a[1]*a[3]^7+9604*a[2]^6*a[3]^2+
237356*a[2]^5*a[3]^3+2438681*a[2]^4*a[3]^4+13331612*a[2]^3*a[3]^5+40894394*a[2]
^2*a[3]^6+66731840*a[2]*a[3]^7+45252529*a[3]^8-128*a[1]^4*a[2]^2*a[3]+160*a[1]^
4*a[2]*a[3]^2+2128*a[1]^4*a[3]^3-1792*a[1]^3*a[2]^3*a[3]-7056*a[1]^3*a[2]^2*a[3
]^2+56728*a[1]^3*a[2]*a[3]^3+208152*a[1]^3*a[3]^4-9408*a[1]^2*a[2]^4*a[3]-85848
*a[1]^2*a[2]^3*a[3]^2+191484*a[1]^2*a[2]^2*a[3]^3+3207480*a[1]^2*a[2]*a[3]^4+
6611556*a[1]^2*a[3]^5-21952*a[1]*a[2]^5*a[3]-314188*a[1]*a[2]^4*a[3]^2-379414*a
[1]*a[2]^3*a[3]^3+11915930*a[1]*a[2]^2*a[3]^4+60928454*a[1]*a[2]*a[3]^5+
86630306*a[1]*a[3]^6-19208*a[2]^6*a[3]-374556*a[2]^5*a[3]^2-1560748*a[2]^4*a[3]
^3+11338642*a[2]^3*a[3]^4+119805390*a[2]^2*a[3]^5+375806738*a[2]*a[3]^6+
406647150*a[3]^7+96*a[1]^4*a[2]^2-1392*a[1]^4*a[2]*a[3]-1656*a[1]^4*a[3]^2+1392
*a[1]^3*a[2]^3-11168*a[1]^3*a[2]^2*a[3]-157416*a[1]^3*a[2]*a[3]^2-78448*a[1]^3*
a[3]^3+7596*a[1]^2*a[2]^4-13500*a[1]^2*a[2]^3*a[3]-1275530*a[1]^2*a[2]^2*a[3]^2
-5242880*a[1]^2*a[2]*a[3]^3+719930*a[1]^2*a[3]^4+18480*a[1]*a[2]^5+78312*a[1]*a
[2]^4*a[3]-3417260*a[1]*a[2]^3*a[3]^2-30459586*a[1]*a[2]^2*a[3]^3-55941768*a[1]
*a[2]*a[3]^4+54192278*a[1]*a[3]^5+16905*a[2]^6+169526*a[2]^5*a[3]-2827226*a[2]^
4*a[3]^2-45755378*a[2]^3*a[3]^3-188544047*a[2]^2*a[3]^4-113346664*a[2]*a[3]^5+
470471004*a[3]^6+1224*a[1]^4*a[2]-4572*a[1]^4*a[3]+16388*a[1]^3*a[2]^2+42770*a[
1]^3*a[2]*a[3]-595006*a[1]^3*a[3]^2+81636*a[1]^2*a[2]^3+750104*a[1]^2*a[2]^2*a[
3]-2681558*a[1]^2*a[2]*a[3]^2-24886362*a[1]^2*a[3]^3+178308*a[1]*a[2]^4+2912420
*a[1]*a[2]^3*a[3]+2167248*a[1]*a[2]^2*a[3]^2-121923856*a[1]*a[2]*a[3]^3-\
400178496*a[1]*a[3]^4+142996*a[2]^5+3512130*a[2]^4*a[3]+15310056*a[2]^3*a[3]^2-\
130635650*a[2]^2*a[3]^3-1120028360*a[2]*a[3]^4-2175650060*a[3]^5+5589*a[1]^4+
68492*a[1]^3*a[2]+509506*a[1]^3*a[3]+311839*a[1]^2*a[2]^2+4826948*a[1]^2*a[2]*a
[3]+15888455*a[1]^2*a[3]^2+624214*a[1]*a[2]^3+15166540*a[1]*a[2]^2*a[3]+
102271858*a[1]*a[2]*a[3]^2+206574588*a[1]*a[3]^3+467072*a[2]^4+15533834*a[2]^3*
a[3]+169874633*a[2]^2*a[3]^2+616597884*a[2]*a[3]^3+1169575269*a[3]^4+15348*a[1]
^3+141710*a[1]^2*a[2]+1004736*a[1]^2*a[3]+432144*a[1]*a[2]^2+6328246*a[1]*a[2]*
a[3]+20670518*a[1]*a[3]^2+428194*a[2]^3+10182866*a[2]^2*a[3]+62112530*a[2]*a[3]
^2+155219962*a[3]^3+15125*a[1]^2+94250*a[1]*a[2]+637510*a[1]*a[3]+148978*a[2]^2
+1928324*a[2]*a[3]+7104762*a[3]^2+6424*a[1]+19516*a[2]+142116*a[3]+1060)/(4*a[1
]*a[2]+14*a[1]*a[3]+14*a[2]^2+111*a[2]*a[3]+217*a[3]^2+33*a[1]+71*a[2]+1185*a[3
]+32)^2/(1+2*a[1]+7*a[2]+31*a[3])^2, (12288*a[1]^8*a[2]^4*a[3]^2+184320*a[1]^8*
a[2]^3*a[3]^3+1041408*a[1]^8*a[2]^2*a[3]^4+2623488*a[1]^8*a[2]*a[3]^5+2483712*a
[1]^8*a[3]^6+356352*a[1]^7*a[2]^5*a[3]^2+6680576*a[1]^7*a[2]^4*a[3]^3+50452480*
a[1]^7*a[2]^3*a[3]^4+191754240*a[1]^7*a[2]^2*a[3]^5+366494464*a[1]^7*a[2]*a[3]^
6+281550080*a[1]^7*a[3]^7+4525056*a[1]^6*a[2]^6*a[3]^2+101650432*a[1]^6*a[2]^5*
a[3]^3+959971840*a[1]^6*a[2]^4*a[3]^4+4877168128*a[1]^6*a[2]^3*a[3]^5+
14052065216*a[1]^6*a[2]^2*a[3]^6+21753974144*a[1]^6*a[2]*a[3]^7+14124377792*a[1
]^6*a[3]^8+32858112*a[1]^5*a[2]^7*a[3]^2+859490304*a[1]^5*a[2]^6*a[3]^3+
9730354176*a[1]^5*a[2]^5*a[3]^4+61827686400*a[1]^5*a[2]^4*a[3]^5+238136054016*a
[1]^5*a[2]^3*a[3]^6+555753356928*a[1]^5*a[2]^2*a[3]^7+727124337408*a[1]^5*a[2]*
a[3]^8+411022442880*a[1]^5*a[3]^9+149210880*a[1]^4*a[2]^8*a[3]^2+4451686400*a[1
]^4*a[2]^7*a[3]^3+58690497280*a[1]^4*a[2]^6*a[3]^4+447112425600*a[1]^4*a[2]^5*a
[3]^5+2154244709520*a[1]^4*a[2]^4*a[3]^6+6723197261760*a[1]^4*a[2]^3*a[3]^7+
13267662337440*a[1]^4*a[2]^2*a[3]^8+15123584459200*a[1]^4*a[2]*a[3]^9+
7614317669840*a[1]^4*a[3]^10+433859328*a[1]^3*a[2]^9*a[3]^2+14534920960*a[1]^3*
a[2]^8*a[3]^3+218518695808*a[1]^3*a[2]^7*a[3]^4+1938573891328*a[1]^3*a[2]^6*a[3
]^5+11200316424432*a[1]^3*a[2]^5*a[3]^6+43743615120144*a[1]^3*a[2]^4*a[3]^7+
115509695155680*a[1]^3*a[2]^3*a[3]^8+198753573964064*a[1]^3*a[2]^2*a[3]^9+
201974116950704*a[1]^3*a[2]*a[3]^10+92203026312656*a[1]^3*a[3]^11+788757312*a[1
]^2*a[2]^10*a[3]^2+29312242176*a[1]^2*a[2]^9*a[3]^3+494615086560*a[1]^2*a[2]^8*
a[3]^4+5002619156640*a[1]^2*a[2]^7*a[3]^5+33659272963044*a[1]^2*a[2]^6*a[3]^6+
157684595525496*a[1]^2*a[2]^5*a[3]^7+521357781945228*a[1]^2*a[2]^4*a[3]^8+
1201271076282480*a[1]^2*a[2]^3*a[3]^9+1844269851792060*a[1]^2*a[2]^2*a[3]^10+
1700738363012472*a[1]^2*a[2]*a[3]^11+713810677294260*a[1]^2*a[3]^12+819643776*a
[1]*a[2]^11*a[3]^2+33460028672*a[1]*a[2]^10*a[3]^3+625916058880*a[1]*a[2]^9*a[3
]^4+7102487618688*a[1]*a[2]^8*a[3]^5+54474081565736*a[1]*a[2]^7*a[3]^6+
297234047766112*a[1]*a[2]^6*a[3]^7+1179334344139848*a[1]*a[2]^5*a[3]^8+
3404680512978880*a[1]*a[2]^4*a[3]^9+7004994562148024*a[1]*a[2]^3*a[3]^10+
9766624335657888*a[1]*a[2]^2*a[3]^11+8284983868322392*a[1]*a[2]*a[3]^12+
3230410791962816*a[1]*a[3]^13+372712032*a[2]^12*a[3]^2+16580307184*a[2]^11*a[3]
^3+340455451756*a[2]^10*a[3]^4+4280354486800*a[2]^9*a[3]^5+36819573344281*a[2]^
8*a[3]^6+229004555153584*a[2]^7*a[3]^7+1058599753896640*a[2]^6*a[3]^8+
3669531198618008*a[2]^5*a[3]^9+9466609003268090*a[2]^4*a[3]^10+
17700777333588368*a[2]^3*a[3]^11+22713952785703244*a[2]^2*a[3]^12+
17903792234980904*a[2]*a[3]^13+6533806537040405*a[3]^14+159744*a[1]^8*a[2]^4*a[
3]+2641920*a[1]^8*a[2]^3*a[3]^2+16063488*a[1]^8*a[2]^2*a[3]^3+42687744*a[1]^8*a
[2]*a[3]^4+41930112*a[1]^8*a[3]^5+4644864*a[1]^7*a[2]^5*a[3]+93385728*a[1]^7*a[
2]^4*a[3]^2+757372160*a[1]^7*a[2]^3*a[3]^3+3084771840*a[1]^7*a[2]^2*a[3]^4+
6292732608*a[1]^7*a[2]*a[3]^5+5134314304*a[1]^7*a[3]^6+59126784*a[1]^6*a[2]^6*a
[3]+1398235392*a[1]^6*a[2]^5*a[3]^2+14071514624*a[1]^6*a[2]^4*a[3]^3+
76673156032*a[1]^6*a[2]^3*a[3]^4+237396683328*a[1]^6*a[2]^2*a[3]^5+394585072000
*a[1]^6*a[2]*a[3]^6+274381993088*a[1]^6*a[3]^7+430268160*a[1]^5*a[2]^7*a[3]+
11694859008*a[1]^5*a[2]^6*a[3]^2+139908697920*a[1]^5*a[2]^5*a[3]^3+949613730112
*a[1]^5*a[2]^4*a[3]^4+3929219098816*a[1]^5*a[2]^3*a[3]^5+9869958568512*a[1]^5*a
[2]^2*a[3]^6+13892671104704*a[1]^5*a[2]*a[3]^7+8433944109248*a[1]^5*a[3]^8+
1957334400*a[1]^4*a[2]^8*a[3]+60115769280*a[1]^4*a[2]^7*a[3]^2+831065528960*a[1
]^4*a[2]^6*a[3]^3+6725431102000*a[1]^4*a[2]^5*a[3]^4+34697631686840*a[1]^4*a[2]
^4*a[3]^5+116430265416640*a[1]^4*a[2]^3*a[3]^6+247418580679760*a[1]^4*a[2]^2*a[
3]^7+303723466192720*a[1]^4*a[2]*a[3]^8+164596016665080*a[1]^4*a[3]^9+
5698783104*a[1]^3*a[2]^9*a[3]+195212848320*a[1]^3*a[2]^8*a[3]^2+3056912191088*a
[1]^3*a[2]^7*a[3]^3+28640658168704*a[1]^3*a[2]^6*a[3]^4+176393635105452*a[1]^3*
a[2]^5*a[3]^5+738576449755332*a[1]^3*a[2]^4*a[3]^6+2097776332232040*a[1]^3*a[2]
^3*a[3]^7+3890373678894024*a[1]^3*a[2]^2*a[3]^8+4267529365187452*a[1]^3*a[2]*a[
3]^9+2106075351540916*a[1]^3*a[3]^10+10368632064*a[1]^2*a[2]^10*a[3]+
392079409008*a[1]^2*a[2]^9*a[3]^2+6852712335840*a[1]^2*a[2]^8*a[3]^3+
72792908303780*a[1]^2*a[2]^7*a[3]^4+519533964511208*a[1]^2*a[2]^6*a[3]^5+
2599287058025616*a[1]^2*a[2]^5*a[3]^6+9221501552206284*a[1]^2*a[2]^4*a[3]^7+
22884224471599140*a[1]^2*a[2]^3*a[3]^8+37974287825547360*a[1]^2*a[2]^2*a[3]^9+
37988507731975864*a[1]^2*a[2]*a[3]^10+17358834138421132*a[1]^2*a[3]^11+
10777186224*a[1]*a[2]^11*a[3]+446124892752*a[1]*a[2]^10*a[3]^2+8604748934684*a[
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683833365923522578*a[1]^2*a[3]^5+320183322424168*a[1]*a[2]^6+14946219333273196*
a[1]*a[2]^5*a[3]+256244256540986850*a[1]*a[2]^4*a[3]^2+1905595770063796858*a[1]
*a[2]^3*a[3]^3+5476873210400217978*a[1]*a[2]^2*a[3]^4+3681087582739244186*a[1]*
a[2]*a[3]^5+5296810670513336556*a[1]*a[3]^6+138361164716124*a[2]^7+
7761000580073948*a[2]^6*a[3]+167346607765908732*a[2]^5*a[3]^2+
1679094067757543004*a[2]^4*a[3]^3+7275872875417227184*a[2]^3*a[3]^4+
8287121257711825808*a[2]^2*a[3]^5-2649843009876759304*a[2]*a[3]^6+
53760031941127037624*a[3]^7+103213224342*a[1]^6+2044361891324*a[1]^5*a[2]+
11592860947522*a[1]^5*a[3]+16619091414998*a[1]^4*a[2]^2+199821057830028*a[1]^4*
a[2]*a[3]+473400385972599*a[1]^4*a[3]^2+70877957481182*a[1]^3*a[2]^3+
1358103480343122*a[1]^3*a[2]^2*a[3]+6860020882098920*a[1]^3*a[2]*a[3]^2+
8484203018706172*a[1]^3*a[3]^3+167237838505639*a[1]^2*a[2]^4+4530776720638238*a
[1]^2*a[2]^3*a[3]+37044582094952832*a[1]^2*a[2]^2*a[3]^2+95501707939025498*a[1]
^2*a[2]*a[3]^3+69926079506033329*a[1]^2*a[3]^4+207137441465734*a[1]*a[2]^5+
7407417845772286*a[1]*a[2]^4*a[3]+87269175701492454*a[1]*a[2]^3*a[3]^2+
371101918192212338*a[1]*a[2]^2*a[3]^3+442234113892395796*a[1]*a[2]*a[3]^4+
516348767272902416*a[1]*a[3]^5+105357666928945*a[2]^6+4747296537616916*a[2]^5*a
[3]+75285201541558887*a[2]^4*a[3]^2+475950821770705872*a[2]^3*a[3]^3+
931855951100137355*a[2]^2*a[3]^4+273255639738298676*a[2]*a[3]^5+
4882744868431798885*a[3]^6+118469299278*a[1]^5+2017532956566*a[1]^4*a[2]+
10044757644158*a[1]^4*a[3]+13537314010714*a[1]^3*a[2]^2+143508259919976*a[1]^3*
a[2]*a[3]+287717708899538*a[1]^3*a[3]^2+44573868438854*a[1]^2*a[2]^3+
762203333496980*a[1]^2*a[2]^2*a[3]+3201511335313330*a[1]^2*a[2]*a[3]^2+
3444303637794708*a[1]^2*a[3]^3+71917462651590*a[1]*a[2]^4+1765540590457264*a[1]
*a[2]^3*a[3]+12141625434052764*a[1]*a[2]^2*a[3]^2+23274409058990640*a[1]*a[2]*a
[3]^3+27754993133907838*a[1]*a[3]^4+45478591873170*a[2]^5+1498257369498226*a[2]
^4*a[3]+15166945071917504*a[2]^3*a[3]^2+47351295008528912*a[2]^2*a[3]^3+
35956661175401102*a[2]*a[3]^4+255539489198949502*a[3]^5+79075531229*a[1]^4+
1115234192308*a[1]^3*a[2]+4764345946972*a[1]^3*a[3]+5837462165855*a[1]^2*a[2]^2
+52408180463762*a[1]^2*a[2]*a[3]+91300372720263*a[1]^2*a[3]^2+13325811863308*a[
1]*a[2]^3+194754324313576*a[1]*a[2]^2*a[3]+637411593248460*a[1]*a[2]*a[3]^2+
906932814898304*a[1]*a[3]^3+11151202824365*a[2]^4+238047994694284*a[2]^3*a[3]+
1263388070081962*a[2]^2*a[3]^2+1729416200148596*a[2]*a[3]^3+8500022072779705*a[
3]^4+31041403560*a[1]^3+337602729436*a[1]^2*a[2]+1256078863340*a[1]^2*a[3]+
1232500809772*a[1]*a[2]^2+8911007731048*a[1]*a[2]*a[3]+17989551889756*a[1]*a[3]
^2+1478522129192*a[2]^3+17281706031096*a[2]^2*a[3]+42828529780392*a[2]*a[3]^2+
185129522571480*a[3]^3+7058845876*a[1]^2+50413936560*a[1]*a[2]+199463336096*a[1
]*a[3]+95922426260*a[2]^2+545217451000*a[2]*a[3]+2588980999828*a[3]^2+948782064
*a[1]+2833790512*a[2]+21291873872*a[3]+78763792)/(16*a[1]^2*a[2]*a[3]+56*a[1]^2
*a[3]^2+112*a[1]*a[2]^2*a[3]+888*a[1]*a[2]*a[3]^2+1736*a[1]*a[3]^3+196*a[2]^3*a
[3]+2422*a[2]^2*a[3]^2+9920*a[2]*a[3]^3+13454*a[3]^4+160*a[1]^2*a[2]+506*a[1]^2
*a[3]+1078*a[1]*a[2]^2+8787*a[1]*a[2]*a[3]+20515*a[1]*a[3]^2+1813*a[2]^3+23905*
a[2]^2*a[3]+114660*a[2]*a[3]^2+196416*a[3]^3+2715*a[1]^2+14973*a[1]*a[2]+145068
*a[1]*a[3]+19614*a[2]^2+431384*a[2]*a[3]+1698984*a[3]^2+4117*a[1]+12028*a[2]+
98804*a[3]+1434)^2/(4*a[1]*a[2]+14*a[1]*a[3]+14*a[2]^2+111*a[2]*a[3]+217*a[3]^2
+33*a[1]+71*a[2]+1185*a[3]+32)^2/(1+2*a[1]+7*a[2]+31*a[3])^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+22937600*a[1]^10*a[2]^3*a[3]^3+120422400*a
[1]^10*a[2]^2*a[3]^4+280985600*a[1]^10*a[2]*a[3]^5+245862400*a[1]^10*a[3]^6+
45875200*a[1]^9*a[2]^5*a[3]^2+845414400*a[1]^9*a[2]^4*a[3]^3+6216089600*a[1]^9*
a[2]^3*a[3]^4+22799974400*a[1]^9*a[2]^2*a[3]^5+41726361600*a[1]^9*a[2]*a[3]^6+
30486937600*a[1]^9*a[3]^7+563609600*a[1]^8*a[2]^6*a[3]^2+12867993600*a[1]^8*a[2
]^5*a[3]^3+122132889600*a[1]^8*a[2]^4*a[3]^4+616826470400*a[1]^8*a[2]^3*a[3]^5+
1748412825600*a[1]^8*a[2]^2*a[3]^6+2637401088000*a[1]^8*a[2]*a[3]^7+
1654162227200*a[1]^8*a[3]^8+3979673600*a[1]^7*a[2]^7*a[3]^2+108181913600*a[1]^7
*a[2]^6*a[3]^3+1258534502400*a[1]^7*a[2]^5*a[3]^4+8121638912000*a[1]^7*a[2]^4*a
[3]^5+31396467200000*a[1]^7*a[2]^3*a[3]^6+72702570393600*a[1]^7*a[2]^2*a[3]^7+
93369864089600*a[1]^7*a[2]*a[3]^8+51301894246400*a[1]^7*a[3]^9+17772339200*a[1]
^6*a[2]^8*a[3]^2+558659584000*a[1]^6*a[2]^7*a[3]^3+7680909824000*a[1]^6*a[2]^6*
a[3]^4+60320869683200*a[1]^6*a[2]^5*a[3]^5+295918477657600*a[1]^6*a[2]^4*a[3]^6
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2035182195558400*a[1]^6*a[2]*a[3]^9+995037432972800*a[1]^6*a[3]^10+52122828800*
a[1]^5*a[2]^9*a[3]^2+1849025740800*a[1]^5*a[2]^8*a[3]^3+29187650150400*a[1]^5*a
[2]^7*a[3]^4+269064914944000*a[1]^5*a[2]^6*a[3]^5+1596110999884800*a[1]^5*a[2]^
5*a[3]^6+6317558878848000*a[1]^5*a[2]^4*a[3]^7+16681876678297600*a[1]^5*a[2]^3*
a[3]^8+28332088730265600*a[1]^5*a[2]^2*a[3]^9+28078360919884800*a[1]^5*a[2]*a[3
]^10+12369227013248000*a[1]^5*a[3]^11+101541171200*a[1]^4*a[2]^10*a[3]^2+
3967235686400*a[1]^4*a[2]^9*a[3]^3+69953665689600*a[1]^4*a[2]^8*a[3]^4+
733210279782400*a[1]^4*a[2]^7*a[3]^5+5059411395001600*a[1]^4*a[2]^6*a[3]^6+
24016534157683200*a[1]^4*a[2]^5*a[3]^7+79420328832275200*a[1]^4*a[2]^4*a[3]^8+
180640357549388800*a[1]^4*a[2]^3*a[3]^9+270396383661331200*a[1]^4*a[2]^2*a[3]^
10+240472971447564800*a[1]^4*a[2]*a[3]^11+96457539462636800*a[1]^4*a[3]^12+
132519833600*a[1]^3*a[2]^11*a[3]^2+5536575385600*a[1]^3*a[2]^10*a[3]^3+
105509759129600*a[1]^3*a[2]^9*a[3]^4+1211820189798400*a[1]^3*a[2]^8*a[3]^5+
9329555121017600*a[1]^3*a[2]^7*a[3]^6+50599573241324800*a[1]^3*a[2]^6*a[3]^7+
197428293813856000*a[1]^3*a[2]^5*a[3]^8+554502157861740800*a[1]^3*a[2]^4*a[3]^9
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3]^11+1179399648376096000*a[1]^3*a[2]*a[3]^12+435087789081209600*a[1]^3*a[3]^13
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103734471254400*a[1]^2*a[2]^10*a[3]^4+1241892279600000*a[1]^2*a[2]^9*a[3]^5+
10044719050170000*a[1]^2*a[2]^8*a[3]^6+57991774342454400*a[1]^2*a[2]^7*a[3]^7+
245962697633374400*a[1]^2*a[2]^6*a[3]^8+775647677252649600*a[1]^2*a[2]^5*a[3]^9
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3]^11+3624766453872916800*a[1]^2*a[2]^2*a[3]^12+2654268046230796800*a[1]^2*a[2]
*a[3]^13+914580708304365200*a[1]^2*a[3]^14+84330803200*a[1]*a[2]^13*a[3]^2+
3794886144000*a[1]*a[2]^12*a[3]^3+77614457088000*a[1]*a[2]^11*a[3]^4+
953013801779200*a[1]*a[2]^10*a[3]^5+7815643600889600*a[1]*a[2]^9*a[3]^6+
45051120970195200*a[1]*a[2]^8*a[3]^7+187045697070041600*a[1]*a[2]^7*a[3]^8+
564159080274982400*a[1]*a[2]^6*a[3]^9+1233774753261926400*a[1]*a[2]^5*a[3]^10+
1943526233151948800*a[1]*a[2]^4*a[3]^11+2205881785260428800*a[1]*a[2]^3*a[3]^12
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422769833475270400*a[1]*a[3]^15+36894726400*a[2]^14*a[3]^2+1823653619200*a[2]^
13*a[3]^3+41308916880000*a[2]^12*a[3]^4+567321548886400*a[2]^11*a[3]^5+
5265808316336400*a[2]^10*a[3]^6+34852674901184000*a[2]^9*a[3]^7+
169119039347896400*a[2]^8*a[3]^8+609220635693510400*a[2]^7*a[3]^9+
1632834672584871200*a[2]^6*a[3]^10+3240731311448960000*a[2]^5*a[3]^11+
4730655357783338400*a[2]^4*a[3]^12+5081625472620054400*a[2]^3*a[3]^13+
4080831740610517200*a[2]^2*a[3]^14+2415827619858688000*a[2]*a[3]^15+
819116552358336400*a[3]^16+32768000*a[1]^10*a[2]^4*a[3]+474726400*a[1]^10*a[2]^
3*a[3]^2+2576179200*a[1]^10*a[2]^2*a[3]^3+6206771200*a[1]^10*a[2]*a[3]^4+
5602150400*a[1]^10*a[3]^5+908902400*a[1]^9*a[2]^5*a[3]+17285529600*a[1]^9*a[2]^
4*a[3]^2+132430848000*a[1]^9*a[2]^3*a[3]^3+509729382400*a[1]^9*a[2]^2*a[3]^4+
983779507200*a[1]^9*a[2]*a[3]^5+760478745600*a[1]^9*a[3]^6+11061452800*a[1]^8*a
[2]^6*a[3]+260191027200*a[1]^8*a[2]^5*a[3]^2+2576105574400*a[1]^8*a[2]^4*a[3]^3
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66594244505600*a[1]^8*a[2]*a[3]^6+44823766848000*a[1]^8*a[3]^7+77372108800*a[1]
^7*a[2]^7*a[3]+2164383641600*a[1]^7*a[2]^6*a[3]^2+26248253696000*a[1]^7*a[2]^5*
a[3]^3+178580494182400*a[1]^7*a[2]^4*a[3]^4+734657568812800*a[1]^7*a[2]^3*a[3]^
5+1823667546668800*a[1]^7*a[2]^2*a[3]^6+2524204882796800*a[1]^7*a[2]*a[3]^7+
1500167153721600*a[1]^7*a[3]^8+342328268800*a[1]^6*a[2]^8*a[3]+11063233484800*a
[1]^6*a[2]^7*a[3]^2+158372745216000*a[1]^6*a[2]^6*a[3]^3+1309713738636800*a[1]^
6*a[2]^5*a[3]^4+6831462776566400*a[1]^6*a[2]^4*a[3]^5+22969872204921600*a[1]^6*
a[2]^3*a[3]^6+48526601136652800*a[1]^6*a[2]^2*a[3]^7+58786688121235200*a[1]^6*a
[2]*a[3]^8+31215731892374400*a[1]^6*a[3]^9+995066598400*a[1]^5*a[2]^9*a[3]+
36258706713600*a[1]^5*a[2]^8*a[3]^2+595103457088000*a[1]^5*a[2]^7*a[3]^3+
5766877959360000*a[1]^5*a[2]^6*a[3]^4+36305975265000000*a[1]^5*a[2]^5*a[3]^5+
153728151026923200*a[1]^5*a[2]^4*a[3]^6+437022707484137600*a[1]^5*a[2]^3*a[3]^7
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414034005982833600*a[1]^5*a[3]^10+1923043494400*a[1]^4*a[2]^10*a[3]+
77095134233600*a[1]^4*a[2]^9*a[3]^2+1411178645408000*a[1]^4*a[2]^8*a[3]^3+
15516840605638400*a[1]^4*a[2]^7*a[3]^4+113366606882743200*a[1]^4*a[2]^6*a[3]^5+
574254440668540800*a[1]^4*a[2]^5*a[3]^6+2039482665066240800*a[1]^4*a[2]^4*a[3]^
7+5007205776153952000*a[1]^4*a[2]^3*a[3]^8+8121368019604319200*a[1]^4*a[2]^2*a[
3]^9+7847113666768297600*a[1]^4*a[2]*a[3]^10+3425581258101256800*a[1]^4*a[3]^11
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2109044139350400*a[1]^3*a[2]^9*a[3]^3+25350730457136000*a[1]^3*a[2]^8*a[3]^4+
206047906687467600*a[1]^3*a[2]^7*a[3]^5+1188639716849466000*a[1]^3*a[2]^6*a[3]^
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[3]^8+32105432275818918000*a[1]^3*a[2]^3*a[3]^9+46339157530521802800*a[1]^3*a[2
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3*a[3]^12+2274296348800*a[1]^2*a[2]^12*a[3]+100763827281600*a[1]^2*a[2]^11*a[3]
^2+2061778892740000*a[1]^2*a[2]^10*a[3]^3+25771111853544400*a[1]^2*a[2]^9*a[3]^
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]^6+6052439660043131600*a[1]^2*a[2]^6*a[3]^7+20396381920797363600*a[1]^2*a[2]^5
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[2]^3*a[3]^10+118759935742700792400*a[1]^2*a[2]^2*a[3]^11+94233162151642690400*
a[1]^2*a[2]*a[3]^12+35300110575168952800*a[1]^2*a[3]^13+1575931884800*a[1]*a[2]
^13*a[3]+72677340332800*a[1]*a[2]^12*a[3]^2+1538519502800000*a[1]*a[2]^11*a[3]^
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31277418215857951200*a[3]^15+163840000*a[1]^10*a[2]^4+3279872000*a[1]^10*a[2]^3
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66225107200*a[1]^10*a[3]^4+4501504000*a[1]^9*a[2]^5+110132582400*a[1]^9*a[2]^4*
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                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



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3]^4+1600550200074072004298*a[3]^5+125038992881371*a[1]^4+1500219285752692*a[1]
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]^2*a[3]^2+30457410512374831804*a[2]*a[3]^3+58923487888421614195*a[3]^4+
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168160827530500644*a[1]*a[3]^2+2350946461927608*a[2]^3+63557732419706504*a[2]^2
*a[3]+533641843320068408*a[2]*a[3]^2+1386029417836937320*a[3]^3+32525945166124*
a[1]^2+200038006037040*a[1]*a[2]+1653625896336704*a[1]*a[3]+306665848699340*a[2
]^2+5183057044025800*a[2]*a[3]+20346305813786572*a[3]^2+6940069920336*a[1]+
21537110422288*a[2]+170431994338928*a[3]+623757420208


We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 6, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                            1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
                x[n + 1] = -------------------------------------
                           1 + 2 x[n - 2] + 8 x[n - 1] + 30 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], -----------------------------]
                                            1 + 2 x[1] + 8 x[2] + 30 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], -----------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 8 x[2] + 30 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+8*a[2]+30*a
[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+8*a[2]+30*a[3]), (2*a[1]*a[2]+4
*a[1]*a[3]+8*a[2]^2+46*a[2]*a[3]+60*a[3]^2+39*a[1]+83*a[2]+1401*a[3]+38)/(4*a[1
]*a[2]+16*a[1]*a[3]+16*a[2]^2+124*a[2]*a[3]+240*a[3]^2+32*a[1]+70*a[2]+1148*a[3
]+31)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+8*a[2]+30*a[3]), (2*a[1]*a[2]+4*a[1]
*a[3]+8*a[2]^2+46*a[2]*a[3]+60*a[3]^2+39*a[1]+83*a[2]+1401*a[3]+38)/(4*a[1]*a[2
]+16*a[1]*a[3]+16*a[2]^2+124*a[2]*a[3]+240*a[3]^2+32*a[1]+70*a[2]+1148*a[3]+31)
, (8*a[1]^2*a[2]*a[3]+32*a[1]^2*a[3]^2+64*a[1]*a[2]^2*a[3]+496*a[1]*a[2]*a[3]^2
+960*a[1]*a[3]^3+128*a[2]^3*a[3]+1472*a[2]^2*a[3]^2+5640*a[2]*a[3]^3+7200*a[3]^
4+164*a[1]^2*a[2]+424*a[1]^2*a[3]+1296*a[1]*a[2]^2+8312*a[1]*a[2]*a[3]+14776*a[
1]*a[3]^2+2560*a[2]^3+26224*a[2]^2*a[3]+96004*a[2]*a[3]^2+126240*a[3]^3+3014*a[
1]^2+18436*a[1]*a[2]+155174*a[1]*a[3]+25752*a[2]^2+530274*a[2]*a[3]+1679520*a[3
]^2+4475*a[1]+14901*a[2]+100716*a[3]+1499)/(16*a[1]^2*a[2]*a[3]+64*a[1]^2*a[3]^
2+128*a[1]*a[2]^2*a[3]+992*a[1]*a[2]*a[3]^2+1920*a[1]*a[3]^3+256*a[2]^3*a[3]+
2944*a[2]^2*a[3]^2+11280*a[2]*a[3]^3+14400*a[3]^4+160*a[1]^2*a[2]+528*a[1]^2*a[
3]+1216*a[1]*a[2]^2+9248*a[1]*a[2]*a[3]+21360*a[1]*a[3]^2+2304*a[2]^3+27584*a[2
]^2*a[3]+124800*a[2]*a[3]^2+201600*a[3]^3+2660*a[1]^2+15904*a[1]*a[2]+141524*a[
1]*a[3]+21984*a[2]^2+464444*a[2]*a[3]+1643264*a[3]^2+4048*a[1]+12984*a[2]+96730
*a[3]+1419)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
32*a[1]*a[2]^3+120*a[1]*a[2]^2*a[3]+32*a[1]*a[2]*a[3]^2+120*a[1]*a[3]^3+64*a[2]
^4+480*a[2]^3*a[3]+964*a[2]^2*a[3]^2+480*a[2]*a[3]^3+900*a[3]^4-8*a[1]^2*a[2]-8
*a[1]^2*a[3]-60*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-236*a[1]*a[3]^2-112*a[2]^3-1028*
a[2]^2*a[3]-2744*a[2]*a[3]^2-1740*a[3]^3+9*a[1]^2+68*a[1]*a[2]+218*a[1]*a[3]+
133*a[2]^2+724*a[2]*a[3]+1730*a[3]^2+8*a[1]+30*a[2]+118*a[3]+2)/(1+2*a[1]+8*a[2
]+30*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+512*a[1]^4*a[2]*a[3]^3+1024*a[1]^4*a[3]^
4+1024*a[1]^3*a[2]^3*a[3]^2+12032*a[1]^3*a[2]^2*a[3]^3+47104*a[1]^3*a[2]*a[3]^4
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553344*a[1]^2*a[2]^2*a[3]^4+1428480*a[1]^2*a[2]*a[3]^5+1382400*a[1]^2*a[3]^6+
16384*a[1]*a[2]^5*a[3]^2+315392*a[1]*a[2]^4*a[3]^3+2427904*a[1]*a[2]^3*a[3]^4+
9342720*a[1]*a[2]^2*a[3]^5+17971200*a[1]*a[2]*a[3]^6+13824000*a[1]*a[3]^7+16384
*a[2]^6*a[3]^2+376832*a[2]^5*a[3]^3+3610624*a[2]^4*a[3]^4+18447360*a[2]^3*a[3]^
5+53006400*a[2]^2*a[3]^6+81216000*a[2]*a[3]^7+51840000*a[3]^8-128*a[1]^4*a[2]^2
*a[3]+2048*a[1]^4*a[3]^3-2048*a[1]^3*a[2]^3*a[3]-9472*a[1]^3*a[2]^2*a[3]^2+
47232*a[1]^3*a[2]*a[3]^3+209408*a[1]^3*a[3]^4-12288*a[1]^2*a[2]^4*a[3]-113664*a
[1]^2*a[2]^3*a[3]^2+116736*a[1]^2*a[2]^2*a[3]^3+3163776*a[1]^2*a[2]*a[3]^4+
6658560*a[1]^2*a[3]^5-32768*a[1]*a[2]^5*a[3]-454656*a[1]*a[2]^4*a[3]^2-808960*a
[1]*a[2]^3*a[3]^3+12321024*a[1]*a[2]^2*a[3]^4+63031680*a[1]*a[2]*a[3]^5+
86054400*a[1]*a[3]^6-32768*a[2]^6*a[3]-606208*a[2]^5*a[3]^2-2605056*a[2]^4*a[3]
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a[2]^3+27584*a[2]^2*a[3]+124800*a[2]*a[3]^2+201600*a[3]^3+2660*a[1]^2+15904*a[1
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12984*a[2]+96730*a[3]+1419)^2/(4*a[1]*a[2]+16*a[1]*a[3]+16*a[2]^2+124*a[2]*a[3]
+240*a[3]^2+32*a[1]+70*a[2]+1148*a[3]+31)^2/(1+2*a[1]+8*a[2]+30*a[3])^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+26214400*a[1]^10*a[2]^3*a[3]^3+157286400*a
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2]^3*a[3]+2944*a[2]^2*a[3]^2+11280*a[2]*a[3]^3+14400*a[3]^4+160*a[1]^2*a[2]+528
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a[2]+96730*a[3]+1419)^2/(4*a[1]*a[2]+16*a[1]*a[3]+16*a[2]^2+124*a[2]*a[3]+240*a
[3]^2+32*a[1]+70*a[2]+1148*a[3]+31)^2/(1+2*a[1]+8*a[2]+30*a[3])^2


                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



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2]+8630075993293024*a[1]^5*a[3]+6407984261597456*a[1]^4*a[2]^2+
133144113196186048*a[1]^4*a[2]*a[3]+615403950031696336*a[1]^4*a[3]^2+
27505548144242752*a[1]^3*a[2]^3+870202745951013600*a[1]^3*a[2]^2*a[3]+
8258887255462828896*a[1]^3*a[2]*a[3]^2+23418007533834549760*a[1]^3*a[3]^3+
72017455487429248*a[1]^2*a[2]^4+3008757426265597600*a[1]^2*a[2]^3*a[3]+
43232068053727480448*a[1]^2*a[2]^2*a[3]^2+250362842111272261088*a[1]^2*a[2]*a[3
]^3+490248956741955162960*a[1]^2*a[3]^4+106388609594530176*a[1]*a[2]^5+
5435109734975180096*a[1]*a[2]^4*a[3]+103840575750944124928*a[1]*a[2]^3*a[3]^2+
912258894678565424672*a[1]*a[2]^2*a[3]^3+3648913373012860492320*a[1]*a[2]*a[3]^
4+5288112104838399963296*a[1]*a[3]^5+67223246840081664*a[2]^6+
4030952680013932032*a[2]^5*a[3]+95520405631333286816*a[2]^4*a[3]^2+
1124735932846104600928*a[2]^3*a[3]^3+6851134395226281077808*a[2]^2*a[3]^4+
20330270883458199026528*a[2]*a[3]^5+22841270154931818243376*a[3]^6+
99605199730752*a[1]^5+1581368492458384*a[1]^4*a[2]+14071975969874896*a[1]^4*a[3
]+10263935657782432*a[1]^3*a[2]^2+187598130719028192*a[1]^3*a[2]*a[3]+
780128080464168816*a[1]^3*a[3]^2+34041055615762560*a[1]^2*a[2]^3+
951011932327013744*a[1]^2*a[2]^2*a[3]+8115284344315365216*a[1]^2*a[2]*a[3]^2+
21003974722351085200*a[1]^2*a[3]^3+57525846498807072*a[1]*a[2]^4+
2169011229044962496*a[1]*a[2]^3*a[3]+28332816556712298320*a[1]*a[2]^2*a[3]^2+
150556500448533618560*a[1]*a[2]*a[3]^3+273055195250463120336*a[1]*a[3]^4+
39429346388314176*a[2]^5+1872359015582751504*a[2]^4*a[3]+33135232077002505200*a
[2]^3*a[3]^2+270049312009298732928*a[2]^2*a[3]^3+1006959337282124574000*a[2]*a[
3]^4+1367236579648910162976*a[3]^5+112491242821540*a[1]^4+1484260790400120*a[1]
^3*a[2]+12270378822057520*a[1]^3*a[3]+7371802024090780*a[1]^2*a[2]^2+
125288693478092256*a[1]^2*a[2]*a[3]+486758309616998056*a[1]^2*a[3]^2+
16329682860963120*a[1]*a[2]^3+426425032893055608*a[1]*a[2]^2*a[3]+
3408400482015578672*a[1]*a[2]*a[3]^2+8291246269306337920*a[1]*a[3]^3+
13601602699837452*a[2]^4+483560995787384032*a[2]^3*a[3]+5947138922224720000*a[2
]^2*a[3]^2+29795948279223899096*a[2]*a[3]^3+51051399720006228852*a[3]^4+
75123430402932*a[1]^3+756353755916784*a[1]^2*a[2]+5903181060587860*a[1]^2*a[3]+
2529051849035644*a[1]*a[2]^2+40648917669727296*a[1]*a[2]*a[3]+
149497567817880348*a[1]*a[3]^2+2808538703600876*a[2]^3+69576855527546580*a[2]^2
*a[3]+527471392683734904*a[2]*a[3]^2+1217634345563735964*a[3]^3+29503274021881*
a[1]^2+200521620039236*a[1]*a[2]+1487949171529852*a[1]*a[3]+338241676040286*a[2
]^2+5175898068928370*a[2]*a[3]+18117296585164398*a[3]^2+6318514279770*a[1]+
21724919505546*a[2]+153760756566978*a[3]+569923332511


We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 7, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                            1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
                x[n + 1] = -------------------------------------
                           1 + 2 x[n - 2] + 9 x[n - 1] + 29 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], -----------------------------]
                                            1 + 2 x[1] + 9 x[2] + 29 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], -----------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 9 x[2] + 29 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+9*a[2]+29*a
[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+9*a[2]+29*a[3]), (2*a[1]*a[2]+4
*a[1]*a[3]+9*a[2]^2+47*a[2]*a[3]+58*a[3]^2+39*a[1]+84*a[2]+1400*a[3]+38)/(4*a[1
]*a[2]+18*a[1]*a[3]+18*a[2]^2+139*a[2]*a[3]+261*a[3]^2+31*a[1]+69*a[2]+1111*a[3
]+30)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+9*a[2]+29*a[3]), (2*a[1]*a[2]+4*a[1]
*a[3]+9*a[2]^2+47*a[2]*a[3]+58*a[3]^2+39*a[1]+84*a[2]+1400*a[3]+38)/(4*a[1]*a[2
]+18*a[1]*a[3]+18*a[2]^2+139*a[2]*a[3]+261*a[3]^2+31*a[1]+69*a[2]+1111*a[3]+30)
, (8*a[1]^2*a[2]*a[3]+36*a[1]^2*a[3]^2+72*a[1]*a[2]^2*a[3]+556*a[1]*a[2]*a[3]^2
+1044*a[1]*a[3]^3+162*a[2]^3*a[3]+1773*a[2]^2*a[3]^2+6380*a[2]*a[3]^3+7569*a[3]
^4+164*a[1]^2*a[2]+430*a[1]^2*a[3]+1456*a[1]*a[2]^2+8579*a[1]*a[2]*a[3]+14621*a
[1]*a[3]^2+3231*a[2]^3+29608*a[2]^2*a[3]+99594*a[2]*a[3]^2+121597*a[3]^3+3010*a
[1]^2+19968*a[1]*a[2]+153377*a[1]*a[3]+29256*a[2]^2+580377*a[2]*a[3]+1621543*a[
3]^2+4468*a[1]+16359*a[2]+99027*a[3]+1496)/(16*a[1]^2*a[2]*a[3]+72*a[1]^2*a[3]^
2+144*a[1]*a[2]^2*a[3]+1112*a[1]*a[2]*a[3]^2+2088*a[1]*a[3]^3+324*a[2]^3*a[3]+
3546*a[2]^2*a[3]^2+12760*a[2]*a[3]^3+15138*a[3]^4+160*a[1]^2*a[2]+554*a[1]^2*a[
3]+1350*a[1]*a[2]^2+9757*a[1]*a[2]*a[3]+22393*a[1]*a[3]^2+2835*a[2]^3+31383*a[2
]^2*a[3]+136308*a[2]*a[3]^2+208220*a[3]^3+2603*a[1]^2+16745*a[1]*a[2]+137920*a[
1]*a[3]+24228*a[2]^2+494450*a[2]*a[3]+1587836*a[3]^2+3975*a[1]+13854*a[2]+94588
*a[3]+1402)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
36*a[1]*a[2]^3+116*a[1]*a[2]^2*a[3]+36*a[1]*a[2]*a[3]^2+116*a[1]*a[3]^3+81*a[2]
^4+522*a[2]^3*a[3]+922*a[2]^2*a[3]^2+522*a[2]*a[3]^3+841*a[3]^4-8*a[1]^2*a[2]-8
*a[1]^2*a[3]-68*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-228*a[1]*a[3]^2-144*a[2]^3-1148*
a[2]^2*a[3]-2708*a[2]*a[3]^2-1624*a[3]^3+9*a[1]^2+78*a[1]*a[2]+208*a[1]*a[3]+
176*a[2]^2+780*a[2]*a[3]+1631*a[3]^2+8*a[1]+34*a[2]+114*a[3]+2)/(1+2*a[1]+9*a[2
]+29*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+576*a[1]^4*a[2]*a[3]^3+1296*a[1]^4*a[3]^
4+1152*a[1]^3*a[2]^3*a[3]^2+14080*a[1]^3*a[2]^2*a[3]^3+56736*a[1]^3*a[2]*a[3]^4
+75168*a[1]^3*a[3]^5+7776*a[1]^2*a[2]^4*a[3]^2+120096*a[1]^2*a[2]^3*a[3]^3+
689208*a[1]^2*a[2]^2*a[3]^4+1741392*a[1]^2*a[2]*a[3]^5+1634904*a[1]^2*a[3]^6+
23328*a[1]*a[2]^5*a[3]^2+435456*a[1]*a[2]^4*a[3]^3+3228552*a[1]*a[2]^3*a[3]^4+
11886520*a[1]*a[2]^2*a[3]^5+21738168*a[1]*a[2]*a[3]^6+15804072*a[1]*a[3]^7+
26244*a[2]^6*a[3]^2+574452*a[2]^5*a[3]^3+5210649*a[2]^4*a[3]^4+25075836*a[2]^3*
a[3]^5+67544074*a[2]^2*a[3]^6+96580440*a[2]*a[3]^7+57289761*a[3]^8-128*a[1]^4*a
[2]^2*a[3]-160*a[1]^4*a[2]*a[3]^2+1872*a[1]^4*a[3]^3-2304*a[1]^3*a[2]^3*a[3]-\
12496*a[1]^3*a[2]^2*a[3]^2+36008*a[1]^3*a[2]*a[3]^3+205128*a[1]^3*a[3]^4-15552*
a[1]^2*a[2]^4*a[3]-149256*a[1]^2*a[2]^3*a[3]^2+1836*a[1]^2*a[2]^2*a[3]^3+
3071640*a[1]^2*a[2]*a[3]^4+6561540*a[1]^2*a[3]^5-46656*a[1]*a[2]^5*a[3]-642492*
a[1]*a[2]^4*a[3]^2-1488618*a[1]*a[2]^3*a[3]^3+12239734*a[1]*a[2]^2*a[3]^4+
64414394*a[1]*a[2]*a[3]^5+83728278*a[1]*a[3]^6-52488*a[2]^6*a[3]-941868*a[2]^5*
a[3]^2-4222368*a[2]^4*a[3]^3+11570382*a[2]^3*a[3]^4+144652754*a[2]^2*a[3]^5+
405044102*a[2]*a[3]^6+377102718*a[3]^7+96*a[1]^4*a[2]^2-1040*a[1]^4*a[2]*a[3]-\
2680*a[1]^4*a[3]^2+1808*a[1]^3*a[2]^3-8768*a[1]^3*a[2]^2*a[3]-152136*a[1]^3*a[2
]*a[3]^2-190288*a[1]^3*a[3]^3+12844*a[1]^2*a[2]^4+10572*a[1]^2*a[2]^3*a[3]-\
1404842*a[1]^2*a[2]^2*a[3]^2-6101880*a[1]^2*a[2]*a[3]^3-2886774*a[1]^2*a[3]^4+
40752*a[1]*a[2]^5+251624*a[1]*a[2]^4*a[3]-4353816*a[1]*a[2]^3*a[3]^2-39085630*a
[1]*a[2]^2*a[3]^3-75587612*a[1]*a[2]*a[3]^4+3626450*a[1]*a[3]^5+48681*a[2]^6+
545706*a[2]^5*a[3]-3887522*a[2]^4*a[3]^2-69607594*a[2]^3*a[3]^3-261646471*a[2]^
2*a[3]^4-230932452*a[2]*a[3]^5+197880572*a[3]^6+1112*a[1]^4*a[2]-3004*a[1]^4*a[
3]+18852*a[1]^3*a[2]^2+50062*a[1]^3*a[2]*a[3]-457066*a[1]^3*a[3]^2+118124*a[1]^
2*a[2]^3+973132*a[1]^2*a[2]^2*a[3]-2357554*a[1]^2*a[2]*a[3]^2-19963050*a[1]^2*a
[3]^3+319656*a[1]*a[2]^4+4665796*a[1]*a[2]^3*a[3]+4892732*a[1]*a[2]^2*a[3]^2-\
122872608*a[1]*a[2]*a[3]^3-313436368*a[1]*a[3]^4+308340*a[2]^5+7032774*a[2]^4*a
[3]+29010212*a[2]^3*a[3]^2-162004142*a[2]^2*a[3]^3-1120277940*a[2]*a[3]^4-\
1608226900*a[3]^5+5061*a[1]^4+74640*a[1]^3*a[2]+451934*a[1]^3*a[3]+402795*a[1]^
2*a[2]^2+5255204*a[1]^2*a[2]*a[3]+13606531*a[1]^2*a[3]^2+934542*a[1]*a[2]^3+
20132940*a[1]*a[2]^2*a[3]+108586378*a[1]*a[2]*a[3]^2+169519340*a[1]*a[3]^3+
806148*a[2]^4+24206706*a[2]^3*a[3]+230959929*a[2]^2*a[3]^2+592824500*a[2]*a[3]^
3+973276353*a[3]^4+13912*a[1]^3+156838*a[1]^2*a[2]+886432*a[1]^2*a[3]+578028*a[
1]*a[2]^2+6901794*a[1]*a[2]*a[3]+17579878*a[1]*a[3]^2+666450*a[2]^3+13990602*a[
2]^2*a[3]+63594122*a[2]*a[3]^2+130205962*a[3]^3+13733*a[1]^2+105334*a[1]*a[2]+
561306*a[1]*a[3]+206946*a[2]^2+2067736*a[2]*a[3]+6096230*a[3]^2+5844*a[1]+21732
*a[2]+125396*a[3]+964)/(4*a[1]*a[2]+18*a[1]*a[3]+18*a[2]^2+139*a[2]*a[3]+261*a[
3]^2+31*a[1]+69*a[2]+1111*a[3]+30)^2/(1+2*a[1]+9*a[2]+29*a[3])^2, (12288*a[1]^8
*a[2]^4*a[3]^2+241664*a[1]^8*a[2]^3*a[3]^3+1795072*a[1]^8*a[2]^2*a[3]^4+5953536
*a[1]^8*a[2]*a[3]^5+7423488*a[1]^8*a[3]^6+462848*a[1]^7*a[2]^5*a[3]^2+10309632*
a[1]^7*a[2]^4*a[3]^3+91826176*a[1]^7*a[2]^3*a[3]^4+407625728*a[1]^7*a[2]^2*a[3]
^5+899069184*a[1]^7*a[2]*a[3]^6+785541888*a[1]^7*a[3]^7+7638016*a[1]^6*a[2]^6*a
[3]^2+189808640*a[1]^6*a[2]^5*a[3]^3+1962656256*a[1]^6*a[2]^4*a[3]^4+
10793712128*a[1]^6*a[2]^3*a[3]^5+33258241984*a[1]^6*a[2]^2*a[3]^6+54393070464*a
[1]^6*a[2]*a[3]^7+36880733376*a[1]^6*a[3]^8+72105984*a[1]^5*a[2]^7*a[3]^2+
1976100864*a[1]^5*a[2]^6*a[3]^3+23193314304*a[1]^5*a[2]^5*a[3]^4+151109339136*a
[1]^5*a[2]^4*a[3]^5+590309026048*a[1]^5*a[2]^3*a[3]^6+1383321583232*a[1]^5*a[2]
^2*a[3]^7+1801945693440*a[1]^5*a[2]*a[3]^8+1007620124544*a[1]^5*a[3]^9+
425813760*a[1]^4*a[2]^8*a[3]^2+12752363520*a[1]^4*a[2]^7*a[3]^3+167115160320*a[
1]^4*a[2]^6*a[3]^4+1252734102400*a[1]^4*a[2]^5*a[3]^5+5882025570960*a[1]^4*a[2]
^4*a[3]^6+17738091108800*a[1]^4*a[2]^3*a[3]^7+33600755032480*a[1]^4*a[2]^2*a[3]
^8+36608934317760*a[1]^4*a[2]*a[3]^9+17587735914960*a[1]^4*a[3]^10+1610378496*a
[1]^3*a[2]^9*a[3]^2+52315123968*a[1]^3*a[2]^8*a[3]^3+755956381824*a[1]^3*a[2]^7
*a[3]^4+6387566147072*a[1]^3*a[2]^6*a[3]^5+34847216903056*a[1]^3*a[2]^5*a[3]^6+
127556744392176*a[1]^3*a[2]^4*a[3]^7+313956898048288*a[1]^3*a[2]^3*a[3]^8+
502012326309344*a[1]^3*a[2]^2*a[3]^9+473888789322192*a[1]^3*a[2]*a[3]^10+
201367867861296*a[1]^3*a[3]^11+3808109376*a[1]^2*a[2]^10*a[3]^2+133382225664*a[
1]^2*a[2]^9*a[3]^3+2104634407392*a[1]^2*a[2]^8*a[3]^4+19743523017696*a[1]^2*a[2
]^7*a[3]^5+122251772582628*a[1]^2*a[2]^6*a[3]^6+523537525338072*a[1]^2*a[2]^5*a
[3]^7+1574714681763564*a[1]^2*a[2]^4*a[3]^8+3292912281330032*a[1]^2*a[2]^3*a[3]
^9+4589412706816636*a[1]^2*a[2]^2*a[3]^10+3852370933500888*a[1]^2*a[2]*a[3]^11+
1478293305732756*a[1]^2*a[3]^12+5147183232*a[1]*a[2]^11*a[3]^2+193390986240*a[1
]*a[2]^10*a[3]^3+3306299461632*a[1]*a[2]^9*a[3]^4+34037808875136*a[1]*a[2]^8*a[
3]^5+235162260143928*a[1]*a[2]^7*a[3]^6+1148719023527280*a[1]*a[2]^6*a[3]^7+
4062138368145272*a[1]*a[2]^5*a[3]^8+10430259571956912*a[1]*a[2]^4*a[3]^9+
19097393604829096*a[1]*a[2]^3*a[3]^10+23765564712723728*a[1]*a[2]^2*a[3]^11+
18078759740910888*a[1]*a[2]*a[3]^12+6356209152604752*a[1]*a[3]^13+3044094048*a[
2]^12*a[3]^2+122157736848*a[2]^11*a[3]^3+2248536240108*a[2]^10*a[3]^4+
25172894331792*a[2]^9*a[3]^5+191551036601241*a[2]^8*a[3]^6+1047786437482272*a[2
]^7*a[3]^7+4241648640004528*a[2]^6*a[3]^8+12850718986288616*a[2]^5*a[3]^9+
28993652113357002*a[2]^4*a[3]^10+47557904937875456*a[2]^3*a[3]^11+
53792774725054684*a[2]^2*a[3]^12+37582879416367320*a[2]*a[3]^13+
12223103534730693*a[3]^14+159744*a[1]^8*a[2]^4*a[3]+3354624*a[1]^8*a[2]^3*a[3]^
2+25883648*a[1]^8*a[2]^2*a[3]^3+86854912*a[1]^8*a[2]*a[3]^4+106888320*a[1]^8*a[
3]^5+5996544*a[1]^7*a[2]^5*a[3]+140555264*a[1]^7*a[2]^4*a[3]^2+1306568960*a[1]^
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a[2]*a[3]+339252241377510*a[1]^3*a[3]^2+81677385886382*a[1]^2*a[2]^3+
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1599789696*a[1]+5825874432*a[2]+35160984832*a[3]+133751056)/(16*a[1]^2*a[2]*a[3
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324*a[2]^3*a[3]+3546*a[2]^2*a[3]^2+12760*a[2]*a[3]^3+15138*a[3]^4+160*a[1]^2*a[
2]+554*a[1]^2*a[3]+1350*a[1]*a[2]^2+9757*a[1]*a[2]*a[3]+22393*a[1]*a[3]^2+2835*
a[2]^3+31383*a[2]^2*a[3]+136308*a[2]*a[3]^2+208220*a[3]^3+2603*a[1]^2+16745*a[1
]*a[2]+137920*a[1]*a[3]+24228*a[2]^2+494450*a[2]*a[3]+1587836*a[3]^2+3975*a[1]+
13854*a[2]+94588*a[3]+1402)^2/(4*a[1]*a[2]+18*a[1]*a[3]+18*a[2]^2+139*a[2]*a[3]
+261*a[3]^2+31*a[1]+69*a[2]+1111*a[3]+30)^2/(1+2*a[1]+9*a[2]+29*a[3])^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+29491200*a[1]^10*a[2]^3*a[3]^3+199065600*a
[1]^10*a[2]^2*a[3]^4+597196800*a[1]^10*a[2]*a[3]^5+671846400*a[1]^10*a[3]^6+
58982400*a[1]^9*a[2]^5*a[3]^2+1251737600*a[1]^9*a[2]^4*a[3]^3+10587340800*a[1]^
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18310075609556786908*a[1]^2*a[3]^3+69684645757981146*a[1]*a[2]^4+
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144505109599879243404*a[1]*a[2]*a[3]^3+235200608958214698218*a[1]*a[3]^4+
51213329353567134*a[2]^5+2294997193013332542*a[2]^4*a[3]+37882565043849769256*a
[2]^3*a[3]^2+283920527727809857720*a[2]^2*a[3]^3+958902368325170338778*a[2]*a[3
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3*a[2]+10899928571152620*a[1]^3*a[3]+7782345193945169*a[1]^2*a[2]^2+
121759214289917798*a[1]^2*a[2]*a[3]+429083030395933057*a[1]^2*a[3]^2+
18624815387182236*a[1]*a[2]^3+451294963628941204*a[1]*a[2]^2*a[3]+
3300052279850420092*a[1]*a[2]*a[3]^2+7232584319701147156*a[1]*a[3]^3+
16698350855240883*a[2]^4+554972352542167620*a[2]^3*a[3]+6294802805229485210*a[2
]^2*a[3]^2+28664366364828311268*a[2]*a[3]^3+43926407213215356251*a[3]^4+
67399777528072*a[1]^3+740957138451448*a[1]^2*a[2]+5261239255907048*a[1]^2*a[3]+
2693398236108684*a[1]*a[2]^2+39710889837138744*a[1]*a[2]*a[3]+
132031929399394780*a[1]*a[3]^2+3238123167861984*a[2]^3+74172872566405608*a[2]^2
*a[3]+512666264400762656*a[2]*a[3]^2+1062518977860313912*a[3]^3+26578484576328*
a[1]^2+197649764203184*a[1]*a[2]+1329983873648624*a[1]*a[3]+363210393636900*a[2
]^2+5081663168135384*a[2]*a[3]+16025324504644724*a[3]^2+5713664872704*a[1]+
21540479890368*a[2]+137795122283968*a[3]+517204223344)/(16*a[1]^2*a[2]*a[3]+72*
a[1]^2*a[3]^2+144*a[1]*a[2]^2*a[3]+1112*a[1]*a[2]*a[3]^2+2088*a[1]*a[3]^3+324*a
[2]^3*a[3]+3546*a[2]^2*a[3]^2+12760*a[2]*a[3]^3+15138*a[3]^4+160*a[1]^2*a[2]+
554*a[1]^2*a[3]+1350*a[1]*a[2]^2+9757*a[1]*a[2]*a[3]+22393*a[1]*a[3]^2+2835*a[2
]^3+31383*a[2]^2*a[3]+136308*a[2]*a[3]^2+208220*a[3]^3+2603*a[1]^2+16745*a[1]*a
[2]+137920*a[1]*a[3]+24228*a[2]^2+494450*a[2]*a[3]+1587836*a[3]^2+3975*a[1]+
13854*a[2]+94588*a[3]+1402)^2/(4*a[1]*a[2]+18*a[1]*a[3]+18*a[2]^2+139*a[2]*a[3]
+261*a[3]^2+31*a[1]+69*a[2]+1111*a[3]+30)^2/(1+2*a[1]+9*a[2]+29*a[3])^2


                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



1638400*a[1]^10*a[2]^4*a[3]^2+29491200*a[1]^10*a[2]^3*a[3]^3+199065600*a[1]^10*
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a[2]^5*a[3]^2+4491750912000*a[1]^8*a[2]^4*a[3]^3+24720501964800*a[1]^8*a[2]^3*a
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We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 8, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                           1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
               x[n + 1] = --------------------------------------
                          1 + 2 x[n - 2] + 10 x[n - 1] + 28 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], ------------------------------]
                                            1 + 2 x[1] + 10 x[2] + 28 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], ------------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 10 x[2] + 28 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+10*a[2]+28*
a[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+10*a[2]+28*a[3]), (2*a[1]*a[2]
+4*a[1]*a[3]+10*a[2]^2+48*a[2]*a[3]+56*a[3]^2+39*a[1]+85*a[2]+1399*a[3]+38)/(4*
a[1]*a[2]+20*a[1]*a[3]+20*a[2]^2+156*a[2]*a[3]+280*a[3]^2+30*a[1]+68*a[2]+1074*
a[3]+29)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+10*a[2]+28*a[3]), (2*a[1]*a[2]+4*
a[1]*a[3]+10*a[2]^2+48*a[2]*a[3]+56*a[3]^2+39*a[1]+85*a[2]+1399*a[3]+38)/(4*a[1
]*a[2]+20*a[1]*a[3]+20*a[2]^2+156*a[2]*a[3]+280*a[3]^2+30*a[1]+68*a[2]+1074*a[3
]+29), (8*a[1]^2*a[2]*a[3]+40*a[1]^2*a[3]^2+80*a[1]*a[2]^2*a[3]+624*a[1]*a[2]*a
[3]^2+1120*a[1]*a[3]^3+200*a[2]^3*a[3]+2120*a[2]^2*a[3]^2+7168*a[2]*a[3]^3+7840
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3+3006*a[1]^2+21498*a[1]*a[2]+151582*a[1]*a[3]+32832*a[2]^2+630264*a[2]*a[3]+
1563710*a[3]^2+4461*a[1]+17815*a[2]+97340*a[3]+1493)/(16*a[1]^2*a[2]*a[3]+80*a[
1]^2*a[3]^2+160*a[1]*a[2]^2*a[3]+1248*a[1]*a[2]*a[3]^2+2240*a[1]*a[3]^3+400*a[2
]^3*a[3]+4240*a[2]^2*a[3]^2+14336*a[2]*a[3]^3+15680*a[3]^4+160*a[1]^2*a[2]+584*
a[1]^2*a[3]+1480*a[1]*a[2]^2+10320*a[1]*a[2]*a[3]+23608*a[1]*a[3]^2+3400*a[2]^3
+35320*a[2]^2*a[3]+149400*a[2]*a[3]^2+216048*a[3]^3+2544*a[1]^2+17496*a[1]*a[2]
+134256*a[1]*a[3]+26340*a[2]^2+521420*a[2]*a[3]+1532688*a[3]^2+3898*a[1]+14638*
a[2]+92378*a[3]+1383)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
40*a[1]*a[2]^3+112*a[1]*a[2]^2*a[3]+40*a[1]*a[2]*a[3]^2+112*a[1]*a[3]^3+100*a[2
]^4+560*a[2]^3*a[3]+884*a[2]^2*a[3]^2+560*a[2]*a[3]^3+784*a[3]^4-8*a[1]^2*a[2]-\
8*a[1]^2*a[3]-76*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-220*a[1]*a[3]^2-180*a[2]^3-1264
*a[2]^2*a[3]-2668*a[2]*a[3]^2-1512*a[3]^3+9*a[1]^2+88*a[1]*a[2]+198*a[1]*a[3]+
225*a[2]^2+824*a[2]*a[3]+1538*a[3]^2+8*a[1]+38*a[2]+110*a[3]+2)/(1+2*a[1]+10*a[
2]+28*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+640*a[1]^4*a[2]*a[3]^3+1600*a[1]^4*a[3]
^4+1280*a[1]^3*a[2]^3*a[3]^2+16384*a[1]^3*a[2]^2*a[3]^3+67840*a[1]^3*a[2]*a[3]^
4+89600*a[1]^3*a[3]^5+9600*a[1]^2*a[2]^4*a[3]^2+149760*a[1]^2*a[2]^3*a[3]^3+
852864*a[1]^2*a[2]^2*a[3]^4+2096640*a[1]^2*a[2]*a[3]^5+1881600*a[1]^2*a[3]^6+
32000*a[1]*a[2]^5*a[3]^2+588800*a[1]*a[2]^4*a[3]^3+4240640*a[1]*a[2]^3*a[3]^4+
14948864*a[1]*a[2]^2*a[3]^5+25840640*a[1]*a[2]*a[3]^6+17561600*a[1]*a[3]^7+
40000*a[2]^6*a[3]^2+848000*a[2]^5*a[3]^3+7361600*a[2]^4*a[3]^4+33528320*a[2]^3*
a[3]^5+84621824*a[2]^2*a[3]^6+112394240*a[2]*a[3]^7+61465600*a[3]^8-128*a[1]^4*
a[2]^2*a[3]-320*a[1]^4*a[2]*a[3]^2+1600*a[1]^4*a[3]^3-2560*a[1]^3*a[2]^3*a[3]-\
16128*a[1]^3*a[2]^2*a[3]^2+22528*a[1]^3*a[2]*a[3]^3+195840*a[1]^3*a[3]^4-19200*
a[1]^2*a[2]^4*a[3]-193920*a[1]^2*a[2]^3*a[3]^2-160128*a[1]^2*a[2]^2*a[3]^3+
2916096*a[1]^2*a[2]*a[3]^4+6343680*a[1]^2*a[3]^5-64000*a[1]*a[2]^5*a[3]-889600*
a[1]*a[2]^4*a[3]^2-2490880*a[1]*a[2]^3*a[3]^3+11562752*a[1]*a[2]^2*a[3]^4+
64891904*a[1]*a[2]*a[3]^5+80030720*a[1]*a[3]^6-80000*a[2]^6*a[3]-1416000*a[2]^5
*a[3]^2-6635200*a[2]^4*a[3]^3+9391360*a[2]^3*a[3]^4+153925632*a[2]^2*a[3]^5+
411041792*a[2]*a[3]^6+352988160*a[3]^7+96*a[1]^4*a[2]^2-864*a[1]^4*a[2]*a[3]-\
2976*a[1]^4*a[3]^2+2016*a[1]^3*a[2]^3-6560*a[1]^3*a[2]^2*a[3]-147552*a[1]^3*a[2
]*a[3]^2-231136*a[1]^3*a[3]^3+15984*a[1]^2*a[2]^4+34464*a[1]^2*a[2]^3*a[3]-\
1415264*a[1]^2*a[2]^2*a[3]^2-6539936*a[1]^2*a[2]*a[3]^3-4180432*a[1]^2*a[3]^4+
56640*a[1]*a[2]^5+408672*a[1]*a[2]^4*a[3]-4539296*a[1]*a[2]^3*a[3]^2-43468576*a
[1]*a[2]^2*a[3]^3-85565856*a[1]*a[2]*a[3]^4-13808704*a[1]*a[3]^5+75600*a[2]^6+
901760*a[2]^5*a[3]-3870656*a[2]^4*a[3]^2-82623584*a[2]^3*a[3]^3-302169296*a[2]^
2*a[3]^4-287785792*a[2]*a[3]^5+106131648*a[3]^6+1056*a[1]^4*a[2]-2352*a[1]^4*a[
3]+19808*a[1]^3*a[2]^2+54896*a[1]^3*a[2]*a[3]-401200*a[1]^3*a[3]^2+136944*a[1]^
2*a[2]^3+1092128*a[1]^2*a[2]^2*a[3]-2145440*a[1]^2*a[2]*a[3]^2-17970288*a[1]^2*
a[3]^3+405936*a[1]*a[2]^4+5686448*a[1]*a[2]^3*a[3]+6697200*a[1]*a[2]^2*a[3]^2-\
122623456*a[1]*a[2]*a[3]^3-277502352*a[1]*a[3]^4+422080*a[2]^5+9378384*a[2]^4*a
[3]+38137248*a[2]^3*a[3]^2-178111856*a[2]^2*a[3]^3-1108276496*a[2]*a[3]^4-\
1373394848*a[3]^5+4824*a[1]^4+77096*a[1]^3*a[2]+426448*a[1]^3*a[3]+447412*a[1]^
2*a[2]^2+5419088*a[1]^2*a[2]*a[3]+12614468*a[1]^2*a[3]^2+1102072*a[1]*a[2]^3+
22620520*a[1]*a[2]^2*a[3]+110444128*a[1]*a[2]*a[3]^2+153816648*a[1]*a[3]^3+
1007576*a[2]^4+29012312*a[2]^3*a[3]+262002680*a[2]^2*a[3]^2+571831200*a[2]*a[3]
^3+892541352*a[3]^4+13272*a[1]^3+163016*a[1]^2*a[2]+834612*a[1]^2*a[3]+651372*a
[1]*a[2]^2+7118452*a[1]*a[2]*a[3]+16250696*a[1]*a[3]^2+797656*a[2]^3+15943796*a
[2]^2*a[3]+63593888*a[2]*a[3]^2+119592688*a[3]^3+13118*a[1]^2+109958*a[1]*a[2]+
528160*a[1]*a[3]+236917*a[2]^2+2118746*a[2]*a[3]+5660295*a[3]^2+5590*a[1]+22690
*a[2]+118092*a[3]+922)/(4*a[1]*a[2]+20*a[1]*a[3]+20*a[2]^2+156*a[2]*a[3]+280*a[
3]^2+30*a[1]+68*a[2]+1074*a[3]+29)^2/(1+2*a[1]+10*a[2]+28*a[3])^2, (12288*a[1]^
8*a[2]^4*a[3]^2+270336*a[1]^8*a[2]^3*a[3]^3+2248704*a[1]^8*a[2]^2*a[3]^4+
8355840*a[1]^8*a[2]*a[3]^5+11673600*a[1]^8*a[3]^6+516096*a[1]^7*a[2]^5*a[3]^2+
12492800*a[1]^7*a[2]^4*a[3]^3+120143872*a[1]^7*a[2]^3*a[3]^4+570114048*a[1]^7*a
[2]^2*a[3]^5+1323950080*a[1]^7*a[2]*a[3]^6+1189683200*a[1]^7*a[3]^7+9498624*a[1
]^6*a[2]^6*a[3]^2+250593280*a[1]^6*a[2]^5*a[3]^3+2724474880*a[1]^6*a[2]^4*a[3]^
4+15557459968*a[1]^6*a[2]^3*a[3]^5+49007562752*a[1]^6*a[2]^2*a[3]^6+80498769920
*a[1]^6*a[2]*a[3]^7+53878476800*a[1]^6*a[3]^8+100024320*a[1]^5*a[2]^7*a[3]^2+
2854526976*a[1]^5*a[2]^6*a[3]^3+34515247104*a[1]^5*a[2]^5*a[3]^4+228737482752*a
[1]^5*a[2]^4*a[3]^5+896138379264*a[1]^5*a[2]^3*a[3]^6+2075670061056*a[1]^5*a[2]
^2*a[3]^7+2637137141760*a[1]^5*a[2]*a[3]^8+1422868070400*a[1]^5*a[3]^9+
658944000*a[1]^4*a[2]^8*a[3]^2+20219494400*a[1]^4*a[2]^7*a[3]^3+268560117760*a[
1]^4*a[2]^6*a[3]^4+2015936593920*a[1]^4*a[2]^5*a[3]^5+9358729850880*a[1]^4*a[2]
^4*a[3]^6+27560997765120*a[1]^4*a[2]^3*a[3]^7+50429228052480*a[1]^4*a[2]^2*a[3]
^8+52615525580800*a[1]^4*a[2]*a[3]^9+24063506432000*a[1]^4*a[3]^10+2780160000*a
[1]^3*a[2]^9*a[3]^2+91270144000*a[1]^3*a[2]^8*a[3]^3+1318953779200*a[1]^3*a[2]^
7*a[3]^4+11021061406720*a[1]^3*a[2]^6*a[3]^5+58782866104320*a[1]^3*a[2]^5*a[3]^
6+208104639455232*a[1]^3*a[2]^4*a[3]^7+490765028573184*a[1]^3*a[2]^3*a[3]^8+
746432741605376*a[1]^3*a[2]^2*a[3]^9+666985200680960*a[1]^3*a[2]*a[3]^10+
267572471398400*a[1]^3*a[3]^11+7334400000*a[1]^2*a[2]^10*a[3]^2+256542720000*a[
1]^2*a[2]^9*a[3]^3+4003029504000*a[1]^2*a[2]^8*a[3]^4+36754152652800*a[1]^2*a[2
]^7*a[3]^5+220458853724160*a[1]^2*a[2]^6*a[3]^6+905781195915264*a[1]^2*a[2]^5*a
[3]^7+2592336513085440*a[1]^2*a[2]^4*a[3]^8+5125551507406848*a[1]^2*a[2]^3*a[3]
^9+6726625309458432*a[1]^2*a[2]^2*a[3]^10+5305410274590720*a[1]^2*a[2]*a[3]^11+
1911460179148800*a[1]^2*a[3]^12+11059200000*a[1]*a[2]^11*a[3]^2+410700800000*a[
1]*a[2]^10*a[3]^3+6877281280000*a[1]*a[2]^9*a[3]^4+68694134784000*a[1]*a[2]^8*a
[3]^5+456186831257600*a[1]*a[2]^7*a[3]^6+2123343235833856*a[1]*a[2]^6*a[3]^7+
7101620344872960*a[1]*a[2]^5*a[3]^8+17149053047848960*a[1]*a[2]^4*a[3]^9+
29423492340580352*a[1]*a[2]^3*a[3]^10+34251709359587328*a[1]*a[2]^2*a[3]^11+
24361357333626880*a[1]*a[2]*a[3]^12+8007750169395200*a[1]*a[3]^13+7296000000*a[
2]^12*a[3]^2+286796800000*a[2]^11*a[3]^3+5127884800000*a[2]^10*a[3]^4+
55284643840000*a[2]^9*a[3]^5+401677786624000*a[2]^8*a[3]^6+2081262270791680*a[2
]^7*a[3]^7+7926529872891904*a[2]^6*a[3]^8+22476215450157056*a[2]^5*a[3]^9+
47308202379296768*a[2]^4*a[3]^10+72286845989421056*a[2]^3*a[3]^11+
76147662111309824*a[2]^2*a[3]^12+49555968697303040*a[2]*a[3]^13+
15011075725721600*a[3]^14+159744*a[1]^8*a[2]^4*a[3]+3723264*a[1]^8*a[2]^3*a[3]^
2+31850496*a[1]^8*a[2]^2*a[3]^3+118136832*a[1]^8*a[2]*a[3]^4+159989760*a[1]^8*a
[3]^5+6660096*a[1]^7*a[2]^5*a[3]+168787968*a[1]^7*a[2]^4*a[3]^2+1679507456*a[1]
^7*a[2]^3*a[3]^3+8177442816*a[1]^7*a[2]^2*a[3]^4+19499753472*a[1]^7*a[2]*a[3]^5
+18321448960*a[1]^7*a[3]^6+121651200*a[1]^6*a[2]^6*a[3]+3327074304*a[1]^6*a[2]^
5*a[3]^2+37522423808*a[1]^6*a[2]^4*a[3]^3+222829993984*a[1]^6*a[2]^3*a[3]^4+
734961745920*a[1]^6*a[2]^2*a[3]^5+1277550346240*a[1]^6*a[2]*a[3]^6+912249651200
*a[1]^6*a[3]^7+1270640640*a[1]^5*a[2]^7*a[3]+37291032576*a[1]^5*a[2]^6*a[3]^2+
468036808704*a[1]^5*a[2]^5*a[3]^3+3248664346624*a[1]^5*a[2]^4*a[3]^4+
13451499753472*a[1]^5*a[2]^3*a[3]^5+33194278084608*a[1]^5*a[2]^2*a[3]^6+
45152007213056*a[1]^5*a[2]*a[3]^7+26136779694080*a[1]^5*a[3]^8+8295936000*a[1]^
4*a[2]^8*a[3]+260138803200*a[1]^4*a[2]^7*a[3]^2+3585388113920*a[1]^4*a[2]^6*a[3
]^3+28293998632960*a[1]^4*a[2]^5*a[3]^4+139520806400000*a[1]^4*a[2]^4*a[3]^5+
439481995632640*a[1]^4*a[2]^3*a[3]^6+863119070842880*a[1]^4*a[2]^2*a[3]^7+
968070353551360*a[1]^4*a[2]*a[3]^8+476920898764800*a[1]^4*a[3]^9+34652160000*a[
1]^3*a[2]^9*a[3]+1157038080000*a[1]^3*a[2]^8*a[3]^2+17338147020800*a[1]^3*a[2]^
7*a[3]^3+152598884802560*a[1]^3*a[2]^6*a[3]^4+866905459015680*a[1]^3*a[2]^5*a[3
]^5+3290145312301056*a[1]^3*a[2]^4*a[3]^6+8341390601527296*a[1]^3*a[2]^3*a[3]^7
+13656355457777664*a[1]^3*a[2]^2*a[3]^8+13165442252505088*a[1]^3*a[2]*a[3]^9+
5728050204835840*a[1]^3*a[3]^10+90393600000*a[1]^2*a[2]^10*a[3]+3205002240000*a
[1]^2*a[2]^9*a[3]^2+51820360704000*a[1]^2*a[2]^8*a[3]^3+501627364352000*a[1]^2*
a[2]^7*a[3]^4+3209421060055040*a[1]^2*a[2]^6*a[3]^5+14153403181891584*a[1]^2*a[
2]^5*a[3]^6+43582511623053312*a[1]^2*a[2]^4*a[3]^7+92814862111309824*a[1]^2*a[2
]^3*a[3]^8+131519116419661824*a[1]^2*a[2]^2*a[3]^9+112623014050398208*a[1]^2*a[
2]*a[3]^10+44435479317053440*a[1]^2*a[3]^11+134592000000*a[1]*a[2]^11*a[3]+
5055590400000*a[1]*a[2]^10*a[3]^2+87677488640000*a[1]*a[2]^9*a[3]^3+
923770615808000*a[1]*a[2]^8*a[3]^4+6548402232217600*a[1]*a[2]^7*a[3]^5+
32733107380756480*a[1]*a[2]^6*a[3]^6+117816018165927936*a[1]*a[2]^5*a[3]^7+
306447470960320512*a[1]*a[2]^4*a[3]^8+567747164413198336*a[1]*a[2]^3*a[3]^9+
717769962486169600*a[1]*a[2]^2*a[3]^10+559347497521971200*a[1]*a[2]*a[3]^11+
203466163109888000*a[1]*a[3]^12+87552000000*a[2]^12*a[3]+3477004800000*a[2]^11*
a[3]^2+64376307200000*a[2]^10*a[3]^3+732353372160000*a[2]^9*a[3]^4+
5682244726272000*a[2]^8*a[3]^5+31623834782965760*a[2]^7*a[3]^6+
129589889847988224*a[2]^6*a[3]^7+395620183241588736*a[2]^5*a[3]^8+
898659748999806976*a[2]^4*a[3]^9+1490663749831753728*a[2]^3*a[3]^10+
1720256586812424192*a[2]^2*a[3]^11+1239001150646124544*a[2]*a[3]^12+
418936661358673920*a[3]^13+820224*a[1]^8*a[2]^4+19126272*a[1]^8*a[2]^3*a[3]+
172274688*a[1]^8*a[2]^2*a[3]^2+707622912*a[1]^8*a[2]*a[3]^3+1116109824*a[1]^8*a
[3]^4+34086912*a[1]^7*a[2]^5+863778816*a[1]^7*a[2]^4*a[3]+8930631680*a[1]^7*a[2
]^3*a[3]^2+47137271808*a[1]^7*a[2]^2*a[3]^3+126870306816*a[1]^7*a[2]*a[3]^4+
138052556800*a[1]^7*a[3]^5+619935744*a[1]^6*a[2]^6+16946909184*a[1]^6*a[2]^5*a[
3]+197110961152*a[1]^6*a[2]^4*a[3]^2+1251166105600*a[1]^6*a[2]^3*a[3]^3+
4561511636992*a[1]^6*a[2]^2*a[3]^4+8965785378816*a[1]^6*a[2]*a[3]^5+
7310900462592*a[1]^6*a[3]^6+6439864320*a[1]^5*a[2]^7+188855629824*a[1]^5*a[2]^6
*a[3]+2435143206912*a[1]^5*a[2]^5*a[3]^2+17895137630208*a[1]^5*a[2]^4*a[3]^3+
80655538255872*a[1]^5*a[2]^3*a[3]^4+221103040530432*a[1]^5*a[2]^2*a[3]^5+
337985081112576*a[1]^5*a[2]*a[3]^6+220552694360064*a[1]^5*a[3]^7+41768217600*a[
1]^4*a[2]^8+1308290365440*a[1]^4*a[2]^7*a[3]+18495896827904*a[1]^4*a[2]^6*a[3]^
2+153590245976064*a[1]^4*a[2]^5*a[3]^3+815090350797824*a[1]^4*a[2]^4*a[3]^4+
2811796461185024*a[1]^4*a[2]^3*a[3]^5+6120323596191744*a[1]^4*a[2]^2*a[3]^6+
7662217964369920*a[1]^4*a[2]*a[3]^7+4235647278110720*a[1]^4*a[3]^8+173125632000
*a[1]^3*a[2]^9+5771224934400*a[1]^3*a[2]^8*a[3]+88693196677120*a[1]^3*a[2]^7*a[
3]^2+818573039632384*a[1]^3*a[2]^6*a[3]^3+4964295812562944*a[1]^3*a[2]^5*a[3]^4
+20397012758784000*a[1]^3*a[2]^4*a[3]^5+56598314606723072*a[1]^3*a[2]^3*a[3]^6+
102351353614135296*a[1]^3*a[2]^2*a[3]^7+110079132197675008*a[1]^3*a[2]*a[3]^8+
54130394643562496*a[1]^3*a[3]^9+447690240000*a[1]^2*a[2]^10+15834823680000*a[1]
^2*a[2]^9*a[3]+262732140364800*a[1]^2*a[2]^8*a[3]^2+2663010121728000*a[1]^2*a[2
]^7*a[3]^3+18092196542490624*a[1]^2*a[2]^6*a[3]^4+85600161048932352*a[1]^2*a[2]
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[3]^4+42110923260*a[1]^3+585255474436*a[1]^2*a[2]+2100508589564*a[1]^2*a[3]+
2779328350780*a[1]*a[2]^2+18475439736520*a[1]*a[2]*a[3]+38483283368020*a[1]*a[3
]^2+4318692013496*a[2]^3+45919541387964*a[2]^2*a[3]+128625489179880*a[2]*a[3]^2
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a[1]*a[3]+257838545318*a[2]^2+1548727267774*a[2]*a[3]+4971296900746*a[3]^2+
1968822822*a[1]+7802477302*a[2]+42803773382*a[3]+165103909)/(16*a[1]^2*a[2]*a[3
]+80*a[1]^2*a[3]^2+160*a[1]*a[2]^2*a[3]+1248*a[1]*a[2]*a[3]^2+2240*a[1]*a[3]^3+
400*a[2]^3*a[3]+4240*a[2]^2*a[3]^2+14336*a[2]*a[3]^3+15680*a[3]^4+160*a[1]^2*a[
2]+584*a[1]^2*a[3]+1480*a[1]*a[2]^2+10320*a[1]*a[2]*a[3]+23608*a[1]*a[3]^2+3400
*a[2]^3+35320*a[2]^2*a[3]+149400*a[2]*a[3]^2+216048*a[3]^3+2544*a[1]^2+17496*a[
1]*a[2]+134256*a[1]*a[3]+26340*a[2]^2+521420*a[2]*a[3]+1532688*a[3]^2+3898*a[1]
+14638*a[2]+92378*a[3]+1383)^2/(4*a[1]*a[2]+20*a[1]*a[3]+20*a[2]^2+156*a[2]*a[3
]+280*a[3]^2+30*a[1]+68*a[2]+1074*a[3]+29)^2/(1+2*a[1]+10*a[2]+28*a[3])^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+32768000*a[1]^10*a[2]^3*a[3]^3+245760000*a
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                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



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]^4+975681183944083934192*a[3]^5+88961546214148*a[1]^4+1383487002915088*a[1]^3*
a[2]+9608330064180240*a[1]^3*a[3]+8027590888342288*a[1]^2*a[2]^2+
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]^2*a[3]^2+27153630413284730992*a[2]*a[3]^3+37511237734577302864*a[3]^4+
59999030034340*a[1]^3+714592484561164*a[1]^2*a[2]+4654490755265236*a[1]^2*a[3]+
2802231546460420*a[1]*a[2]^2+38197556898429080*a[1]*a[2]*a[3]+
115766807020277580*a[1]*a[3]^2+3619856038667304*a[2]^3+77236832071902836*a[2]^2
*a[3]+490599148102294520*a[2]*a[3]^2+920371358859910588*a[3]^3+23763263607229*a
[1]^2+191778136818948*a[1]*a[2]+1180144256667716*a[1]*a[3]+380858386508582*a[2]
^2+4911428607472426*a[2]*a[3]+14072225093928754*a[3]^2+5128409298378*a[1]+
21020058849898*a[2]+122589749258618*a[3]+465895939891


We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 9, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                           1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
               x[n + 1] = --------------------------------------
                          1 + 2 x[n - 2] + 11 x[n - 1] + 27 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], ------------------------------]
                                            1 + 2 x[1] + 11 x[2] + 27 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], ------------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 11 x[2] + 27 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+11*a[2]+27*
a[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+11*a[2]+27*a[3]), (2*a[1]*a[2]
+4*a[1]*a[3]+11*a[2]^2+49*a[2]*a[3]+54*a[3]^2+39*a[1]+86*a[2]+1398*a[3]+38)/(4*
a[1]*a[2]+22*a[1]*a[3]+22*a[2]^2+175*a[2]*a[3]+297*a[3]^2+29*a[1]+67*a[2]+1037*
a[3]+28)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+11*a[2]+27*a[3]), (2*a[1]*a[2]+4*
a[1]*a[3]+11*a[2]^2+49*a[2]*a[3]+54*a[3]^2+39*a[1]+86*a[2]+1398*a[3]+38)/(4*a[1
]*a[2]+22*a[1]*a[3]+22*a[2]^2+175*a[2]*a[3]+297*a[3]^2+29*a[1]+67*a[2]+1037*a[3
]+28), (8*a[1]^2*a[2]*a[3]+44*a[1]^2*a[3]^2+88*a[1]*a[2]^2*a[3]+700*a[1]*a[2]*a
[3]^2+1188*a[1]*a[3]^3+242*a[2]^3*a[3]+2519*a[2]^2*a[3]^2+7992*a[2]*a[3]^3+8019
*a[3]^4+164*a[1]^2*a[2]+442*a[1]^2*a[3]+1776*a[1]*a[2]^2+9143*a[1]*a[2]*a[3]+
14281*a[1]*a[3]^2+4807*a[2]^3+36538*a[2]^2*a[3]+106450*a[2]*a[3]^2+112239*a[3]^
3+3002*a[1]^2+23026*a[1]*a[2]+149789*a[1]*a[3]+36480*a[2]^2+679935*a[2]*a[3]+
1506021*a[3]^2+4454*a[1]+19269*a[2]+95655*a[3]+1490)/(16*a[1]^2*a[2]*a[3]+88*a[
1]^2*a[3]^2+176*a[1]*a[2]^2*a[3]+1400*a[1]*a[2]*a[3]^2+2376*a[1]*a[3]^3+484*a[2
]^3*a[3]+5038*a[2]^2*a[3]^2+15984*a[2]*a[3]^3+16038*a[3]^4+160*a[1]^2*a[2]+618*
a[1]^2*a[3]+1606*a[1]*a[2]^2+10943*a[1]*a[2]*a[3]+24999*a[1]*a[3]^2+3993*a[2]^3
+39413*a[2]^2*a[3]+164292*a[2]*a[3]^2+224856*a[3]^3+2483*a[1]^2+18157*a[1]*a[2]
+130532*a[1]*a[3]+28314*a[2]^2+545372*a[2]*a[3]+1477808*a[3]^2+3817*a[1]+15336*
a[2]+90100*a[3]+1362)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
44*a[1]*a[2]^3+108*a[1]*a[2]^2*a[3]+44*a[1]*a[2]*a[3]^2+108*a[1]*a[3]^3+121*a[2
]^4+594*a[2]^3*a[3]+850*a[2]^2*a[3]^2+594*a[2]*a[3]^3+729*a[3]^4-8*a[1]^2*a[2]-\
8*a[1]^2*a[3]-84*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-212*a[1]*a[3]^2-220*a[2]^3-1376
*a[2]^2*a[3]-2624*a[2]*a[3]^2-1404*a[3]^3+9*a[1]^2+98*a[1]*a[2]+188*a[1]*a[3]+
280*a[2]^2+856*a[2]*a[3]+1451*a[3]^2+8*a[1]+42*a[2]+106*a[3]+2)/(1+2*a[1]+11*a[
2]+27*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+704*a[1]^4*a[2]*a[3]^3+1936*a[1]^4*a[3]
^4+1408*a[1]^3*a[2]^3*a[3]^2+18944*a[1]^3*a[2]^2*a[3]^3+80608*a[1]^3*a[2]*a[3]^
4+104544*a[1]^3*a[3]^5+11616*a[1]^2*a[2]^4*a[3]^2+184800*a[1]^2*a[2]^3*a[3]^3+
1048632*a[1]^2*a[2]^2*a[3]^4+2494800*a[1]^2*a[2]*a[3]^5+2117016*a[1]^2*a[3]^6+
42592*a[1]*a[2]^5*a[3]^2+782144*a[1]*a[2]^4*a[3]^3+5508184*a[1]*a[2]^3*a[3]^4+
18585288*a[1]*a[2]^2*a[3]^5+30215592*a[1]*a[2]*a[3]^6+19053144*a[1]*a[3]^7+
58564*a[2]^6*a[3]^2+1219196*a[2]^5*a[3]^3+10213489*a[2]^4*a[3]^4+44144892*a[2]^
3*a[3]^5+104271786*a[2]^2*a[3]^6+128175696*a[2]*a[3]^7+64304361*a[3]^8-128*a[1]
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]-20368*a[1]^3*a[2]^2*a[3]^2+6264*a[1]^3*a[2]*a[3]^3+182072*a[1]^3*a[3]^4-23232
*a[1]^2*a[2]^4*a[3]-248952*a[1]^2*a[2]^3*a[3]^2-377508*a[1]^2*a[2]^2*a[3]^3+
2685048*a[1]^2*a[2]*a[3]^4+6026724*a[1]^2*a[3]^5-85184*a[1]*a[2]^5*a[3]-1209516
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64347534*a[1]*a[2]*a[3]^5+75298410*a[1]*a[3]^6-117128*a[2]^6*a[3]-2071036*a[2]^
5*a[3]^2-10121892*a[2]^4*a[3]^3+5001194*a[2]^3*a[3]^4+160226262*a[2]^2*a[3]^5+
410696730*a[2]*a[3]^6+325202526*a[3]^7+96*a[1]^4*a[2]^2-688*a[1]^4*a[2]*a[3]-\
3128*a[1]^4*a[3]^2+2224*a[1]^3*a[2]^3-3680*a[1]^3*a[2]^2*a[3]-140648*a[1]^3*a[2
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1376874*a[1]^2*a[2]^2*a[3]^2-6951248*a[1]^2*a[2]*a[3]^3-5181350*a[1]^2*a[3]^4+
76208*a[1]*a[2]^5+631720*a[1]*a[2]^4*a[3]-4415108*a[1]*a[2]^3*a[3]^2-47764570*a
[1]*a[2]^2*a[3]^3-95026672*a[1]*a[2]*a[3]^4-26907282*a[1]*a[3]^5+112409*a[2]^6+
1428350*a[2]^5*a[3]-3134930*a[2]^4*a[3]^2-95991042*a[2]^3*a[3]^3-344175359*a[2]
^2*a[3]^4-339189984*a[2]*a[3]^5+37969236*a[3]^6+1000*a[1]^4*a[2]-1788*a[1]^4*a[
3]+20580*a[1]^3*a[2]^2+59994*a[1]^3*a[2]*a[3]-353542*a[1]^3*a[3]^2+155732*a[1]^
2*a[2]^3+1214256*a[1]^2*a[2]^2*a[3]-1933278*a[1]^2*a[2]*a[3]^2-16254490*a[1]^2*
a[3]^3+501852*a[1]*a[2]^4+6794452*a[1]*a[2]^3*a[3]+8682536*a[1]*a[2]^2*a[3]^2-\
122506656*a[1]*a[2]*a[3]^3-245788656*a[1]*a[3]^4+558228*a[2]^5+12133498*a[2]^4*
a[3]+48813232*a[2]^3*a[3]^2-196153706*a[2]^2*a[3]^3-1091516784*a[2]*a[3]^4-\
1166223420*a[3]^5+4605*a[1]^4+79220*a[1]^3*a[2]+403082*a[1]^3*a[3]+491239*a[1]^
2*a[2]^2+5556308*a[1]^2*a[2]*a[3]+11715055*a[1]^2*a[3]^2+1275958*a[1]*a[2]^3+
25091820*a[1]*a[2]^2*a[3]+111646706*a[1]*a[2]*a[3]^2+139807164*a[1]*a[3]^3+
1228968*a[2]^4+34077466*a[2]^3*a[3]+292982489*a[2]^2*a[3]^2+547005420*a[2]*a[3]
^3+821628269*a[3]^4+12684*a[1]^3+168462*a[1]^2*a[2]+787488*a[1]^2*a[3]+724456*a
[1]*a[2]^2+7299918*a[1]*a[2]*a[3]+15054582*a[1]*a[3]^2+935378*a[2]^3+17908882*a
[2]^2*a[3]+63273954*a[2]*a[3]^2+110090106*a[3]^3+12557*a[1]^2+114114*a[1]*a[2]+
498174*a[1]*a[3]+267234*a[2]^2+2161484*a[2]*a[3]+5266034*a[3]^2+5360*a[1]+23580
*a[2]+111460*a[3]+884)/(4*a[1]*a[2]+22*a[1]*a[3]+22*a[2]^2+175*a[2]*a[3]+297*a[
3]^2+29*a[1]+67*a[2]+1037*a[3]+28)^2/(1+2*a[1]+11*a[2]+27*a[3])^2, (12288*a[1]^
8*a[2]^4*a[3]^2+299008*a[1]^8*a[2]^3*a[3]^3+2753536*a[1]^8*a[2]^2*a[3]^4+
11331584*a[1]^8*a[2]*a[3]^5+17532416*a[1]^8*a[3]^6+569344*a[1]^7*a[2]^5*a[3]^2+
14921728*a[1]^7*a[2]^4*a[3]^3+154417152*a[1]^7*a[2]^3*a[3]^4+780111872*a[1]^7*a
[2]^2*a[3]^5+1893357312*a[1]^7*a[2]*a[3]^6+1717340416*a[1]^7*a[3]^7+11561984*a[
1]^6*a[2]^6*a[3]^2+324366336*a[1]^6*a[2]^5*a[3]^3+3713483264*a[1]^6*a[2]^4*a[3]
^4+22014699008*a[1]^6*a[2]^3*a[3]^5+70604913088*a[1]^6*a[2]^2*a[3]^6+
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)^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+36044800*a[1]^10*a[2]^3*a[3]^3+297369600*a
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2]+2053164566315270*a[1]^6*a[3]+1113600969348714*a[1]^5*a[2]^2+
30268349505133940*a[1]^5*a[2]*a[3]+138650060629336204*a[1]^5*a[3]^2+
1681023452450040*a[1]^4*a[2]^3+192372184215600570*a[1]^4*a[2]^2*a[3]+
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]^3+186775018829730761366*a[1]^3*a[3]^4+52444009417490362*a[1]^2*a[2]^5+
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10161348945711448852*a[1]*a[2]^5*a[3]+238598318059739078678*a[1]*a[2]^4*a[3]^2+
2671858190262714842006*a[1]*a[2]^3*a[3]^3+14536025005634541148614*a[1]*a[2]^2*a
[3]^4+35949782612293717763942*a[1]*a[2]*a[3]^5+31318823913882080422372*a[1]*a[3
]^6+167809079811781980*a[2]^7+10423244635093131684*a[2]^6*a[3]+
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403465085720380885*a[1]^4*a[3]^2+34037202039925978*a[1]^3*a[2]^3+
913154097747929166*a[1]^3*a[2]^2*a[3]+6975579356993586456*a[1]^3*a[2]*a[3]^2+
15098994863980116004*a[1]^3*a[3]^3+108870372822906673*a[1]^2*a[2]^4+
3910113297552303130*a[1]^2*a[2]^3*a[3]+46290279601284765728*a[1]^2*a[2]^2*a[3]^
2+210440769056315998406*a[1]^2*a[2]*a[3]^3+307156165044066600783*a[1]^2*a[3]^4+
196677265820647650*a[1]*a[2]^5+8686827774230857634*a[1]*a[2]^4*a[3]+
139138894409896115770*a[1]*a[2]^3*a[3]^2+982583388353965155206*a[1]*a[2]^2*a[3]
^3+3013775334714552475284*a[1]*a[2]*a[3]^4+3178530400289276244392*a[1]*a[3]^5+
151319743496886795*a[2]^6+7877436920467519812*a[2]^5*a[3]+158555506520656034021
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67694627645994*a[1]^5+1353922563926578*a[1]^4*a[2]+9492371769499162*a[1]^4*a[3]
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1029229865350318772*a[1]^2*a[2]^2*a[3]+6954071995854907422*a[1]^2*a[2]*a[3]^2+
13601888336092400260*a[1]^2*a[3]^3+91925839598266818*a[1]*a[2]^4+
2917957462764048536*a[1]*a[2]^3*a[3]+30878205178841300132*a[1]*a[2]^2*a[3]^2+
127256691258176431160*a[1]*a[2]*a[3]^3+170565395589339190986*a[1]*a[3]^4+
76341621727084518*a[2]^5+3099131623640144262*a[2]^4*a[3]+45359389828959501824*a
[2]^3*a[3]^2+293464291791022552848*a[2]^2*a[3]^3+830940253519531231962*a[2]*a[3
]^4+813835887759204187338*a[3]^5+78042875543611*a[1]^4+1304425557473700*a[1]^3*
a[2]+8396967523450876*a[1]^3*a[3]+8101887612308521*a[1]^2*a[2]^2+
109819082783401854*a[1]^2*a[2]*a[3]+325915034836161073*a[1]^2*a[3]^2+
22182760731898596*a[1]*a[2]^3+472439213346626112*a[1]*a[2]^2*a[3]+
2956222484072283188*a[1]*a[2]*a[3]^2+5381090981682572584*a[1]*a[3]^3+
22596262155703863*a[2]^4+669016018853805124*a[2]^3*a[3]+6590460820563048094*a[2
]^2*a[3]^2+25347878701298265212*a[2]*a[3]^3+31767427217534867163*a[3]^4+
52944351111192*a[1]^3+678689004993276*a[1]^2*a[2]+4083916548915484*a[1]^2*a[3]+
2852950278306532*a[1]*a[2]^2+36196673320030200*a[1]*a[2]*a[3]+
100697804268343860*a[1]*a[3]^2+3935529653518856*a[2]^3+78714550429526088*a[2]^2
*a[3]+462583517972136744*a[2]*a[3]^2+790818216777487272*a[3]^3+21068281288780*a
[1]^2+183272997408080*a[1]*a[2]+1038773886088960*a[1]*a[3]+390732423934284*a[2]
^2+4676222417737928*a[2]*a[3]+12258767279232204*a[3]^2+4565405852112*a[1]+
20201654145648*a[2]+108191104008528*a[3]+416273053744)/(16*a[1]^2*a[2]*a[3]+88*
a[1]^2*a[3]^2+176*a[1]*a[2]^2*a[3]+1400*a[1]*a[2]*a[3]^2+2376*a[1]*a[3]^3+484*a
[2]^3*a[3]+5038*a[2]^2*a[3]^2+15984*a[2]*a[3]^3+16038*a[3]^4+160*a[1]^2*a[2]+
618*a[1]^2*a[3]+1606*a[1]*a[2]^2+10943*a[1]*a[2]*a[3]+24999*a[1]*a[3]^2+3993*a[
2]^3+39413*a[2]^2*a[3]+164292*a[2]*a[3]^2+224856*a[3]^3+2483*a[1]^2+18157*a[1]*
a[2]+130532*a[1]*a[3]+28314*a[2]^2+545372*a[2]*a[3]+1477808*a[3]^2+3817*a[1]+
15336*a[2]+90100*a[3]+1362)^2/(4*a[1]*a[2]+22*a[1]*a[3]+22*a[2]^2+175*a[2]*a[3]
+297*a[3]^2+29*a[1]+67*a[2]+1037*a[3]+28)^2/(1+2*a[1]+11*a[2]+27*a[3])^2


                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



1638400*a[1]^10*a[2]^4*a[3]^2+36044800*a[1]^10*a[2]^3*a[3]^3+297369600*a[1]^10*
a[2]^2*a[3]^4+1090355200*a[1]^10*a[2]*a[3]^5+1499238400*a[1]^10*a[3]^6+72089600
*a[1]^9*a[2]^5*a[3]^2+1762918400*a[1]^9*a[2]^4*a[3]^3+16977100800*a[1]^9*a[2]^3
*a[3]^4+80091545600*a[1]^9*a[2]^2*a[3]^5+183724851200*a[1]^9*a[2]*a[3]^6+
161917747200*a[1]^9*a[3]^7+1389363200*a[1]^8*a[2]^6*a[3]^2+37378457600*a[1]^8*a
[2]^5*a[3]^3+410406092800*a[1]^8*a[2]^4*a[3]^4+2345056665600*a[1]^8*a[2]^3*a[3]
^5+7322825523200*a[1]^8*a[2]^2*a[3]^6+11799006208000*a[1]^8*a[2]*a[3]^7+
7652112793600*a[1]^8*a[3]^8+15337062400*a[1]^7*a[2]^7*a[3]^2+449998028800*a[1]^
7*a[2]^6*a[3]^3+5536472268800*a[1]^7*a[2]^5*a[3]^4+36937978675200*a[1]^7*a[2]^4
*a[3]^5+144018137190400*a[1]^7*a[2]^3*a[3]^6+327686864384000*a[1]^7*a[2]^2*a[3]
^7+402884201984000*a[1]^7*a[2]*a[3]^8+206728483737600*a[1]^7*a[3]^9+
106334412800*a[1]^6*a[2]^8*a[3]^2+3376555212800*a[1]^6*a[2]^7*a[3]^3+
45921495961600*a[1]^6*a[2]^6*a[3]^4+348907391078400*a[1]^6*a[2]^5*a[3]^5+
1618423099328000*a[1]^6*a[2]^4*a[3]^6+4692244444518400*a[1]^6*a[2]^3*a[3]^7+
8309169352422400*a[1]^6*a[2]^2*a[3]^8+8227513608652800*a[1]^6*a[2]*a[3]^9+
3493461414643200*a[1]^6*a[3]^10+477030400000*a[1]^5*a[2]^9*a[3]^2+
16274195660800*a[1]^5*a[2]^8*a[3]^3+241842191360000*a[1]^5*a[2]^7*a[3]^4+
2053527292723200*a[1]^5*a[2]^6*a[3]^5+10978863575680000*a[1]^5*a[2]^5*a[3]^6+
38343425616204800*a[1]^5*a[2]^4*a[3]^7+87564495898880000*a[1]^5*a[2]^3*a[3]^8+
126260495736883200*a[1]^5*a[2]^2*a[3]^9+104469997484160000*a[1]^5*a[2]*a[3]^10+
37853323217740800*a[1]^5*a[3]^11+1374801612800*a[1]^4*a[2]^10*a[3]^2+
49977520947200*a[1]^4*a[2]^9*a[3]^3+802285815654400*a[1]^4*a[2]^8*a[3]^4+
7489554045388800*a[1]^4*a[2]^7*a[3]^5+45046905237689600*a[1]^4*a[2]^6*a[3]^6+
182566526810483200*a[1]^4*a[2]^5*a[3]^7+505563420435270400*a[1]^4*a[2]^4*a[3]^8
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2457251737600*a[1]^3*a[2]^11*a[3]^2+94012742348800*a[1]^3*a[2]^10*a[3]^3+
1605714106726400*a[1]^3*a[2]^9*a[3]^4+16170147363532800*a[1]^3*a[2]^8*a[3]^5+
106790024404729600*a[1]^3*a[2]^7*a[3]^6+486353900057043200*a[1]^3*a[2]^6*a[3]^7
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]^9+5573577438336441600*a[1]^3*a[2]^3*a[3]^10+5807153206798099200*a[1]^3*a[2]^2
*a[3]^11+3612182760194976000*a[1]^3*a[2]*a[3]^12+1017806956620537600*a[1]^3*a[3
]^13+2641751763200*a[1]^2*a[2]^12*a[3]^2+103759103782400*a[1]^2*a[2]^11*a[3]^3+
1831030493142400*a[1]^2*a[2]^10*a[3]^4+19216364134281600*a[1]^2*a[2]^9*a[3]^5+
133815796059981200*a[1]^2*a[2]^8*a[3]^6+653087662578102400*a[1]^2*a[2]^7*a[3]^7
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]^9+11031598821451183200*a[1]^2*a[2]^4*a[3]^10+14702156271016790400*a[1]^2*a[2]
^3*a[3]^11+13324662516581064000*a[1]^2*a[2]^2*a[3]^12+7399787817197952000*a[1]^
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3]^2+78186781798400*a[1]*a[2]^12*a[3]^3+1370940324300800*a[1]*a[2]^11*a[3]^4+
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460496650956320000*a[1]*a[2]^8*a[3]^7+1559420631171392000*a[1]*a[2]^7*a[3]^8+
3824514459147456000*a[1]*a[2]^6*a[3]^9+6846188604330240000*a[1]*a[2]^5*a[3]^10+
9021935872491264000*a[1]*a[2]^4*a[3]^11+8890607626530854400*a[1]*a[2]^3*a[3]^12
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980160199968787200*a[1]*a[3]^15+1371896838400*a[2]^14*a[3]^2+57120795635200*a[2
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90741511689360400*a[2]^10*a[3]^6+480436112882176000*a[2]^9*a[3]^7+
1849189782419122000*a[2]^8*a[3]^8+5260876005765600000*a[2]^7*a[3]^9+
11160622815288372000*a[2]^6*a[3]^10+17755524182145024000*a[2]^5*a[3]^11+
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1654020337447328400*a[3]^16+32768000*a[1]^10*a[2]^4*a[3]+690995200*a[1]^10*a[2]
^3*a[3]^2+5454028800*a[1]^10*a[2]^2*a[3]^3+19093606400*a[1]^10*a[2]*a[3]^4+
25010022400*a[1]^10*a[3]^5+1410252800*a[1]^9*a[2]^5*a[3]+33309081600*a[1]^9*a[2
]^4*a[3]^2+312096153600*a[1]^9*a[2]^3*a[3]^3+1450296320000*a[1]^9*a[2]^2*a[3]^4
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232906437632000*a[1]^8*a[2]*a[3]^6+162547410291200*a[1]^8*a[3]^7+286674432000*a
[1]^7*a[2]^7*a[3]+8217708851200*a[1]^7*a[2]^6*a[3]^2+100341294950400*a[1]^7*a[2
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a[3]^5+6568949573472000*a[1]^7*a[2]^2*a[3]^6+8691902239187200*a[1]^7*a[2]*a[3]^
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193706583459065600*a[1]^6*a[2]*a[3]^8+89981117576380800*a[1]^6*a[3]^9+
8510460851200*a[1]^5*a[2]^9*a[3]+285932303641600*a[1]^5*a[2]^8*a[3]^2+
4263692744819200*a[1]^5*a[2]^7*a[3]^3+37067906397939200*a[1]^5*a[2]^6*a[3]^4+
206942784078990400*a[1]^5*a[2]^5*a[3]^5+767811954900088000*a[1]^5*a[2]^4*a[3]^6
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23979006374400*a[1]^4*a[2]^10*a[3]+860770111308800*a[1]^4*a[2]^9*a[3]^2+
13912359231724800*a[1]^4*a[2]^8*a[3]^3+133414690676960000*a[1]^4*a[2]^7*a[3]^4+
839981507452768800*a[1]^4*a[2]^6*a[3]^5+3620742486533827200*a[1]^4*a[2]^5*a[3]^
6+10793081586562301600*a[1]^4*a[2]^4*a[3]^7+21915205393582009600*a[1]^4*a[2]^3*
a[3]^8+28954705875111324000*a[1]^4*a[2]^2*a[3]^9+22453108827613833600*a[1]^4*a[
2]*a[3]^10+7756903328276440800*a[1]^4*a[3]^11+42023746054400*a[1]^3*a[2]^11*a[3
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^5+9507209180152518800*a[1]^3*a[2]^6*a[3]^6+32829356653711416400*a[1]^3*a[2]^5*
a[3]^7+80678180846020944400*a[1]^3*a[2]^4*a[3]^8+138131579495951622000*a[1]^3*a
[2]^3*a[3]^9+156860395330782884400*a[1]^3*a[2]^2*a[3]^10+106345221212029734000*
a[1]^3*a[2]*a[3]^11+32622807218972929200*a[1]^3*a[3]^12+44700027152000*a[1]^2*a
[2]^12*a[3]+1734434492753600*a[1]^2*a[2]^11*a[3]^2+30833252239143200*a[1]^2*a[2
]^10*a[3]^3+332353848979358800*a[1]^2*a[2]^9*a[3]^4+2418901424787306800*a[1]^2*
a[2]^8*a[3]^5+12514481360087102000*a[1]^2*a[2]^7*a[3]^6+47174370854813694800*a[
1]^2*a[2]^6*a[3]^7+130629345627038989200*a[1]^2*a[2]^5*a[3]^8+
264217054865475495600*a[1]^2*a[2]^4*a[3]^9+381832551744495339600*a[1]^2*a[2]^3*
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109819082783401854*a[1]^2*a[2]*a[3]+325915034836161073*a[1]^2*a[3]^2+
22182760731898596*a[1]*a[2]^3+472439213346626112*a[1]*a[2]^2*a[3]+
2956222484072283188*a[1]*a[2]*a[3]^2+5381090981682572584*a[1]*a[3]^3+
22596262155703863*a[2]^4+669016018853805124*a[2]^3*a[3]+6590460820563048094*a[2
]^2*a[3]^2+25347878701298265212*a[2]*a[3]^3+31767427217534867163*a[3]^4+
52944351111192*a[1]^3+678689004993276*a[1]^2*a[2]+4083916548915484*a[1]^2*a[3]+
2852950278306532*a[1]*a[2]^2+36196673320030200*a[1]*a[2]*a[3]+
100697804268343860*a[1]*a[3]^2+3935529653518856*a[2]^3+78714550429526088*a[2]^2
*a[3]+462583517972136744*a[2]*a[3]^2+790818216777487272*a[3]^3+21068281288780*a
[1]^2+183272997408080*a[1]*a[2]+1038773886088960*a[1]*a[3]+390732423934284*a[2]
^2+4676222417737928*a[2]*a[3]+12258767279232204*a[3]^2+4565405852112*a[1]+
20201654145648*a[2]+108191104008528*a[3]+416273053744


We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.







                             Theorem number, 10, :

      For any positive initial conditions x[1], x[2], x[3], the recurrence



                           1 + x[n - 2] + 2 x[n - 1] + 37 x[n]
               x[n + 1] = --------------------------------------
                          1 + 2 x[n - 2] + 12 x[n - 1] + 26 x[n]



                                converges to, 1



Proof: Introducing the corresponding transformation from [0,infinity]^3 into\
     [0,infinity]^3

                                             1 + x[1] + 2 x[2] + 37 x[3]
     [x[1],x[2],x[3]] goes to, [x[2], x[3], ------------------------------]
                                            1 + 2 x[1] + 12 x[2] + 26 x[3]



We have to prove that its orbit always converges to, [1, 1, 1],

    no matter where you start



                 So let's (semi-rigorousl) prove the following



                                     Lemma

 The transformation, [x[1], x[2], x[3]], goes to ,

                  1 + x[1] + 2 x[2] + 37 x[3]
    [x[2], x[3], ------------------------------],  has the Global Attractor,
                 1 + 2 x[1] + 12 x[2] + 26 x[3]

                                             3
    [1, 1, 1], for the positive orthant in, R



                              Semi-Rigorous Proof:



Starting at any point in the positive orthant, [a[1], a[2], a[3]], the first, 4,

    terms of the orbit are



[[a[1], a[2], a[3]], [a[2], a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+12*a[2]+26*
a[3])], [a[3], (1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+12*a[2]+26*a[3]), (2*a[1]*a[2]
+4*a[1]*a[3]+12*a[2]^2+50*a[2]*a[3]+52*a[3]^2+39*a[1]+87*a[2]+1397*a[3]+38)/(4*
a[1]*a[2]+24*a[1]*a[3]+24*a[2]^2+196*a[2]*a[3]+312*a[3]^2+28*a[1]+66*a[2]+1000*
a[3]+27)], [(1+a[1]+2*a[2]+37*a[3])/(1+2*a[1]+12*a[2]+26*a[3]), (2*a[1]*a[2]+4*
a[1]*a[3]+12*a[2]^2+50*a[2]*a[3]+52*a[3]^2+39*a[1]+87*a[2]+1397*a[3]+38)/(4*a[1
]*a[2]+24*a[1]*a[3]+24*a[2]^2+196*a[2]*a[3]+312*a[3]^2+28*a[1]+66*a[2]+1000*a[3
]+27), (8*a[1]^2*a[2]*a[3]+48*a[1]^2*a[3]^2+96*a[1]*a[2]^2*a[3]+784*a[1]*a[2]*a
[3]^2+1248*a[1]*a[3]^3+288*a[2]^3*a[3]+2976*a[2]^2*a[3]^2+8840*a[2]*a[3]^3+8112
*a[3]^4+164*a[1]^2*a[2]+448*a[1]^2*a[3]+1936*a[1]*a[2]^2+9440*a[1]*a[2]*a[3]+
14096*a[1]*a[3]^2+5712*a[2]^3+40096*a[2]^2*a[3]+109692*a[2]*a[3]^2+107536*a[3]^
3+2998*a[1]^2+24552*a[1]*a[2]+147998*a[1]*a[3]+40200*a[2]^2+729390*a[2]*a[3]+
1448476*a[3]^2+4447*a[1]+20721*a[2]+93972*a[3]+1487)/(16*a[1]^2*a[2]*a[3]+96*a[
1]^2*a[3]^2+192*a[1]*a[2]^2*a[3]+1568*a[1]*a[2]*a[3]^2+2496*a[1]*a[3]^3+576*a[2
]^3*a[3]+5952*a[2]^2*a[3]^2+17680*a[2]*a[3]^3+16224*a[3]^4+160*a[1]^2*a[2]+656*
a[1]^2*a[3]+1728*a[1]*a[2]^2+11632*a[1]*a[2]*a[3]+26560*a[1]*a[3]^2+4608*a[2]^3
+43680*a[2]^2*a[3]+181200*a[2]*a[3]^2+234416*a[3]^3+2420*a[1]^2+18728*a[1]*a[2]
+126748*a[1]*a[3]+30144*a[2]^2+566324*a[2]*a[3]+1423184*a[3]^2+3732*a[1]+15948*
a[2]+87754*a[3]+1339)]]


Let L be the distance-squared of the above members of the orbit to the fixed\

     point, [1, 1, 1], Then L is



[a[1]^2+a[2]^2+a[3]^2-2*a[1]-2*a[2]-2*a[3]+3, (4*a[1]^2*a[2]^2+4*a[1]^2*a[3]^2+
48*a[1]*a[2]^3+104*a[1]*a[2]^2*a[3]+48*a[1]*a[2]*a[3]^2+104*a[1]*a[3]^3+144*a[2
]^4+624*a[2]^3*a[3]+820*a[2]^2*a[3]^2+624*a[2]*a[3]^3+676*a[3]^4-8*a[1]^2*a[2]-\
8*a[1]^2*a[3]-92*a[1]*a[2]^2-304*a[1]*a[2]*a[3]-204*a[1]*a[3]^2-264*a[2]^3-1484
*a[2]^2*a[3]-2576*a[2]*a[3]^2-1300*a[3]^3+9*a[1]^2+108*a[1]*a[2]+178*a[1]*a[3]+
341*a[2]^2+876*a[2]*a[3]+1370*a[3]^2+8*a[1]+46*a[2]+102*a[3]+2)/(1+2*a[1]+12*a[
2]+26*a[3])^2, (64*a[1]^4*a[2]^2*a[3]^2+768*a[1]^4*a[2]*a[3]^3+2304*a[1]^4*a[3]
^4+1536*a[1]^3*a[2]^3*a[3]^2+21760*a[1]^3*a[2]^2*a[3]^3+95232*a[1]^3*a[2]*a[3]^
4+119808*a[1]^3*a[3]^5+13824*a[1]^2*a[2]^4*a[3]^2+225792*a[1]^2*a[2]^3*a[3]^3+
1281408*a[1]^2*a[2]^2*a[3]^4+2935296*a[1]^2*a[2]*a[3]^5+2336256*a[1]^2*a[3]^6+
55296*a[1]*a[2]^5*a[3]^2+1022976*a[1]*a[2]^4*a[3]^3+7082496*a[1]*a[2]^3*a[3]^4+
22846720*a[1]*a[2]^2*a[3]^5+34784256*a[1]*a[2]*a[3]^6+20247552*a[1]*a[3]^7+
82944*a[2]^6*a[3]^2+1714176*a[2]^5*a[3]^3+13948416*a[2]^4*a[3]^4+57288192*a[2]^
3*a[3]^5+126428224*a[2]^2*a[3]^6+143420160*a[2]*a[3]^7+65804544*a[3]^8-128*a[1]
^4*a[2]^2*a[3]-640*a[1]^4*a[2]*a[3]^2+768*a[1]^4*a[3]^3-3072*a[1]^3*a[2]^3*a[3]
-25216*a[1]^3*a[2]^2*a[3]^2-13312*a[1]^3*a[2]*a[3]^3+164352*a[1]^3*a[3]^4-27648
*a[1]^2*a[2]^4*a[3]-315648*a[1]^2*a[2]^3*a[3]^2-660096*a[1]^2*a[2]^2*a[3]^3+
2369280*a[1]^2*a[2]*a[3]^4+5630976*a[1]^2*a[3]^5-110592*a[1]*a[2]^5*a[3]-\
1617408*a[1]*a[2]^4*a[3]^2-5819904*a[1]*a[2]^3*a[3]^3+7934080*a[1]*a[2]^2*a[3]^
4+62726144*a[1]*a[2]*a[3]^5+69828096*a[1]*a[3]^6-165888*a[2]^6*a[3]-2958336*a[2
]^5*a[3]^2-15026688*a[2]^4*a[3]^3-2189568*a[2]^3*a[3]^4+162955520*a[2]^2*a[3]^5
+404064128*a[2]*a[3]^6+295276800*a[3]^7+96*a[1]^4*a[2]^2-512*a[1]^4*a[2]*a[3]-\
3136*a[1]^4*a[3]^2+2432*a[1]^3*a[2]^3-128*a[1]^3*a[2]^2*a[3]-130656*a[1]^3*a[2]
*a[3]^2-286528*a[1]^3*a[3]^3+23296*a[1]^2*a[2]^4+113280*a[1]^2*a[2]^3*a[3]-\
1277600*a[1]^2*a[2]^2*a[3]^2-7316640*a[1]^2*a[2]*a[3]^3-5922240*a[1]^2*a[3]^4+
99840*a[1]*a[2]^5+937472*a[1]*a[2]^4*a[3]-3864960*a[1]*a[2]^3*a[3]^2-51858496*a
[1]*a[2]^2*a[3]^3-103620128*a[1]*a[2]*a[3]^4-36267712*a[1]*a[3]^5+161280*a[2]^6
+2182656*a[2]^5*a[3]-1310336*a[2]^4*a[3]^2-109360768*a[2]^3*a[3]^3-386660992*a[
2]^2*a[3]^4-382977504*a[2]*a[3]^5-10556416*a[3]^6+944*a[1]^4*a[2]-1312*a[1]^4*a
[3]+21168*a[1]^3*a[2]^2+64960*a[1]^3*a[2]*a[3]-313696*a[1]^3*a[3]^2+174176*a[1]
^2*a[2]^3+1337104*a[1]^2*a[2]^2*a[3]-1744048*a[1]^2*a[2]*a[3]^2-14789952*a[1]^2
*a[3]^3+606528*a[1]*a[2]^4+7979200*a[1]*a[2]^3*a[3]+10722608*a[1]*a[2]^2*a[3]^2
-122887776*a[1]*a[2]*a[3]^3-217792096*a[1]*a[3]^4+717696*a[2]^5+15301440*a[2]^4
*a[3]+60942176*a[2]^3*a[3]^2-217590320*a[2]^2*a[3]^3-1071645600*a[2]*a[3]^4-\
983513440*a[3]^5+4404*a[1]^4+81072*a[1]^3*a[2]+381776*a[1]^3*a[3]+534204*a[1]^2
*a[2]^2+5671808*a[1]^2*a[2]*a[3]+10903420*a[1]^2*a[3]^2+1455024*a[1]*a[2]^3+
27537240*a[1]*a[2]^2*a[3]+112342168*a[1]*a[2]*a[3]^2+127353608*a[1]*a[3]^3+
1469232*a[2]^4+39367344*a[2]^3*a[3]+323688288*a[2]^2*a[3]^2+519838904*a[2]*a[3]
^3+759292344*a[3]^4+12148*a[1]^3+173320*a[1]^2*a[2]+744916*a[1]^2*a[3]+797064*a
[1]*a[2]^2+7454496*a[1]*a[2]*a[3]+13983448*a[1]*a[3]^2+1078704*a[2]^3+19875936*
a[2]^2*a[3]+62752700*a[2]*a[3]^2+101590672*a[3]^3+12050*a[1]^2+117910*a[1]*a[2]
+471240*a[1]*a[3]+297753*a[2]^2+2198914*a[2]*a[3]+4910627*a[3]^2+5154*a[1]+
24426*a[2]+105476*a[3]+850)/(4*a[1]*a[2]+24*a[1]*a[3]+24*a[2]^2+196*a[2]*a[3]+
312*a[3]^2+28*a[1]+66*a[2]+1000*a[3]+27)^2/(1+2*a[1]+12*a[2]+26*a[3])^2, (12288
*a[1]^8*a[2]^4*a[3]^2+327680*a[1]^8*a[2]^3*a[3]^3+3309568*a[1]^8*a[2]^2*a[3]^4+
14942208*a[1]^8*a[2]*a[3]^5+25362432*a[1]^8*a[3]^6+622592*a[1]^7*a[2]^5*a[3]^2+
17596416*a[1]^7*a[2]^4*a[3]^3+195297280*a[1]^7*a[2]^3*a[3]^4+1047265280*a[1]^7*
a[2]^2*a[3]^5+2643984384*a[1]^7*a[2]*a[3]^6+2382888960*a[1]^7*a[3]^7+13828096*a
[1]^6*a[2]^6*a[3]^2+412516352*a[1]^6*a[2]^5*a[3]^3+4975902720*a[1]^6*a[2]^4*a[3
]^4+30632050688*a[1]^6*a[2]^3*a[3]^5+99769778176*a[1]^6*a[2]^2*a[3]^6+
160213499904*a[1]^6*a[2]*a[3]^7+99885514752*a[1]^6*a[3]^8+175767552*a[1]^5*a[2]
^7*a[3]^2+5516427264*a[1]^5*a[2]^6*a[3]^3+71697924096*a[1]^5*a[2]^5*a[3]^4+
495183298560*a[1]^5*a[2]^4*a[3]^5+1943915462656*a[1]^5*a[2]^3*a[3]^6+
4316540960768*a[1]^5*a[2]^2*a[3]^7+5061597462528*a[1]^5*a[2]*a[3]^8+
2454587965440*a[1]^5*a[3]^9+1397882880*a[1]^4*a[2]^8*a[3]^2+46035763200*a[1]^4*
a[2]^7*a[3]^3+640319815680*a[1]^4*a[2]^6*a[3]^4+4880473292800*a[1]^4*a[2]^5*a[3
]^5+22193366507520*a[1]^4*a[2]^4*a[3]^6+61688578703360*a[1]^4*a[2]^3*a[3]^7+
103222648176640*a[1]^4*a[2]^2*a[3]^8+96298283827200*a[1]^4*a[2]*a[3]^9+
38864335011840*a[1]^4*a[3]^10+7120355328*a[1]^3*a[2]^9*a[3]^2+245531934720*a[1]
^3*a[2]^8*a[3]^3+3634809274368*a[1]^3*a[2]^7*a[3]^4+30201431195648*a[1]^3*a[2]^
6*a[3]^5+155017074737152*a[1]^3*a[2]^5*a[3]^6+511347057721344*a[1]^3*a[2]^4*a[3
]^7+1093357565378560*a[1]^3*a[2]^3*a[3]^8+1478438394527744*a[1]^3*a[2]^2*a[3]^9
+1160857388580864*a[1]^3*a[2]*a[3]^10+406988105711616*a[1]^3*a[3]^11+
22677553152*a[1]^2*a[2]^10*a[3]^2+817368662016*a[1]^2*a[2]^9*a[3]^3+
12821204828160*a[1]^2*a[2]^8*a[3]^4+115046598574080*a[1]^2*a[2]^7*a[3]^5+
654520572248064*a[1]^2*a[2]^6*a[3]^6+2478400728170496*a[1]^2*a[2]^5*a[3]^7+
6380869204328448*a[1]^2*a[2]^4*a[3]^8+11151795570606080*a[1]^2*a[2]^3*a[3]^9+
12801264872980480*a[1]^2*a[2]^2*a[3]^10+8788509662380032*a[1]^2*a[2]*a[3]^11+
2750945801011200*a[1]^2*a[3]^12+41278242816*a[1]*a[2]^11*a[3]^2+1552775380992*a
[1]*a[2]^10*a[3]^3+25713754767360*a[1]*a[2]^9*a[3]^4+247349154152448*a[1]*a[2]^
8*a[3]^5+1539052744605696*a[1]*a[2]^7*a[3]^6+6539907656712192*a[1]*a[2]^6*a[3]^
7+19536147569082368*a[1]*a[2]^5*a[3]^8+41467028713144320*a[1]*a[2]^4*a[3]^9+
61939429956911104*a[1]*a[2]^3*a[3]^10+62501259954618368*a[1]*a[2]^2*a[3]^11+
38478111765430272*a[1]*a[2]*a[3]^12+10934108299984896*a[1]*a[3]^13+32869711872*
a[2]^12*a[3]^2+1288798470144*a[2]^11*a[3]^3+22462753406976*a[2]^10*a[3]^4+
230279581532160*a[2]^9*a[3]^5+1551236149149696*a[2]^8*a[3]^6+7277144057708544*a
[2]^7*a[3]^7+24593315765616640*a[2]^6*a[3]^8+60970377234710528*a[2]^5*a[3]^9+
111203708481515520*a[2]^4*a[3]^10+146672674775564288*a[2]^3*a[3]^11+
133236120734040064*a[2]^2*a[3]^12+74718742690136064*a[2]*a[3]^13+
19459024855695360*a[3]^14+159744*a[1]^8*a[2]^4*a[3]+4485120*a[1]^8*a[2]^3*a[3]^
2+46112768*a[1]^8*a[2]^2*a[3]^3+204488704*a[1]^8*a[2]*a[3]^4+328630272*a[1]^8*a
[3]^5+7962624*a[1]^7*a[2]^5*a[3]+234795008*a[1]^7*a[2]^4*a[3]^2+2670264320*a[1]
^7*a[2]^3*a[3]^3+14477393920*a[1]^7*a[2]^2*a[3]^4+37104386048*a[1]^7*a[2]*a[3]^
5+35840262144*a[1]^7*a[3]^6+173932544*a[1]^6*a[2]^6*a[3]+5368635392*a[1]^6*a[2]
^5*a[3]^2+66641874944*a[1]^6*a[2]^4*a[3]^3+422188531712*a[1]^6*a[2]^3*a[3]^4+
1434585202688*a[1]^6*a[2]^2*a[3]^5+2480009052160*a[1]^6*a[2]*a[3]^6+
1691351973888*a[1]^6*a[3]^7+2172518400*a[1]^5*a[2]^7*a[3]+70042697728*a[1]^5*a[
2]^6*a[3]^2+939496488960*a[1]^5*a[2]^5*a[3]^3+6755386847232*a[1]^5*a[2]^4*a[3]^
4+28073908371456*a[1]^5*a[2]^3*a[3]^5+67431679885312*a[1]^5*a[2]^2*a[3]^6+
86451131777024*a[1]^5*a[2]*a[3]^7+45927872790528*a[1]^5*a[3]^8+16957440000*a[1]
^4*a[2]^8*a[3]+570275758080*a[1]^4*a[2]^7*a[3]^2+8202415554560*a[1]^4*a[2]^6*a[
3]^3+65615125094400*a[1]^4*a[2]^5*a[3]^4+318773628989440*a[1]^4*a[2]^4*a[3]^5+
962296759459840*a[1]^4*a[2]^3*a[3]^6+1761608216821760*a[1]^4*a[2]^2*a[3]^7+
1799838338744320*a[1]^4*a[2]*a[3]^8+796650791239680*a[1]^4*a[3]^9+84637384704*a
[1]^3*a[2]^9*a[3]+2966522757120*a[1]^3*a[2]^8*a[3]^2+45486370848768*a[1]^3*a[2]
^7*a[3]^3+399025341169664*a[1]^3*a[2]^6*a[3]^4+2202065013440512*a[1]^3*a[2]^5*a
[3]^5+7914748372652032*a[1]^3*a[2]^4*a[3]^6+18532031592202240*a[1]^3*a[2]^3*a[3
]^7+27442736928292864*a[1]^3*a[2]^2*a[3]^8+23621748281311232*a[1]^3*a[2]*a[3]^9
+9131576109367296*a[1]^3*a[3]^10+263637172224*a[1]^2*a[2]^10*a[3]+9625669337088
*a[1]^2*a[2]^9*a[3]^2+156630520627200*a[1]^2*a[2]^8*a[3]^3+1490808116674560*a[1
]^2*a[2]^7*a[3]^4+9162929730715648*a[1]^2*a[2]^6*a[3]^5+37909681673281536*a[1]^
2*a[2]^5*a[3]^6+106986203025076224*a[1]^2*a[2]^4*a[3]^7+204764211903774720*a[1]
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681110482633600*a[1]^3*a[2]^4+15183123117811328*a[1]^3*a[2]^3*a[3]+
101804763020580672*a[1]^3*a[2]^2*a[3]^2+229746272514478336*a[1]^3*a[2]*a[3]^3+
174458727797402816*a[1]^3*a[3]^4+1700570052537600*a[1]^2*a[2]^5+
51352083532497152*a[1]^2*a[2]^4*a[3]+511478216915141504*a[1]^2*a[2]^3*a[3]^2+
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1303805795226624*a[2]^7+61769868622092288*a[2]^6*a[3]+1111556744151305472*a[2]^
5*a[3]^2+9079125512813424384*a[2]^4*a[3]^3+30684298795048639744*a[2]^3*a[3]^4+
35596267018444193408*a[2]^2*a[3]^5+32179444057744016256*a[2]*a[3]^6+
95397210437556002304*a[3]^7+105885221392*a[1]^6+3109783549504*a[1]^5*a[2]+
12457918268192*a[1]^5*a[3]+37043848113968*a[1]^4*a[2]^2+318948725780288*a[1]^4*
a[2]*a[3]+562955527020144*a[1]^4*a[3]^2+226377786144512*a[1]^3*a[2]^3+
3221731793744032*a[1]^3*a[2]^2*a[3]+11869152636821280*a[1]^3*a[2]*a[3]^2+
12745373857225472*a[1]^3*a[3]^3+744118340423104*a[1]^2*a[2]^4+15692675058497248
*a[1]^2*a[2]^3*a[3]+95778378595019712*a[1]^2*a[2]^2*a[3]^2+194817686905731168*a
[1]^2*a[2]*a[3]^3+170753362266055344*a[1]^2*a[3]^4+1246653821927424*a[1]*a[2]^5
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3897277183612911760*a[2]^2*a[3]^4+4087650965399568096*a[2]*a[3]^5+
9170441250287426000*a[3]^6+135462807968*a[1]^5+3405873955536*a[1]^4*a[2]+
12682923676848*a[1]^4*a[3]+33615863903584*a[1]^3*a[2]^2+263901968668416*a[1]^3*
a[2]*a[3]+445138296004848*a[1]^3*a[3]^2+159755517617984*a[1]^2*a[2]^3+
2074101212081680*a[1]^2*a[2]^2*a[3]+6884839516433440*a[1]^2*a[2]*a[3]^2+
7878486506097648*a[1]^2*a[3]^3+361583196023520*a[1]*a[2]^4+7040190483413824*a[1
]*a[2]^3*a[3]+38116306581070064*a[1]*a[2]^2*a[3]^2+70849783555165120*a[1]*a[2]*
a[3]^3+85120074675457648*a[1]*a[3]^4+309965406840000*a[2]^5+8536553539906896*a[
2]^4*a[3]+70169231014062224*a[2]^3*a[3]^2+198319124146401632*a[2]^2*a[3]^3+
263140196725617552*a[2]*a[3]^4+514225454094205152*a[3]^5+105266399004*a[1]^4+
2160489697528*a[1]^3*a[2]+7605548975872*a[1]^3*a[3]+16590430759460*a[1]^2*a[2]^
2+117906135220512*a[1]^2*a[2]*a[3]+202407140416568*a[1]^2*a[3]^2+54861140069568
*a[1]*a[2]^3+637805514558856*a[1]*a[2]^2*a[3]+1924507541191200*a[1]*a[2]*a[3]^2
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*a[3]^4+51211390380*a[1]^3+791163958536*a[1]^2*a[2]+2753755129340*a[1]^2*a[3]+
4187619707172*a[1]*a[2]^2+27055497702928*a[1]*a[2]*a[3]+53108687821396*a[1]*a[3
]^2+7253493881292*a[2]^3+74278122741516*a[2]^2*a[3]+213736662353752*a[2]*a[3]^2
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a[1]*a[3]+419269639710*a[2]^2+2538419615990*a[2]*a[3]+6612940790798*a[3]^2+
2762246934*a[1]+12693965382*a[2]+58730959342*a[3]+232911457)/(16*a[1]^2*a[2]*a[
3]+96*a[1]^2*a[3]^2+192*a[1]*a[2]^2*a[3]+1568*a[1]*a[2]*a[3]^2+2496*a[1]*a[3]^3
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[2]+656*a[1]^2*a[3]+1728*a[1]*a[2]^2+11632*a[1]*a[2]*a[3]+26560*a[1]*a[3]^2+
4608*a[2]^3+43680*a[2]^2*a[3]+181200*a[2]*a[3]^2+234416*a[3]^3+2420*a[1]^2+
18728*a[1]*a[2]+126748*a[1]*a[3]+30144*a[2]^2+566324*a[2]*a[3]+1423184*a[3]^2+
3732*a[1]+15948*a[2]+87754*a[3]+1339)^2/(4*a[1]*a[2]+24*a[1]*a[3]+24*a[2]^2+196
*a[2]*a[3]+312*a[3]^2+28*a[1]+66*a[2]+1000*a[3]+27)^2/(1+2*a[1]+12*a[2]+26*a[3]
)^2]


We claim that the, 4,

    -th entry  times (101/100) is always less than the first entry



The difference between the distance-square to the point of the originating p\

    oint and (101/100) times that distance, 4, -th point is





1/100*(1638400*a[1]^10*a[2]^4*a[3]^2+39321600*a[1]^10*a[2]^3*a[3]^3+353894400*a
[1]^10*a[2]^2*a[3]^4+1415577600*a[1]^10*a[2]*a[3]^5+2123366400*a[1]^10*a[3]^6+
78643200*a[1]^9*a[2]^5*a[3]^2+2057830400*a[1]^9*a[2]^4*a[3]^3+21076377600*a[1]^
9*a[2]^3*a[3]^4+104752742400*a[1]^9*a[2]^2*a[3]^5+249141657600*a[1]^9*a[2]*a[3]
^6+220830105600*a[1]^9*a[3]^7+1653145600*a[1]^8*a[2]^6*a[3]^2+46832025600*a[1]^
8*a[2]^5*a[3]^3+536590745600*a[1]^8*a[2]^4*a[3]^4+3160237670400*a[1]^8*a[2]^3*a
[3]^5+10000701849600*a[1]^8*a[2]^2*a[3]^6+15974793216000*a[1]^8*a[2]*a[3]^7+
10049893171200*a[1]^8*a[3]^8+19896729600*a[1]^7*a[2]^7*a[3]^2+606509465600*a[1]
^7*a[2]^6*a[3]^3+7672587878400*a[1]^7*a[2]^5*a[3]^4+51954319360000*a[1]^7*a[2]^
4*a[3]^5+202377265152000*a[1]^7*a[2]^3*a[3]^6+451959953817600*a[1]^7*a[2]^2*a[3
]^7+536130197913600*a[1]^7*a[2]*a[3]^8+261462845030400*a[1]^7*a[3]^9+
150287155200*a[1]^6*a[2]^8*a[3]^2+4902223872000*a[1]^6*a[2]^7*a[3]^3+
67745972224000*a[1]^6*a[2]^6*a[3]^4+516397675315200*a[1]^6*a[2]^5*a[3]^5+
2368859330969600*a[1]^6*a[2]^4*a[3]^6+6690629458329600*a[1]^6*a[2]^3*a[3]^7+
11378212444569600*a[1]^6*a[2]^2*a[3]^8+10683355653734400*a[1]^6*a[2]*a[3]^9+
4255230512332800*a[1]^6*a[3]^10+733269196800*a[1]^5*a[2]^9*a[3]^2+
25456331980800*a[1]^5*a[2]^8*a[3]^3+380766545510400*a[1]^5*a[2]^7*a[3]^4+
3214738128896000*a[1]^5*a[2]^6*a[3]^5+16866915699916800*a[1]^5*a[2]^5*a[3]^6+
57053247569920000*a[1]^5*a[2]^4*a[3]^7+124640758962585600*a[1]^5*a[2]^3*a[3]^8+
170041242392985600*a[1]^5*a[2]^2*a[3]^9+131853533891788800*a[1]^5*a[2]*a[3]^10+
44411142537216000*a[1]^5*a[3]^11+2288988979200*a[1]^4*a[2]^10*a[3]^2+
84048504422400*a[1]^4*a[2]^9*a[3]^3+1348179827097600*a[1]^4*a[2]^8*a[3]^4+
12430025451110400*a[1]^4*a[2]^7*a[3]^5+72951864711577600*a[1]^4*a[2]^6*a[3]^6+
285105508397875200*a[1]^4*a[2]^5*a[3]^7+752998280885043200*a[1]^4*a[2]^4*a[3]^8
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]^10+996681827273932800*a[1]^4*a[2]*a[3]^11+291219536137420800*a[1]^4*a[3]^12+
4382628249600*a[1]^3*a[2]^11*a[3]^2+168561318297600*a[1]^3*a[2]^10*a[3]^3+
2863651199385600*a[1]^3*a[2]^9*a[3]^4+28365014591078400*a[1]^3*a[2]^8*a[3]^5+
182177024940441600*a[1]^3*a[2]^7*a[3]^6+798102688091340800*a[1]^3*a[2]^6*a[3]^7
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]^9+7695538834322227200*a[1]^3*a[2]^3*a[3]^10+7455113140292812800*a[1]^3*a[2]^2
*a[3]^11+4285613054361600000*a[1]^3*a[2]*a[3]^12+1110054641703321600*a[1]^3*a[3
]^13+4892236185600*a[1]^2*a[2]^12*a[3]^2+192937564569600*a[1]^2*a[2]^11*a[3]^3+
3383873136230400*a[1]^2*a[2]^10*a[3]^4+34920572387328000*a[1]^2*a[2]^9*a[3]^5+
236581283561472000*a[1]^2*a[2]^8*a[3]^6+1111989279286886400*a[1]^2*a[2]^7*a[3]^
7+3734406058993254400*a[1]^2*a[2]^6*a[3]^8+9076177123776921600*a[1]^2*a[2]^5*a[
3]^9+15934828110356480000*a[1]^2*a[2]^4*a[3]^10+19821111126196224000*a[1]^2*a[2
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[3]^2+143709437952000*a[1]*a[2]^12*a[3]^3+2493511630848000*a[1]*a[2]^11*a[3]^4+
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5735756897936998400*a[1]*a[2]^6*a[3]^9+9739935949022822400*a[1]*a[2]^5*a[3]^10+
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[2]^4*a[3]+747110400*a[1]^10*a[2]^3*a[3]^2+6370099200*a[1]^10*a[2]^2*a[3]^3+
24064819200*a[1]^10*a[2]*a[3]^4+33973862400*a[1]^10*a[3]^5+1533542400*a[1]^9*a[
2]^5*a[3]+38394265600*a[1]^9*a[2]^4*a[3]^2+378667008000*a[1]^9*a[2]^3*a[3]^3+
1835296358400*a[1]^9*a[2]^2*a[3]^4+4362810163200*a[1]^9*a[2]*a[3]^5+
4064123289600*a[1]^9*a[3]^6+31411404800*a[1]^8*a[2]^6*a[3]+856617779200*a[1]^8*
a[2]^5*a[3]^2+9564202598400*a[1]^8*a[2]^4*a[3]^3+55942643712000*a[1]^8*a[2]^3*a
[3]^4+180756519321600*a[1]^8*a[2]^2*a[3]^5+305391049113600*a[1]^8*a[2]*a[3]^6+
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1]^7*a[2]^6*a[3]^2+135095779328000*a[1]^7*a[2]^5*a[3]^3+919350437478400*a[1]^7*
a[2]^4*a[3]^4+3697322282188800*a[1]^7*a[2]^3*a[3]^5+8769953385676800*a[1]^7*a[2
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1104826844164915200*a[1]^5*a[2]^4*a[3]^6+2618139128207769600*a[1]^5*a[2]^3*a[3]
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[1]^2*a[3]^2+192*a[1]*a[2]^2*a[3]+1568*a[1]*a[2]*a[3]^2+2496*a[1]*a[3]^3+576*a[
2]^3*a[3]+5952*a[2]^2*a[3]^2+17680*a[2]*a[3]^3+16224*a[3]^4+160*a[1]^2*a[2]+656
*a[1]^2*a[3]+1728*a[1]*a[2]^2+11632*a[1]*a[2]*a[3]+26560*a[1]*a[3]^2+4608*a[2]^
3+43680*a[2]^2*a[3]+181200*a[2]*a[3]^2+234416*a[3]^3+2420*a[1]^2+18728*a[1]*a[2
]+126748*a[1]*a[3]+30144*a[2]^2+566324*a[2]*a[3]+1423184*a[3]^2+3732*a[1]+15948
*a[2]+87754*a[3]+1339)^2/(4*a[1]*a[2]+24*a[1]*a[3]+24*a[2]^2+196*a[2]*a[3]+312*
a[3]^2+28*a[1]+66*a[2]+1000*a[3]+27)^2/(1+2*a[1]+12*a[2]+26*a[3])^2


                     The denominator is obviously positive



  We have to prove that the numerator is non-negative in the positive orthant



                        The numerator is the polynomial



1638400*a[1]^10*a[2]^4*a[3]^2+39321600*a[1]^10*a[2]^3*a[3]^3+353894400*a[1]^10*
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a[2]^5*a[3]^2+9564202598400*a[1]^8*a[2]^4*a[3]^3+55942643712000*a[1]^8*a[2]^3*a
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1104826844164915200*a[1]^5*a[2]^4*a[3]^6+2618139128207769600*a[1]^5*a[2]^3*a[3]
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a[3]^6+49880498372319641600*a[1]^3*a[2]^5*a[3]^7+116615573115581235200*a[1]^3*a
[2]^4*a[3]^8+188361467339735040000*a[1]^3*a[2]^3*a[3]^9+200202431059250380800*a
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a[3]^7+196458575442254233600*a[1]^2*a[2]^5*a[3]^8+376126016075445043200*a[1]^2*
a[2]^4*a[3]^9+510667262126417510400*a[1]^2*a[2]^3*a[3]^10+468457857847944806400
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19124578805418*a[2]+94637540069458*a[3]+368587765543


We claim that this polynomial is always non-negative in the positive orthant\

          3
     of, R



This is a routine exercise in multi-variable calculus, that we prefer not to\
     do rigorously,since it would take too long, and checked it numerically \
    for many random values

and since we know that there exists a proof, we did not bother. Semi-Rigorou\
    s QED.





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                         This took, 33.841,  seconds.