Solutions of a selection of , 50, diophantine equations with, 4, variables of degree, 4 By Shalosh B. Ekhad Theorem Number, 1 Let 4 P(X[1], X[2], X[3], X[4]) = -7796191113086208 X[1] 3 3 - 14511183219555008 X[1] X[2] + 23693172881677376 X[1] X[3] 3 2 2 - 905497100671296 X[1] X[4] + 10936764354126960 X[1] X[2] 2 2 + 10376174283286944 X[1] X[2] X[3] - 87829573086456672 X[1] X[2] X[4] 2 2 2 - 20486756711459280 X[1] X[3] + 52393421823324384 X[1] X[3] X[4] 2 2 3 + 99293496673283568 X[1] X[4] + 17561777948115912 X[1] X[2] 2 2 - 41926980661293288 X[1] X[2] X[3] - 97727275784993640 X[1] X[2] X[4] 2 + 19522795407549720 X[1] X[2] X[3] + 215906464065105264 X[1] X[2] X[3] X[4] 2 3 + 168318131580752184 X[1] X[2] X[4] + 2622024195232200 X[1] X[3] 2 2 - 93301754339915880 X[1] X[3] X[4] - 216452455802683848 X[1] X[3] X[4] 3 4 - 92118603907943064 X[1] X[4] - 7649464596523397 X[2] 3 3 + 4619507294204792 X[2] X[3] + 62433201739441836 X[2] X[4] 2 2 2 + 10725708308525430 X[2] X[3] - 27060822020780064 X[2] X[3] X[4] 2 2 3 - 168687677430788310 X[2] X[4] - 9326306741184400 X[2] X[3] 2 2 - 60840651244235940 X[2] X[3] X[4] + 47162085348322296 X[2] X[3] X[4] 3 4 + 189298969124965884 X[2] X[4] + 1405725723568375 X[3] 3 2 2 + 32333080364514600 X[3] X[4] + 63274634125101630 X[3] X[4] 3 4 - 25822012659928848 X[3] X[4] - 76323772822026573 X[4] + 600339407985504000 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -16 t + 17 t + 15 t + 13 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 15 t + 11 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j 19 t - 4 t + 11 t + 6 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 15 t + 11 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j -3 t + 13 t + 5 t + 16 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 15 t + 11 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j 8 t + 5 t + 4 t - 2 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 15 t + 11 t + 8 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 2 Let 4 P(X[1], X[2], X[3], X[4]) = 84899507968778855 X[1] 3 3 - 48318716199356009 X[1] X[2] - 302284094864872065 X[1] X[3] 3 2 2 - 195096413575729391 X[1] X[4] + 7224666386844214 X[1] X[2] 2 2 + 143217125562148803 X[1] X[2] X[3] + 84132198283656996 X[1] X[2] X[4] 2 2 2 + 388325133130544390 X[1] X[3] + 515786370245609342 X[1] X[3] X[4] 2 2 3 + 166964471076542205 X[1] X[4] - 924284113480376 X[1] X[2] 2 2 - 12801252610347454 X[1] X[2] X[3] - 7671915835514062 X[1] X[2] X[4] 2 - 142284214733828904 X[1] X[2] X[3] - 166817827218661487 X[1] X[2] X[3] X[4] 2 3 - 48716247614231219 X[1] X[2] X[4] - 208862006772327205 X[1] X[3] 2 2 - 435649543443909586 X[1] X[3] X[4] - 291021624462905094 X[1] X[3] X[4] 3 4 - 62962343593837468 X[1] X[4] + 44230879380760 X[2] 3 3 + 754589848123920 X[2] X[3] + 418449905056644 X[2] X[4] 2 2 2 + 5871511476609410 X[2] X[3] + 7120063031265956 X[2] X[3] X[4] 2 2 3 + 2122983802478504 X[2] X[4] + 47069517638278145 X[2] X[3] 2 2 + 82511847332067126 X[2] X[3] X[4] + 48064591869199465 X[2] X[3] X[4] 3 4 + 9306096137244871 X[2] X[4] + 38081559063792400 X[3] 3 2 2 + 115091720687655025 X[3] X[4] + 121353677174845284 X[3] X[4] 3 4 + 54304311778818389 X[3] X[4] + 8819191428957051 X[4] + 27188576791513203 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -15 t - 20 t - 5 t - 8 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 12 t - 13 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j -14 t + 7 t + 18 t - 8 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 12 t - 13 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j -2 t - 9 t - 7 t - 18 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 12 t - 13 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j -18 t - 18 t + 5 t + 19 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 12 t - 13 t - 7 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 3 Let 2 P(X[1], X[2], X[3], X[4]) = 34167660689316 X[1] c[10] + 45280705600042 X[1] X[2] c[10] - 25379784353660 X[1] X[3] c[10] 2 + 51460973992032 X[1] X[4] c[10] + 15307131413277 X[2] c[10] - 16315718823712 X[2] X[3] c[10] + 30949260525788 X[2] X[4] c[10] 2 + 5140453565808 X[3] c[10] - 21295505116380 X[3] X[4] c[10] 2 2 + 24344874390232 X[4] c[10] - 515285477268 X[1] - 1109772849146 X[1] X[2] + 349724633356 X[1] X[3] 2 - 1307895583152 X[1] X[4] - 530707203549 X[2] + 367468236848 X[2] X[3] 2 - 1357402221028 X[2] X[4] + 427348601340 X[3] X[4] - 619567509512 X[4] + 5140453565808 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -19 t - 13 t + 13 t + 6 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 11 t + 5 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j 18 t + 20 t - 12 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 11 t + 5 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j -11 t - 2 t - 8 t - 17 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 11 t + 5 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j 4 t - 9 t - 12 t - 6 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 11 t + 5 t - 11 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 4 Let 4 3 P(X[1], X[2], X[3], X[4]) = -252460670724 X[1] - 240990462738 X[1] X[2] 3 3 - 2038874241330 X[1] X[3] - 981973660140 X[1] X[4] 2 2 2 + 4868190728829 X[1] X[2] + 2932653752280 X[1] X[2] X[3] 2 2 2 + 1676401477740 X[1] X[2] X[4] - 5851445485725 X[1] X[3] 2 2 2 - 5810216151600 X[1] X[3] X[4] - 1366728476400 X[1] X[4] 3 2 - 6724036008792 X[1] X[2] + 15048965367090 X[1] X[2] X[3] 2 2 + 9350400796470 X[1] X[2] X[4] + 8026814163150 X[1] X[2] X[3] 2 + 10644906267900 X[1] X[2] X[3] X[4] + 3280360195350 X[1] X[2] X[4] 3 2 - 8355604369500 X[1] X[3] - 12885074736750 X[1] X[3] X[4] 2 3 - 6211444250250 X[1] X[3] X[4] - 958012471500 X[1] X[4] 4 3 - 10289148844869 X[2] - 8604261560160 X[2] X[3] 3 2 2 - 12717914109030 X[2] X[4] + 11664816923475 X[2] X[3] 2 2 2 + 12268420793100 X[2] X[3] X[4] + 2354253608025 X[2] X[4] 3 2 + 3711253666500 X[2] X[3] + 9002460264750 X[2] X[3] X[4] 2 3 + 5339058205500 X[2] X[3] X[4] + 881836698000 X[2] X[4] 4 3 - 4580639808750 X[3] - 10469326005000 X[3] X[4] 2 2 3 - 8343688888125 X[3] X[4] - 2854864132500 X[3] X[4] 4 - 360927452500 X[4] + 230660156250000 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -15 t - 4 t + 19 t - 3 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 5 t + 20 t - 13 t + 1 j = 0 infinity ----- 2 \ j 13 t + 2 t - 4 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 5 t + 20 t - 13 t + 1 j = 0 infinity ----- 3 2 \ j -2 t - 17 t + 20 t - 12 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 5 t + 20 t - 13 t + 1 j = 0 infinity ----- 3 2 \ j 15 t + 15 t - 15 t + 15 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 5 t + 20 t - 13 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 5 Let 4 P(X[1], X[2], X[3], X[4]) = -10172280824581246723 X[1] 3 3 - 44336405276448065616 X[1] X[2] - 4699278389939165868 X[1] X[3] 3 2 2 - 19719326927175120190 X[1] X[4] - 65127803816504957508 X[1] X[2] 2 - 25410863405227303532 X[1] X[2] X[3] 2 2 2 - 52802552728664128740 X[1] X[2] X[4] + 6157204573929495420 X[1] X[3] 2 2 2 - 19669283610453700844 X[1] X[3] X[4] - 6815639917368808936 X[1] X[4] 3 2 - 36459122449018425240 X[1] X[2] - 32742761439064101128 X[1] X[2] X[3] 2 - 41400609136869599944 X[1] X[2] X[4] 2 + 3438677137675714640 X[1] X[2] X[3] - 37736151029453220488 X[1] X[2] X[3] X[4] 2 3 - 8406314311143486640 X[1] X[2] X[4] + 4583674972413499736 X[1] X[3] 2 - 5228104123203125392 X[1] X[3] X[4] 2 3 - 7814287440238956712 X[1] X[3] X[4] + 1485826245400442456 X[1] X[4] 4 3 - 5893672739268867776 X[2] - 10671412715874219392 X[2] X[3] 3 2 2 - 10075945522028788384 X[2] X[4] - 2413122635435197760 X[2] X[3] 2 2 2 - 13946620641275797280 X[2] X[3] X[4] - 3965479194907932256 X[2] X[4] 3 2 + 2498813669680318528 X[2] X[3] - 5711608154166889472 X[2] X[3] X[4] 2 3 - 4778074068920336672 X[2] X[3] X[4] + 866054500475692096 X[2] X[4] 4 3 + 801707194872633408 X[3] - 198952488210499872 X[3] X[4] 2 2 3 - 1798099129157378496 X[3] X[4] - 118922702185818528 X[3] X[4] 4 + 494634398421689952 X[4] + 783437816241584313616 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -10 t + 18 t - 16 t - 18 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 10 t + 7 t + 13 t + 1 j = 0 infinity ----- 3 2 \ j 6 t - 17 t - 10 t + 16 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 10 t + 7 t + 13 t + 1 j = 0 infinity ----- 3 2 \ j 13 t + 8 t + 19 t + 1 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 10 t + 7 t + 13 t + 1 j = 0 infinity ----- 3 2 \ j 11 t - 15 t + 9 t - 10 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 10 t + 7 t + 13 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 6 Let 4 P(X[1], X[2], X[3], X[4]) = 9790902908835863 X[1] 3 3 + 26246304650871672 X[1] X[2] + 42094832226639516 X[1] X[3] 3 2 2 - 147383877665853492 X[1] X[4] + 19640574441356248 X[1] X[2] 2 2 + 52286803507329144 X[1] X[2] X[3] - 148313521668184272 X[1] X[2] X[4] 2 2 2 + 28741506963938304 X[1] X[3] - 117806613160960692 X[1] X[3] X[4] 2 2 3 + 30831856066942134 X[1] X[4] + 6037012522515168 X[1] X[2] 2 2 + 22816560341780880 X[1] X[2] X[3] - 53737253103291312 X[1] X[2] X[4] 2 + 26847296671784640 X[1] X[2] X[3] - 101699249525491056 X[1] X[2] X[3] X[4] 2 3 + 37372816179106104 X[1] X[2] X[4] + 10824173770450896 X[1] X[3] 2 2 - 57502973979647424 X[1] X[3] X[4] + 57806750021242500 X[1] X[3] X[4] 3 4 - 21229512685222644 X[1] X[4] + 679898288666288 X[2] 3 3 + 3345287603250336 X[2] X[3] - 6845733117057792 X[2] X[4] 2 2 2 + 5727941075160768 X[2] X[3] - 20502723512156688 X[2] X[3] X[4] 2 2 3 + 8544756782292168 X[2] X[4] + 3609516372081120 X[2] X[3] 2 2 - 18274760067517920 X[2] X[3] X[4] + 16605254954860344 X[2] X[3] X[4] 3 4 - 5245123649125104 X[2] X[4] + 83653843236624 X[3] 3 2 2 + 561283157133936 X[3] X[4] - 5213508889162608 X[3] X[4] 3 4 + 6514996030771716 X[3] X[4] - 2488142272997061 X[4] + 24048439055747174656 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -4 t - 18 t + 20 t + 2 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 10 t + 11 t - 5 t + 1 j = 0 infinity ----- 3 2 \ j 15 t - 19 t - 16 t - 20 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 10 t + 11 t - 5 t + 1 j = 0 infinity ----- 3 2 \ j 10 t + 9 t + 11 t - 1 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 10 t + 11 t - 5 t + 1 j = 0 infinity ----- 3 2 \ j 18 t - 18 t + 10 t - 4 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 10 t + 11 t - 5 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 7 Let 4 P(X[1], X[2], X[3], X[4]) = 2915488497984941099 X[1] 3 3 - 3947652366971997550 X[1] X[2] - 5334032057787936082 X[1] X[3] 3 2 2 + 6204312648254850894 X[1] X[4] + 1600128304250504250 X[1] X[2] 2 + 16471429756027510650 X[1] X[2] X[3] 2 2 2 - 15750280492963458300 X[1] X[2] X[4] - 7734713604316401309 X[1] X[3] 2 2 2 + 1380211715854530756 X[1] X[3] X[4] + 4409117708300710974 X[1] X[4] 3 2 + 3854469396830214500 X[1] X[2] - 27948381647143093500 X[1] X[2] X[3] 2 + 12954836692401174000 X[1] X[2] X[4] 2 + 15661258049616429750 X[1] X[2] X[3] + 12065572805352745500 X[1] X[2] X[3] X[4] 2 3 - 12253873952661228000 X[1] X[2] X[4] - 773531208191716948 X[1] X[3] 2 - 4887224952550651152 X[1] X[3] X[4] 2 3 - 2076591613835009016 X[1] X[3] X[4] + 3233368616373410724 X[1] X[4] 4 3 - 4896795328580327500 X[2] + 15394117853353211000 X[2] X[3] 3 2 2 - 497627323717842000 X[2] X[4] - 5583470394309832500 X[2] X[3] 2 2 2 - 16868976163866031500 X[2] X[3] X[4] + 7027871453821165500 X[2] X[4] 3 2 + 915388654800570800 X[2] X[3] + 1146585331980866700 X[2] X[3] X[4] 2 3 + 8488610136455867100 X[2] X[3] X[4] - 4630961765344622400 X[2] X[4] 4 3 + 854824274995319579 X[3] - 1077943776593336622 X[3] X[4] 2 2 3 + 680505343081771086 X[3] X[4] - 1748326267654633908 X[3] X[4] 4 + 853343373094785684 X[4] + 320928737881980200625 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 3 t + 15 t + 5 t - 6 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 14 t + 5 t + 16 t + 1 j = 0 infinity ----- 3 2 \ j -12 t + 2 t - 7 t - 13 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 14 t + 5 t + 16 t + 1 j = 0 infinity ----- 3 2 \ j -12 t + 13 t + 11 t + 1 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 14 t + 5 t + 16 t + 1 j = 0 infinity ----- 3 2 \ j -18 t - 16 t + 13 t - 18 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 14 t + 5 t + 16 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 8 Let 4 P(X[1], X[2], X[3], X[4]) = -1787084565003008 X[1] 3 3 - 12942440661073856 X[1] X[2] + 10489565884559872 X[1] X[3] 3 2 2 + 15451235742112320 X[1] X[4] - 337095413157032320 X[1] X[2] 2 2 + 187136088437136832 X[1] X[2] X[3] + 618156580151805696 X[1] X[2] X[4] 2 2 2 - 34461401344764672 X[1] X[3] - 184094120194523456 X[1] X[3] X[4] 2 2 3 - 285755402364269952 X[1] X[4] + 2662616007349586592 X[1] X[2] 2 - 1705104921834733136 X[1] X[2] X[3] 2 2 - 7037959990626358480 X[1] X[2] X[4] + 331374568663065408 X[1] X[2] X[3] + 2989773653434471968 X[1] X[2] X[3] X[4] 2 3 + 6205461291985209600 X[1] X[2] X[4] - 16703034968976384 X[1] X[3] 2 2 - 274049062196821952 X[1] X[3] X[4] - 1305697677568674448 X[1] X[3] X[4] 3 4 - 1824813129759998096 X[1] X[4] + 12735997980487602835 X[2] 3 3 - 9770416632393929196 X[2] X[3] - 50304624560425379596 X[2] X[4] 2 2 2 + 2796007882240331648 X[2] X[3] + 29232449276366786028 X[2] X[3] X[4] 2 2 3 + 74121169450677711690 X[2] X[4] - 333668827783507840 X[2] X[3] 2 - 5629941331128819072 X[2] X[3] X[4] 2 3 - 28979587369469724004 X[2] X[3] X[4] - 48322808862198853052 X[2] X[4] 4 3 + 13909263705620224 X[3] + 336185610683344768 X[3] X[4] 2 2 3 + 2812305829510650880 X[3] X[4] + 9527197410187581380 X[3] X[4] 4 + 11768421785580052059 X[4] + 3063378198948802816 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -14 t + 19 t - 18 t + 20 ) a[1, j] t = --------------------------- / 4 3 2 ----- t - 18 t + 8 t - 5 t + 1 j = 0 infinity ----- 3 2 \ j 12 t - 16 t - 11 t + 20 ) a[2, j] t = --------------------------- / 4 3 2 ----- t - 18 t + 8 t - 5 t + 1 j = 0 infinity ----- 3 2 \ j -12 t - 2 t - 20 t + 9 ) a[3, j] t = --------------------------- / 4 3 2 ----- t - 18 t + 8 t - 5 t + 1 j = 0 infinity ----- 3 2 \ j 16 t - 12 t - 9 t + 20 ) a[4, j] t = --------------------------- / 4 3 2 ----- t - 18 t + 8 t - 5 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 9 Let 4 P(X[1], X[2], X[3], X[4]) = -93562236488835280 X[1] 3 3 - 324416588956352320 X[1] X[2] - 128171430627583072 X[1] X[3] 3 2 2 + 50323876878341168 X[1] X[4] - 272925125039577760 X[1] X[2] 2 2 - 595219440126279008 X[1] X[2] X[3] + 384427213248509232 X[1] X[2] X[4] 2 2 2 + 128613376413396032 X[1] X[3] - 168463355099309104 X[1] X[3] X[4] 2 2 3 + 9757558548365984 X[1] X[4] + 61116638124328000 X[1] X[2] 2 2 - 979390571690782624 X[1] X[2] X[3] + 704439973298405136 X[1] X[2] X[4] 2 + 377063448966224768 X[1] X[2] X[3] - 462221489404316064 X[1] X[2] X[3] X[4] 2 3 + 90987870248313888 X[1] X[2] X[4] - 2054247343853568 X[1] X[3] 2 2 + 140486261450843968 X[1] X[3] X[4] - 277364482752341448 X[1] X[3] X[4] 3 4 + 106157917265688676 X[1] X[4] + 121581599658214000 X[2] 3 3 - 503399746425067680 X[2] X[3] + 297853191541355920 X[2] X[4] 2 2 2 + 264627595701398848 X[2] X[3] - 300381800806350448 X[2] X[3] X[4] 2 2 3 + 126314878306521424 X[2] X[4] - 59970928846301184 X[2] X[3] 2 2 + 343398625054071744 X[2] X[3] X[4] - 521951939302406552 X[2] X[3] X[4] 3 4 + 219711188054771980 X[2] X[4] - 22820554752618496 X[3] 3 2 2 + 63373488034862080 X[3] X[4] - 75687758872073008 X[3] X[4] 3 4 + 11763944487998588 X[3] X[4] + 11057524952900557 X[4] + 29197013229114532096 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 12 t - 19 t + 7 t + 7 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 13 t + 10 t - 9 t + 1 j = 0 infinity ----- 3 2 \ j -6 t - 7 t + 9 t - 12 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 13 t + 10 t - 9 t + 1 j = 0 infinity ----- 3 2 \ j 8 t - 7 t - 12 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 13 t + 10 t - 9 t + 1 j = 0 infinity ----- 2 \ j -4 t - 16 t - 6 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 13 t + 10 t - 9 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 10 Let 4 P(X[1], X[2], X[3], X[4]) = -162556014562153216 X[1] 3 3 - 45138184706388608 X[1] X[2] - 341016577367745536 X[1] X[3] 3 2 2 + 31992602308439424 X[1] X[4] + 266969354103721056 X[1] X[2] 2 2 - 199786666927832832 X[1] X[2] X[3] - 74061130211749440 X[1] X[2] X[4] 2 2 2 - 218569570054780416 X[1] X[3] + 80316070777467648 X[1] X[3] X[4] 2 2 3 + 5661229186557792 X[1] X[4] - 82754122389782048 X[1] X[2] 2 2 + 473980962228085920 X[1] X[2] X[3] + 30751270151270112 X[1] X[2] X[4] 2 - 229846890016953600 X[1] X[2] X[3] - 131234664898699200 X[1] X[2] X[3] X[4] 2 3 - 3028593745503072 X[1] X[2] X[4] - 39879313646838272 X[1] X[3] 2 2 + 59051955568764672 X[1] X[3] X[4] + 7265149412066208 X[1] X[3] X[4] 3 4 - 155355288388704 X[1] X[4] + 204720823340435 X[2] 3 3 - 85142944775297008 X[2] X[3] - 7396790339180220 X[2] X[4] 2 2 2 + 204788607826973664 X[2] X[3] + 24742725824785392 X[2] X[3] X[4] 2 2 3 + 583901946854010 X[2] X[4] - 74157171058717696 X[2] X[3] 2 2 - 53219737508713920 X[2] X[3] X[4] - 720558751497552 X[2] X[3] X[4] 3 4 + 293376366579060 X[2] X[4] + 1167166676013824 X[3] 3 2 2 + 13807574728209408 X[3] X[4] + 2686955977113696 X[3] X[4] 3 4 - 166481596674864 X[3] X[4] - 25857072020565 X[4] + 77011914302491392 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 7 t - 17 t + 12 t + 1 ) a[1, j] t = ----------------------------- / 4 3 2 ----- t - 18 t - 16 t + 16 t + 1 j = 0 infinity ----- 2 \ j -17 t + 13 t + 3 ) a[2, j] t = ----------------------------- / 4 3 2 ----- t - 18 t - 16 t + 16 t + 1 j = 0 infinity ----- 3 2 \ j -8 t + 12 t - 16 t + 1 ) a[3, j] t = ----------------------------- / 4 3 2 ----- t - 18 t - 16 t + 16 t + 1 j = 0 infinity ----- 3 2 \ j 16 t + 13 t + 19 t + 13 ) a[4, j] t = ----------------------------- / 4 3 2 ----- t - 18 t - 16 t + 16 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 11 Let 2 P(X[1], X[2], X[3], X[4]) = -2863571463081 X[1] c[8] + 320339086644 X[1] X[2] c[8] + 2602439963892 X[1] X[3] c[8] 2 - 18256394872095 X[1] X[4] c[8] - 273681929891 X[2] c[8] - 2803581362448 X[2] X[3] c[8] + 2863418553405 X[2] X[4] c[8] 2 - 6509610844012 X[3] c[8] + 19052279778442 X[3] X[4] c[8] 2 2 - 31549347510234 X[4] c[8] + 207316596960 X[1] + 30197888640 X[1] X[2] 2 + 1056323060640 X[1] X[4] + 10593663696 X[2] + 140811046272 X[2] X[3] 2 + 56345410800 X[2] X[4] + 363263816448 X[3] - 470366260416 X[3] X[4] 2 + 1197860386572 X[4] + 2602439963892 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 11 t - 2 t - 19 t + 13 ) a[1, j] t = --------------------------- / 4 3 2 ----- t + 8 t + 14 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j 19 t - 17 t + 15 t ) a[2, j] t = --------------------------- / 4 3 2 ----- t + 8 t + 14 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j -4 t + 18 t + 7 t - 9 ) a[3, j] t = --------------------------- / 4 3 2 ----- t + 8 t + 14 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j -3 t - 3 t - t - 7 ) a[4, j] t = --------------------------- / 4 3 2 ----- t + 8 t + 14 t + 8 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 12 Let 4 P(X[1], X[2], X[3], X[4]) = 9183499651495002811 X[1] 3 3 - 25252890954368901457 X[1] X[2] + 216690567487290492 X[1] X[3] 3 2 2 + 4295806077253949510 X[1] X[4] + 15393486791126371254 X[1] X[2] 2 + 17571248286782987610 X[1] X[2] X[3] 2 2 2 - 7378646522378289108 X[1] X[2] X[4] - 8177461653612222090 X[1] X[3] 2 2 2 - 27847231886134143 X[1] X[3] X[4] - 1264640238933844377 X[1] X[4] 3 2 - 2892382938655978948 X[1] X[2] - 10850350933406222307 X[1] X[2] X[3] 2 2 + 5555619067586185995 X[1] X[2] X[4] + 574802129447158014 X[1] X[2] X[3] - 2915637142026609171 X[1] X[2] X[3] X[4] 2 3 + 3040643180981828760 X[1] X[2] X[4] + 2667670846125511257 X[1] X[3] 2 2 + 995011862360415885 X[1] X[3] X[4] - 212852931270775893 X[1] X[3] X[4] 3 4 - 692900420143558336 X[1] X[4] + 94722525350433163 X[2] 3 3 + 1345868712235070004 X[2] X[3] - 1045947498427634027 X[2] X[4] 2 2 2 + 1497073625854023564 X[2] X[3] - 753356273866746768 X[2] X[3] X[4] 2 2 3 - 544457184466036431 X[2] X[4] - 1465573275637966971 X[2] X[3] 2 + 2736375158583589536 X[2] X[3] X[4] 2 3 - 1842464137870190817 X[2] X[3] X[4] + 432323988269647675 X[2] X[4] 4 3 + 126511120137649077 X[3] - 1322785294810217244 X[3] X[4] 2 2 3 + 921401023552238070 X[3] X[4] + 198088377927295467 X[3] X[4] 4 - 50266844977430627 X[4] + 138590979289633934001 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -8 t + 2 t - 8 t + 15 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 11 t - 7 t + 13 t + 1 j = 0 infinity ----- 3 2 \ j -16 t + 12 t + 7 t + 16 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 11 t - 7 t + 13 t + 1 j = 0 infinity ----- 3 2 \ j -18 t - 5 t - 7 t + 17 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 11 t - 7 t + 13 t + 1 j = 0 infinity ----- 3 2 \ j -7 t + 20 t - 12 t - 13 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 11 t - 7 t + 13 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 13 Let 4 P(X[1], X[2], X[3], X[4]) = 341789783527088125 X[1] 3 3 + 1379474728711350250 X[1] X[2] + 729811059420113875 X[1] X[3] 3 2 2 + 740547351436149250 X[1] X[4] + 2023448287268448725 X[1] X[2] 2 + 2139759998681911225 X[1] X[2] X[3] 2 2 2 + 2121517681405225900 X[1] X[2] X[4] + 551479816532685775 X[1] X[3] 2 2 2 + 1096609541533329825 X[1] X[3] X[4] + 568744297447931900 X[1] X[4] 3 2 + 1293721457577073000 X[1] X[2] + 2074195160297676600 X[1] X[2] X[3] 2 + 1987937823753697550 X[1] X[2] X[4] 2 + 1078981799719459100 X[1] X[2] X[3] + 2065279793948073950 X[1] X[2] X[3] X[4] 2 3 + 1042767356477638450 X[1] X[2] X[4] + 182513888492821400 X[1] X[3] 2 2 + 517446674861353650 X[1] X[3] X[4] + 528663239654843625 X[1] X[3] X[4] 3 4 + 189707909919116950 X[1] X[4] + 304729199901847916 X[2] 3 3 + 664602778511009052 X[2] X[3] + 614238532117613128 X[2] X[4] 2 2 2 + 528098072923912504 X[2] X[3] + 970137156448506312 X[2] X[3] X[4] 2 2 3 + 472895696468191209 X[2] X[4] + 182991742531860548 X[2] X[3] 2 2 + 491429570777342916 X[2] X[3] X[4] + 482661579301804549 X[2] X[3] X[4] 3 4 + 167707483282832112 X[2] X[4] + 23724901293422316 X[3] 3 2 2 + 80537337330828896 X[3] X[4] + 117310893148385891 X[3] X[4] 3 4 + 84109611143128891 X[3] X[4] + 23565301627489631 X[4] + 13976609925577202500 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 14 t - 2 t - 9 t + 6 ) a[1, j] t = -------------------------- / 4 3 2 ----- t + 5 t - 6 t + 3 t + 1 j = 0 infinity ----- 3 2 \ j -4 t + 15 t + 10 t - 13 ) a[2, j] t = -------------------------- / 4 3 2 ----- t + 5 t - 6 t + 3 t + 1 j = 0 infinity ----- 3 2 \ j -5 t - t + 3 t + 17 ) a[3, j] t = -------------------------- / 4 3 2 ----- t + 5 t - 6 t + 3 t + 1 j = 0 infinity ----- 3 2 \ j -18 t - 6 t - 17 t - 4 ) a[4, j] t = -------------------------- / 4 3 2 ----- t + 5 t - 6 t + 3 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 14 Let 4 3 P(X[1], X[2], X[3], X[4]) = 178433485789183 X[1] - 546903755044734 X[1] X[2] 3 3 + 2746967883097836 X[1] X[3] - 1662564089089804 X[1] X[4] 2 2 2 + 589461983491480 X[1] X[2] - 8040728347960258 X[1] X[2] X[3] 2 2 2 + 5068073193453804 X[1] X[2] X[4] + 20080228873471342 X[1] X[3] 2 2 2 - 24860682646644968 X[1] X[3] X[4] + 7669398006072564 X[1] X[4] 3 2 - 1697571786804506 X[1] X[2] + 18197037774177184 X[1] X[2] X[3] 2 2 - 11434379261576156 X[1] X[2] X[4] - 71003484818429250 X[1] X[2] X[3] + 89051572460372552 X[1] X[2] X[3] X[4] 2 3 - 27940653441547664 X[1] X[2] X[4] + 93445776367455676 X[1] X[3] 2 2 - 174693644099761812 X[1] X[3] X[4] + 108726621310161032 X[1] X[3] X[4] 3 4 - 22529975919853400 X[1] X[4] + 2132063339662753 X[2] 3 3 - 23772876794758562 X[2] X[3] + 15056445513827340 X[2] X[4] 2 2 2 + 107027489910344508 X[2] X[3] - 135235345207565304 X[2] X[3] X[4] 2 2 3 + 42623708484568876 X[2] X[4] - 228584758739639158 X[2] X[3] 2 2 + 431995909443726884 X[2] X[3] X[4] - 271615439821863616 X[2] X[3] X[4] 3 4 + 56808303319509560 X[2] X[4] + 191490975507738283 X[3] 3 2 2 - 480573279205148296 X[3] X[4] + 451223334067823044 X[3] X[4] 3 4 - 187835615636015640 X[3] X[4] + 29246147898357040 X[4] + 9660412212460816 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 9 t + 4 t - 3 t - 4 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 12 t - 11 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j -7 t - 4 t - 11 t + 3 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 12 t - 11 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j -14 t - 6 t + 2 t - 9 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 12 t - 11 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j -18 t - 2 t + 5 t - 16 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 12 t - 11 t + 7 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 15 Let 4 P(X[1], X[2], X[3], X[4]) = -43157565819846656 X[1] 3 3 - 9270562557132800 X[1] X[2] - 144856627024527360 X[1] X[3] 3 2 2 + 182239675041841152 X[1] X[4] + 536610758912507904 X[1] X[2] 2 2 - 373054891117510656 X[1] X[2] X[3] + 1557527314429181952 X[1] X[2] X[4] 2 2 2 - 139679636456742912 X[1] X[3] - 19023021521584128 X[1] X[3] X[4] 2 2 3 + 817203711587844096 X[1] X[4] + 205772949910441984 X[1] X[2] 2 2 + 694753834708288512 X[1] X[2] X[3] - 556845082502344704 X[1] X[2] X[4] 2 - 437186447182285056 X[1] X[2] X[3] + 3018989329562769408 X[1] X[2] X[3] X[4] 2 3 - 2784811986986618880 X[1] X[2] X[4] - 68074772125296960 X[1] X[3] 2 2 - 306951950518721280 X[1] X[3] X[4] + 2298790196054406144 X[1] X[3] X[4] 3 4 - 2031699353762451456 X[1] X[4] - 1354319195780958464 X[2] 3 3 + 1573863052885509888 X[2] X[3] - 8464945263669525504 X[2] X[4] 2 2 2 - 399294657757179360 X[2] X[3] + 6293919083419899648 X[2] X[3] X[4] 2 2 3 - 18668163257954348544 X[2] X[4] + 14417267423244336 X[2] X[3] 2 2 - 678759477577459776 X[2] X[3] X[4] + 7693598598726447360 X[2] X[3] X[4] 3 4 - 17390643068816575488 X[2] X[4] - 31124991956841729 X[3] 3 2 2 + 115246590800230800 X[3] X[4] - 331525929777948000 X[3] X[4] 3 4 + 2960915844117659904 X[3] X[4] - 5833784051173953792 X[4] + 9711443960177688576 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -20 t + 4 t + 17 t + 12 ) a[1, j] t = ------------------------- / 4 2 ----- t + 9 t + 20 t + 1 j = 0 infinity ----- 3 2 \ j 14 t + 5 t + 16 t - 18 ) a[2, j] t = ------------------------ / 4 2 ----- t + 9 t + 20 t + 1 j = 0 infinity ----- 3 2 \ j 12 t + 16 t - 12 t ) a[3, j] t = -------------------- / 4 2 ----- t + 9 t + 20 t + 1 j = 0 infinity ----- 3 2 \ j -7 t + 3 t - 3 t + 14 ) a[4, j] t = ----------------------- / 4 2 ----- t + 9 t + 20 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 16 Let 4 P(X[1], X[2], X[3], X[4]) = -1226413638142346453 X[1] 3 3 - 4176376255112444736 X[1] X[2] - 5551533958741905088 X[1] X[3] 3 2 2 + 3828574792667247244 X[1] X[4] - 13564087645792083714 X[1] X[2] 2 - 15293610367926228216 X[1] X[2] X[3] 2 2 2 + 24513160122325034640 X[1] X[2] X[4] - 8923669213848415608 X[1] X[3] 2 2 2 + 13721961383896236432 X[1] X[3] X[4] - 11077775315476845438 X[1] X[4] 3 2 - 20039965368404889960 X[1] X[2] - 25116312629582243856 X[1] X[2] X[3] 2 + 52712308316045785788 X[1] X[2] X[4] 2 - 17177852483602483968 X[1] X[2] X[3] + 44927099192753242992 X[1] X[2] X[3] X[4] 2 3 - 46186556049631561824 X[1] X[2] X[4] - 5988324888360846592 X[1] X[3] 2 + 15042573918643183248 X[1] X[3] X[4] 2 3 - 20102993451608754144 X[1] X[3] X[4] + 13485018461763179212 X[1] X[4] 4 3 - 9718425505201956993 X[2] - 16151070592712333880 X[2] X[3] 3 2 2 + 33066984378074875560 X[2] X[4] - 12170861096767427592 X[2] X[3] 2 2 2 + 42124547179519588800 X[2] X[3] X[4] - 42292252585360832994 X[2] X[4] 3 2 - 5982906454703050656 X[2] X[3] + 21359065994254217664 X[2] X[3] X[4] 2 3 - 36577426737588393816 X[2] X[3] X[4] + 24110796710669167632 X[2] X[4] 4 3 - 1449294311584813520 X[3] + 5108522482551767744 X[3] X[4] 2 2 3 - 9392183331755163384 X[3] X[4] + 10582652046178543184 X[3] X[4] 4 - 5173009022983350101 X[4] + 398899676777050296576 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -6 t - 19 t + 12 t + 10 ) a[1, j] t = --------------------------- / 4 3 2 ----- t + 4 t + 16 t + 3 t + 1 j = 0 infinity ----- 3 2 \ j 15 t - 12 t - 11 t + 11 ) a[2, j] t = --------------------------- / 4 3 2 ----- t + 4 t + 16 t + 3 t + 1 j = 0 infinity ----- 3 2 \ j 7 t + 18 t + 19 t - 12 ) a[3, j] t = --------------------------- / 4 3 2 ----- t + 4 t + 16 t + 3 t + 1 j = 0 infinity ----- 3 2 \ j 19 t - 5 t + 19 t + 17 ) a[4, j] t = --------------------------- / 4 3 2 ----- t + 4 t + 16 t + 3 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 17 Let 4 P(X[1], X[2], X[3], X[4]) = -27041837001075926825 X[1] 3 3 + 57026937442625105880 X[1] X[2] + 77803726714220675600 X[1] X[3] 3 2 2 - 53254344011238615780 X[1] X[4] + 17885426786112977864 X[1] X[2] 2 - 769359076339724720 X[1] X[2] X[3] 2 2 2 + 91776863218342473912 X[1] X[2] X[4] - 21516667288709899350 X[1] X[3] 2 2 2 + 112727353367226919620 X[1] X[3] X[4] - 33300777210850272076 X[1] X[4] 3 2 - 19725503363568348704 X[1] X[2] - 45038413161090290816 X[1] X[2] X[3] 2 - 10598366983825161392 X[1] X[2] X[4] 2 - 25046842812300165880 X[1] X[2] X[3] - 49298623367026852688 X[1] X[2] X[3] X[4] 2 3 + 42091802693358326256 X[1] X[2] X[4] - 1109246051171936400 X[1] X[3] 2 - 34762178158868652620 X[1] X[3] X[4] 2 3 + 41834387579998765704 X[1] X[3] X[4] - 5463205267610900560 X[1] X[4] 4 3 - 6458400090097912592 X[2] - 12493905820461893312 X[2] X[3] 3 2 2 - 22059043569323658848 X[2] X[4] - 7258233204407312824 X[2] X[3] 2 2 2 - 42920314685665317616 X[2] X[3] X[4] - 17057231642343694640 X[2] X[4] 3 2 - 2046317593752680400 X[2] X[3] - 20965409855670644872 X[2] X[3] X[4] 2 3 - 35354703215824368592 X[2] X[3] X[4] + 3971457806353457312 X[2] X[4] 4 3 - 487156072707211025 X[3] - 1993853264240676500 X[3] X[4] 2 2 3 - 14373570826424603804 X[3] X[4] + 647111795752362320 X[3] X[4] 4 + 717451325286899568 X[4] + 2583638691996481405456 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 3 t + 6 t - 5 t + 9 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 6 t + 15 t + 14 t + 1 j = 0 infinity ----- 3 2 \ j 20 t - 6 t + 13 t + 13 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 6 t + 15 t + 14 t + 1 j = 0 infinity ----- 3 2 \ j -19 t + 2 t + 17 t + 9 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 6 t + 15 t + 14 t + 1 j = 0 infinity ----- 3 2 \ j -18 t + 19 t + 18 t - 19 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 6 t + 15 t + 14 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 18 Let 4 P(X[1], X[2], X[3], X[4]) = 3733808951428688 X[1] 3 3 - 2541714701284968 X[1] X[2] - 10151561733145624 X[1] X[3] 3 2 2 + 7154253567610600 X[1] X[4] - 2085215898143112 X[1] X[2] 2 2 + 2706263720798416 X[1] X[2] X[3] + 12454300922534688 X[1] X[2] X[4] 2 2 2 + 9780996728277528 X[1] X[3] - 7323415760576392 X[1] X[3] X[4] 2 2 3 - 18627680184319660 X[1] X[4] + 2127520844517992 X[1] X[2] 2 2 + 5415127022317720 X[1] X[2] X[3] - 18682764610779262 X[1] X[2] X[4] 2 + 823971487359456 X[1] X[2] X[3] - 32009374080730840 X[1] X[2] X[3] X[4] 2 3 + 54666248238071124 X[1] X[2] X[4] - 3798634823925776 X[1] X[3] 2 2 - 2826720753727946 X[1] X[3] X[4] + 47334245258531420 X[1] X[3] X[4] 3 4 - 53296791812880082 X[1] X[4] - 458061714746933 X[2] 3 3 - 2117869301234874 X[2] X[3] + 5336397117920341 X[2] X[4] 2 2 2 - 3073984574316184 X[2] X[3] + 18576130764856003 X[2] X[3] X[4] 2 2 3 - 23290022889876866 X[2] X[4] - 1093499767781574 X[2] X[3] 2 2 + 18087669767105783 X[2] X[3] X[4] - 54280263692054186 X[2] X[3] X[4] 3 4 + 45128809002076000 X[2] X[4] + 447286143752093 X[3] 3 2 2 + 3296244976825841 X[3] X[4] - 26610791380183808 X[3] X[4] 3 4 + 52838310358374064 X[3] X[4] - 32755927883320463 X[4] + 293198635825936 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 11 t + 17 t + 4 t - 7 ) a[1, j] t = ------------------------- / 4 3 2 ----- t - t - 20 t - 4 t + 1 j = 0 infinity ----- 3 2 \ j 19 t + 14 t + 20 t + 20 ) a[2, j] t = ------------------------- / 4 3 2 ----- t - t - 20 t - 4 t + 1 j = 0 infinity ----- 3 2 \ j 9 t - 12 t - 2 t - 10 ) a[3, j] t = ------------------------- / 4 3 2 ----- t - t - 20 t - 4 t + 1 j = 0 infinity ----- 3 2 \ j 4 t - 12 t + 4 t + 6 ) a[4, j] t = ------------------------- / 4 3 2 ----- t - t - 20 t - 4 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 19 Let 4 P(X[1], X[2], X[3], X[4]) = 312680768902018624 X[1] 3 3 + 2010528778851199488 X[1] X[2] + 1104963891191859264 X[1] X[3] 3 2 2 - 2820386092098014340 X[1] X[4] + 4846463762303994496 X[1] X[2] 2 + 5326096184701203776 X[1] X[2] X[3] 2 2 2 - 13598385633184303760 X[1] X[2] X[4] + 1461597094288096064 X[1] X[3] 2 2 2 - 7473669890412525480 X[1] X[3] X[4] + 9530910259392215300 X[1] X[4] 3 2 + 5190763882885464192 X[1] X[2] + 8555037272354892928 X[1] X[2] X[3] 2 - 21848373644162155880 X[1] X[2] X[4] 2 + 4694311070282691584 X[1] X[2] X[3] - 24011365908323482980 X[1] X[2] X[3] X[4] 2 3 + 30629371903803275800 X[1] X[2] X[4] + 857543758956095584 X[1] X[3] 2 - 6589385132579693020 X[1] X[3] X[4] 2 3 + 16834733667443907650 X[1] X[3] X[4] - 14301525745576496750 X[1] X[4] 4 3 + 2084217646063839744 X[2] + 4579171847656172928 X[2] X[3] 3 2 2 - 11697792174070173480 X[2] X[4] + 3768149159957214976 X[2] X[3] 2 2 2 - 19280281323383209120 X[2] X[3] X[4] + 24601169940975789700 X[2] X[4] 3 2 + 1376343168519225152 X[2] X[3] - 10579801769787554260 X[2] X[3] X[4] 2 3 + 27038345130090207700 X[2] X[3] X[4] - 22976256949599347750 X[2] X[4] 4 3 + 188264276480010464 X[3] - 1932734237650743410 X[3] X[4] 2 2 3 + 7420467593645259550 X[3] X[4] - 12629330159029048250 X[3] X[4] 4 + 8040494551965276875 X[4] + 2606207153550625 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 20 t + 7 t - 3 t + 12 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 8 t - 16 t + 10 t + 1 j = 0 infinity ----- 3 2 \ j t - 3 t - 17 t - 12 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 8 t - 16 t + 10 t + 1 j = 0 infinity ----- 3 2 \ j -7 t + 18 t - 9 t + 6 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 8 t - 16 t + 10 t + 1 j = 0 infinity ----- 3 2 \ j 7 t + 9 t - 17 t - 1 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 8 t - 16 t + 10 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 20 Let 4 P(X[1], X[2], X[3], X[4]) = 33396667496786933 X[1] 3 3 + 587948433299729716 X[1] X[2] + 586995801644866112 X[1] X[3] 3 2 2 + 414705658669440556 X[1] X[4] - 2541258723938160602 X[1] X[2] 2 2 - 7282175413851540448 X[1] X[2] X[3] - 680037026000933804 X[1] X[2] X[4] 2 2 2 - 5146322721529372624 X[1] X[3] - 1176556165137312480 X[1] X[3] X[4] 2 2 3 + 173525870731343734 X[1] X[4] + 4971090382642802116 X[1] X[2] 2 + 21950695586005979392 X[1] X[2] X[3] 2 - 2201012661125183916 X[1] X[2] X[4] 2 + 27650988277001503648 X[1] X[2] X[3] + 7952018008075205248 X[1] X[2] X[3] X[4] 2 3 - 9765860360862669140 X[1] X[2] X[4] + 10312303234592566144 X[1] X[3] 2 + 10520530590922832864 X[1] X[3] X[4] 2 3 - 6201147809094180608 X[1] X[3] X[4] - 3360288071374212164 X[1] X[4] 4 3 - 2956054425239452691 X[2] - 23734284891716881760 X[2] X[3] 3 2 2 + 10356997923915860140 X[2] X[4] - 52529860420435170000 X[2] X[3] 2 2 2 + 18198504614934175328 X[2] X[3] X[4] + 11743807084771895518 X[2] X[4] 3 2 - 45968783175701048704 X[2] X[3] + 10366955537071565344 X[2] X[3] X[4] 2 3 + 7973286005751510752 X[2] X[3] X[4] + 10875184645317390860 X[2] X[4] 4 3 - 14304220451675337984 X[3] + 2877573294755036544 X[3] X[4] 2 2 3 - 2871359682110629968 X[3] X[4] + 5873917400925703200 X[3] X[4] 4 + 4094847783319973757 X[4] + 131609621362108784896 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -15 t - t - 6 t + 11 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 12 t - 3 t + 20 t + 1 j = 0 infinity ----- 3 2 \ j 11 t - 15 t - 13 t + 19 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 12 t - 3 t + 20 t + 1 j = 0 infinity ----- 3 2 \ j -15 t + 9 t - 2 t - 18 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 12 t - 3 t + 20 t + 1 j = 0 infinity ----- 3 2 \ j -10 t + 18 t - 7 t ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 12 t - 3 t + 20 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 21 Let 4 P(X[1], X[2], X[3], X[4]) = 220023907751329792 X[1] 3 3 - 1172445081349820416 X[1] X[2] + 415300126989869056 X[1] X[3] 3 2 2 - 347260709204889600 X[1] X[4] + 2231533128497433216 X[1] X[2] 2 - 1565976875939821056 X[1] X[2] X[3] 2 2 2 + 1466452497173660928 X[1] X[2] X[4] + 273894760673438208 X[1] X[3] 2 2 2 - 543104555386738176 X[1] X[3] X[4] + 172050779906718336 X[1] X[4] 3 2 - 1826095139754659632 X[1] X[2] + 1904837824382672352 X[1] X[2] X[3] 2 2 - 1903653449181386640 X[1] X[2] X[4] - 665920374429979200 X[1] X[2] X[3] + 1390848051026105280 X[1] X[2] X[3] X[4] 2 3 - 552300742641938832 X[1] X[2] X[4] + 76946249286347392 X[1] X[3] 2 2 - 255437121782790720 X[1] X[3] X[4] + 210987604192957920 X[1] X[3] X[4] 3 4 - 26249407135607088 X[1] X[4] + 551290406940448663 X[2] 3 3 - 751868545092262408 X[2] X[3] + 786541169797339980 X[2] X[4] 2 2 2 + 393968631637214376 X[2] X[3] - 850161463368868728 X[2] X[3] X[4] 2 2 3 + 377269773041209050 X[2] X[4] - 92035735743332896 X[2] X[3] 2 2 + 309570711809366160 X[2] X[3] X[4] - 285642853208944344 X[2] X[3] X[4] 3 4 + 60838771019332140 X[2] X[4] + 7826838609633136 X[3] 3 2 2 - 38085081848817312 X[3] X[4] + 54004003359751656 X[3] X[4] 3 4 - 23585907378521448 X[3] X[4] - 471017365524729 X[4] + 111271410663100416 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 12 t + 5 t + 8 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 10 t - 6 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j 3 t + 11 t + 7 t + 4 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 10 t - 6 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j -6 t + 15 t + 13 t - 13 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 10 t - 6 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j 17 t - t + 7 t - 2 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 10 t - 6 t - 18 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 22 Let 4 P(X[1], X[2], X[3], X[4]) = 164443501936878750 X[1] 3 3 + 2008798091245121625 X[1] X[2] + 2071984556371105125 X[1] X[3] 3 2 2 + 520910667576270000 X[1] X[4] + 5406505129939648950 X[1] X[2] 2 + 11207353142962094175 X[1] X[2] X[3] 2 2 2 + 4855936822915243500 X[1] X[2] X[4] + 5810228538037313175 X[1] X[3] 2 2 2 + 5086219481461793250 X[1] X[3] X[4] + 619606516502036250 X[1] X[4] 3 2 + 5587627904761422240 X[1] X[2] + 18142293692614443390 X[1] X[2] X[3] 2 + 8941001390062060050 X[1] X[2] X[4] 2 + 19617659257862389680 X[1] X[2] X[3] + 19307623423682820825 X[1] X[2] X[3] X[4] 2 3 + 4040856274961298375 X[1] X[2] X[4] + 7061680928611773720 X[1] X[3] 2 + 10407685358035853325 X[1] X[3] X[4] 2 3 + 4290820933262839875 X[1] X[3] X[4] + 360301168184568750 X[1] X[4] 4 3 + 2478074131075447256 X[2] + 11124491043184018728 X[2] X[3] 3 2 2 + 5837655612404614100 X[2] X[4] + 18618291218811165174 X[2] X[3] 2 2 2 + 19569497898856817700 X[2] X[3] X[4] + 4739412601458590700 X[2] X[4] 3 2 + 13750533851453228277 X[2] X[3] + 21754267142711063850 X[2] X[3] X[4] 2 3 + 10465325240756087925 X[2] X[3] X[4] + 1435066386289473125 X[2] X[4] 4 3 + 3777704211453833961 X[3] + 8022285786948970950 X[3] X[4] 2 2 3 + 5754989934905868675 X[3] X[4] + 1529573832555399375 X[3] X[4] 4 + 88866492688250000 X[4] + 38138062841424950625 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -2 t - 5 t + 14 t - 3 ) a[1, j] t = --------------------------- / 4 3 2 ----- t - 8 t - 5 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j 18 t - 7 t + 17 t + 15 ) a[2, j] t = --------------------------- / 4 3 2 ----- t - 8 t - 5 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j -19 t + 16 t - 11 t ) a[3, j] t = --------------------------- / 4 3 2 ----- t - 8 t - 5 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j 9 t + t - 8 t - 18 ) a[4, j] t = --------------------------- / 4 3 2 ----- t - 8 t - 5 t - 11 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 23 Let 4 P(X[1], X[2], X[3], X[4]) = 1086385836019157 X[1] 3 3 + 13447832766713868 X[1] X[2] + 13635777346960632 X[1] X[3] 3 2 2 - 11137790928452920 X[1] X[4] + 62201496097292062 X[1] X[2] 2 2 + 126261438506198296 X[1] X[2] X[3] - 103151003025578200 X[1] X[2] X[4] 2 2 2 + 63783003283824320 X[1] X[3] - 104585146972407152 X[1] X[3] X[4] 2 2 3 + 42735644556025592 X[1] X[4] + 127412282273411148 X[1] X[2] 2 2 + 388294443540069096 X[1] X[2] X[3] - 317326749509360040 X[1] X[2] X[4] 2 + 392658367194975360 X[1] X[2] X[3] - 644045628793949984 X[1] X[2] X[3] X[4] 2 3 + 263245353889747344 X[1] X[2] X[4] + 131778051064691712 X[1] X[3] 2 2 - 325286352129906432 X[1] X[3] X[4] + 266860141812745888 X[1] X[3] X[4] 3 4 - 72741328267475168 X[1] X[4] + 97516058954543797 X[2] 3 3 + 396585245102535432 X[2] X[3] - 324253022457586760 X[2] X[4] 2 2 2 + 602120946927592640 X[2] X[3] - 987998798654002032 X[2] X[3] X[4] 2 2 3 + 403995479703519352 X[2] X[4] + 404527217222218752 X[2] X[3] 2 2 - 998902509267014912 X[2] X[3] X[4] + 819777493488895328 X[2] X[3] X[4] 3 4 - 223536436120251168 X[2] X[4] + 101479493132873728 X[3] 3 2 2 - 335147259782227968 X[3] X[4] + 413938367135762176 X[3] X[4] 3 4 - 226550598088279360 X[3] X[4] + 46346973239577424 X[4] + 252356813312000 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 18 t + 18 t - 17 t - 2 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 9 t - 11 t - 10 t + 1 j = 0 infinity ----- 3 2 \ j 2 t + 6 t + 13 t + 12 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 9 t - 11 t - 10 t + 1 j = 0 infinity ----- 3 2 \ j 2 t + t - 12 t + 4 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 9 t - 11 t - 10 t + 1 j = 0 infinity ----- 3 2 \ j 12 t + 14 t - 5 t + 19 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 9 t - 11 t - 10 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 24 Let 4 P(X[1], X[2], X[3], X[4]) = 163438761266518711 X[1] 3 3 - 963618135849740472 X[1] X[2] - 215306896723560260 X[1] X[3] 3 2 2 + 2031216885823800096 X[1] X[4] + 2128679689963331070 X[1] X[2] 2 2 + 944323983495363348 X[1] X[2] X[3] - 8876632872845523348 X[1] X[2] X[4] 2 2 2 + 91609368718936992 X[1] X[3] - 1862764799025511332 X[1] X[3] X[4] 2 2 3 + 8975036958313893606 X[1] X[4] - 2089435321706891904 X[1] X[2] 2 - 1383425006074565004 X[1] X[2] X[3] 2 + 12941058189109559832 X[1] X[2] X[4] 2 - 268522476396721920 X[1] X[2] X[3] + 5438031828230134008 X[1] X[2] X[3] X[4] 2 3 - 25961201958795689232 X[1] X[2] X[4] - 13240498622373056 X[1] X[3] 2 2 + 508466964030782976 X[1] X[3] X[4] - 5250034840231914732 X[1] X[3] X[4] 3 4 + 17014750421214938616 X[1] X[4] + 769308427272753171 X[2] 3 3 + 676730739053270556 X[2] X[3] - 6296321957321790900 X[2] X[4] 2 2 2 + 196351785657468000 X[2] X[3] - 3975520395185671140 X[2] X[3] X[4] 2 2 3 + 18808101831417725178 X[2] X[4] + 18375877691768640 X[2] X[3] 2 2 - 742179273530072256 X[2] X[3] X[4] + 7655509357912068084 X[2] X[3] X[4] 3 4 - 24494930812196712612 X[2] X[4] - 345248183191808 X[3] 3 2 2 - 32298245348922432 X[3] X[4] + 700806650430860640 X[3] X[4] 3 4 - 4845733731551153196 X[3] X[4] + 11796682613596369227 X[4] + 3326878467786813696 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 19 t - 2 t - 5 t + 19 ) a[1, j] t = ------------------------- / 4 3 2 ----- t - 11 t - 2 t - t + 1 j = 0 infinity ----- 3 2 \ j -14 t + 6 t - 7 t + 19 ) a[2, j] t = ------------------------- / 4 3 2 ----- t - 11 t - 2 t - t + 1 j = 0 infinity ----- 3 2 \ j 8 t - 7 t - 16 t - 10 ) a[3, j] t = ------------------------- / 4 3 2 ----- t - 11 t - 2 t - t + 1 j = 0 infinity ----- 3 \ j -15 t - 4 t + 2 ) a[4, j] t = ------------------------- / 4 3 2 ----- t - 11 t - 2 t - t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 25 Let 2 P(X[1], X[2], X[3], X[4]) = -577537024 X[1] + 1059618944 X[1] X[2] 2 + 3550343488 X[1] X[3] - 608586368 X[1] X[4] - 486026116 X[2] 2 - 3256943764 X[2] X[3] + 558292904 X[2] X[4] - 5456333689 X[3] 2 + 1870607908 X[3] X[4] - 160326244 X[4] + 505710144 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 16 t + 16 t + 4 t + 16 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 6 t - 12 t - 16 t + 1 j = 0 infinity ----- 3 2 \ j -10 t + 20 t + 15 t - 19 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 6 t - 12 t - 16 t + 1 j = 0 infinity ----- 3 2 \ j 6 t - 4 t + 2 t + 14 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 6 t - 12 t - 16 t + 1 j = 0 infinity ----- 3 2 \ j -11 t - 10 t + 20 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 6 t - 12 t - 16 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 26 Let 4 P(X[1], X[2], X[3], X[4]) = 137270683363248 X[1] 3 3 - 11228213437382592 X[1] X[2] + 2048102621731008 X[1] X[3] 3 2 2 - 1499595022900224 X[1] X[4] + 55709644812262056 X[1] X[2] 2 2 - 192627204195815424 X[1] X[2] X[3] - 33879272009241888 X[1] X[2] X[4] 2 2 2 + 3253882289425056 X[1] X[3] - 22076646994517760 X[1] X[3] X[4] 2 2 3 - 23619523276473120 X[1] X[4] + 736335361844123760 X[1] X[2] 2 + 1520132840342755920 X[1] X[2] X[3] 2 2 - 1226140667756275392 X[1] X[2] X[4] - 561745048407151680 X[1] X[2] X[3] - 1230971916229942464 X[1] X[2] X[3] X[4] 2 3 + 368787173362754688 X[1] X[2] X[4] - 11987395738062912 X[1] X[3] 2 2 - 152727957008748288 X[1] X[3] X[4] - 302367096172901952 X[1] X[3] X[4] 3 4 + 125692599843987840 X[1] X[4] - 1129619325994889013 X[2] 3 3 - 1419686852304194784 X[2] X[3] + 3408964358616159624 X[2] X[4] 2 2 2 + 1219512144875325768 X[2] X[3] + 3091560172495758720 X[2] X[3] X[4] 2 2 3 - 3068362230793167528 X[2] X[4] - 189301931606483712 X[2] X[3] 2 - 1102522657810667040 X[2] X[3] X[4] 2 3 - 1227353014637771328 X[2] X[3] X[4] + 474372217023797088 X[2] X[4] 4 3 - 5665804120393168 X[3] - 88675148986567168 X[3] X[4] 2 2 3 - 387707443732754592 X[3] X[4] - 399376977018047872 X[3] X[4] 4 + 315843354610506224 X[4] + 150302798433867038976 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 12 t + t + 4 t - 10 ) a[1, j] t = --------------------------- / 4 3 2 ----- t - 14 t + 7 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j -18 t + 14 t + 10 t ) a[2, j] t = --------------------------- / 4 3 2 ----- t - 14 t + 7 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j -3 t + 20 t + 20 t + 6 ) a[3, j] t = --------------------------- / 4 3 2 ----- t - 14 t + 7 t + 8 t + 1 j = 0 infinity ----- 3 2 \ j -18 t - 8 t + 13 t + 6 ) a[4, j] t = --------------------------- / 4 3 2 ----- t - 14 t + 7 t + 8 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 27 Let 4 P(X[1], X[2], X[3], X[4]) = 7699063145154572039 X[1] 3 3 - 8874818700702215702 X[1] X[2] - 24489547108189202321 X[1] X[3] 3 2 2 - 11108455172420914926 X[1] X[4] + 3657787560567644641 X[1] X[2] 2 + 21556888231999005461 X[1] X[2] X[3] 2 2 2 + 9767700581778330431 X[1] X[2] X[4] + 28879885449786338014 X[1] X[3] 2 2 2 + 25780500881013301343 X[1] X[3] X[4] + 4457501591293430854 X[1] X[4] 3 2 - 628457223338167558 X[1] X[2] - 6048708035453058927 X[1] X[2] X[3] 2 - 2668408202716719097 X[1] X[2] X[4] 2 - 17261544994397077581 X[1] X[2] X[3] - 15475213464783348377 X[1] X[2] X[3] X[4] 2 3 - 2623450959001166321 X[1] X[2] X[4] - 14949645421604197996 X[1] X[3] 2 - 19716828653705575828 X[1] X[3] X[4] 2 3 - 6899919194646233318 X[1] X[3] X[4] - 672662179793037111 X[1] X[4] 4 3 + 37002145759050599 X[2] + 531291633033204398 X[2] X[3] 3 2 2 + 214737821878845973 X[2] X[4] + 2479004200976985576 X[2] X[3] 2 2 2 + 2197908105657851022 X[2] X[3] X[4] + 351569681134729696 X[2] X[4] 3 2 + 4547450613250267397 X[2] X[3] + 6034900597201515561 X[2] X[3] X[4] 2 3 + 2084045400268234406 X[2] X[3] X[4] + 202757582543090492 X[2] X[4] 4 3 + 2863936246064259989 X[3] + 4972157227543203211 X[3] X[4] 2 2 3 + 2643457531517823949 X[3] X[4] + 519592266534260471 X[3] X[4] 4 + 33938497703207369 X[4] + 2916993899716223041 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -13 t + t - 15 t - 14 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 11 t - 4 t + 14 t + 1 j = 0 infinity ----- 3 2 \ j -20 t + 19 t - 17 t - 11 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 11 t - 4 t + 14 t + 1 j = 0 infinity ----- 3 2 \ j -11 t - 13 t - 6 t - 11 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 11 t - 4 t + 14 t + 1 j = 0 infinity ----- 3 2 \ j 2 t + 18 t - 3 t - 12 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 11 t - 4 t + 14 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 28 Let 4 P(X[1], X[2], X[3], X[4]) = 190659913730787991 X[1] 3 3 + 602782133528887676 X[1] X[2] + 1643277188840961000 X[1] X[3] 3 2 2 - 408311494273919182 X[1] X[4] - 2171252113520551320 X[1] X[2] 2 - 5672977704395449776 X[1] X[2] X[3] 2 2 2 - 10802379571666162080 X[1] X[2] X[4] - 2446423911780300810 X[1] X[3] 2 2 2 - 17162876974871866986 X[1] X[3] X[4] - 7249322403592343688 X[1] X[4] 3 2 + 3350420044585092920 X[1] X[2] + 8695243162720693728 X[1] X[2] X[3] 2 + 17401773553182171456 X[1] X[2] X[4] 2 + 6061137347795583420 X[1] X[2] X[3] + 37195261943165686440 X[1] X[2] X[3] X[4] 2 3 + 42255243752696252376 X[1] X[2] X[4] + 560334138544365624 X[1] X[3] 2 + 18437537743080599934 X[1] X[3] X[4] 2 3 + 57384336943320448608 X[1] X[3] X[4] + 29861134357530321488 X[1] X[4] 4 3 - 332052023254622384 X[2] - 3945130316748434280 X[2] X[3] 3 2 2 - 5410046784074221616 X[2] X[4] - 6403752848199229128 X[2] X[3] 2 2 2 - 24025894787564920080 X[2] X[3] X[4] - 20361183833585112960 X[2] X[4] 3 2 - 963523830501039816 X[2] X[3] - 21349122256827252792 X[2] X[3] X[4] 2 3 - 57608064571520835144 X[2] X[3] X[4] - 38580157032570380096 X[2] X[4] 4 3 - 42116056773845949 X[3] - 2083425201284903478 X[3] X[4] 2 2 3 - 34729911040845673224 X[3] X[4] - 60223977814292533056 X[3] X[4] 4 - 25845087745635909200 X[4] + 6851657985346139214096 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -15 t + 8 t - 16 t + 13 ) a[1, j] t = ------------------------- / 4 2 ----- t + 10 t + 11 t + 1 j = 0 infinity ----- 3 2 \ j 18 t + 10 t - 10 t - 1 ) a[2, j] t = ------------------------ / 4 2 ----- t + 10 t + 11 t + 1 j = 0 infinity ----- 3 2 \ j -t - 20 t + 10 t + 11 ) a[3, j] t = ----------------------- / 4 2 ----- t + 10 t + 11 t + 1 j = 0 infinity ----- 3 2 \ j -9 t + 19 t + 18 t + 5 ) a[4, j] t = ------------------------ / 4 2 ----- t + 10 t + 11 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 29 Let 4 3 P(X[1], X[2], X[3], X[4]) = -651602168108515 X[1] - 67567271063270 X[1] X[2] 3 3 + 3155193418946520 X[1] X[3] - 138866997775180 X[1] X[4] 2 2 2 + 165732145817799 X[1] X[2] + 377140352608494 X[1] X[2] X[3] 2 2 2 + 494434189041438 X[1] X[2] X[4] - 5705725658830569 X[1] X[3] 2 2 2 + 764203529304144 X[1] X[3] X[4] + 409425962896479 X[1] X[4] 3 2 + 36619531204486 X[1] X[2] - 361767328864578 X[1] X[2] X[3] 2 2 + 102996350425194 X[1] X[2] X[4] - 594465965467920 X[1] X[2] X[3] 2 - 1132723397345760 X[1] X[2] X[3] X[4] + 95721802326390 X[1] X[2] X[4] 3 2 + 4572460636478874 X[1] X[3] - 1218350879508006 X[1] X[3] X[4] 2 3 - 962356231058562 X[1] X[3] X[4] + 46924093310882 X[1] X[4] 4 3 - 3595000165327 X[2] - 57492101812386 X[2] X[3] 3 2 2 - 37236635878946 X[2] X[4] + 180505909173681 X[2] X[3] 2 2 2 - 204672918347604 X[2] X[3] X[4] - 115420780957920 X[2] X[4] 3 2 + 280042417694052 X[2] X[3] + 595080215781372 X[2] X[3] X[4] 2 3 - 259043450109174 X[2] X[3] X[4] - 146119365418910 X[2] X[4] 4 3 - 1372879983290013 X[3] + 590614212202380 X[3] X[4] 2 2 3 + 522969382057275 X[3] X[4] - 137864793468156 X[3] X[4] 4 - 65112510100609 X[4] + 346354283475216 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -6 t - 7 t + 4 t + 20 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 18 t + 10 t - 3 t + 1 j = 0 infinity ----- 3 2 \ j 5 t - 13 t + 7 t - 1 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 18 t + 10 t - 3 t + 1 j = 0 infinity ----- 3 2 \ j -4 t + t + 6 t + 16 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 18 t + 10 t - 3 t + 1 j = 0 infinity ----- 3 \ j -t - 15 t - 3 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 18 t + 10 t - 3 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 30 Let 4 P(X[1], X[2], X[3], X[4]) = -7150583453102858896 X[1] 3 3 - 40553795184721680832 X[1] X[2] - 1800316478437547288 X[1] X[3] 3 2 2 + 12112582483475016848 X[1] X[4] - 85981746725327181408 X[1] X[2] 2 - 9076540306976920320 X[1] X[2] X[3] 2 2 2 + 51199902148949822544 X[1] X[2] X[4] - 319176797043414912 X[1] X[3] 2 2 2 + 3374044245632736552 X[1] X[3] X[4] - 7044360474921699984 X[1] X[4] 3 2 - 80775115906469496688 X[1] X[2] - 14980884641971117152 X[1] X[2] X[3] 2 + 72209265062661280032 X[1] X[2] X[4] 2 - 1079692662459427500 X[1] X[2] X[3] + 10999811674559475528 X[1] X[2] X[3] X[4] 2 3 - 20334997618723292736 X[1] X[2] X[4] - 216395495774090612 X[1] X[3] 2 2 + 872432023984239912 X[1] X[3] X[4] - 2118006019748878944 X[1] X[3] X[4] 3 4 + 1983834912414411440 X[1] X[4] - 28358669520010056784 X[2] 3 3 - 8102058414106320824 X[2] X[3] + 33922035021212237600 X[2] X[4] 2 2 2 - 919823234106104448 X[2] X[3] + 8796833500136010600 X[2] X[3] X[4] 2 2 3 - 14628456589375492368 X[2] X[4] - 362832601989605852 X[2] X[3] 2 + 1468860204909614172 X[2] X[3] X[4] 2 3 - 3395778400833884376 X[2] X[3] X[4] + 2926704678988400768 X[2] X[4] 4 3 + 9596190106114991 X[3] + 180491135528527498 X[3] X[4] 2 2 3 - 491399847055834584 X[3] X[4] + 548939981755428304 X[3] X[4] 4 - 259559698994020528 X[4] + 6006636488796769296 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -19 t + 14 t + 19 t + 3 ) a[1, j] t = ----------------------------- / 4 3 2 ----- t - 19 t + 11 t - 17 t + 1 j = 0 infinity ----- 3 2 \ j 14 t - 7 t - 19 t - 5 ) a[2, j] t = ----------------------------- / 4 3 2 ----- t - 19 t + 11 t - 17 t + 1 j = 0 infinity ----- 3 2 \ j -2 t - 20 t - 12 t - 12 ) a[3, j] t = ----------------------------- / 4 3 2 ----- t - 19 t + 11 t - 17 t + 1 j = 0 infinity ----- 3 2 \ j 6 t - 12 t - 6 t - 19 ) a[4, j] t = ----------------------------- / 4 3 2 ----- t - 19 t + 11 t - 17 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 31 Let 4 P(X[1], X[2], X[3], X[4]) = 14814349996781815 X[1] 3 3 + 153122181165402172 X[1] X[2] + 356895233678842356 X[1] X[3] 3 2 2 - 143440266947948284 X[1] X[4] + 538205697840867922 X[1] X[2] 2 + 2142862430469939132 X[1] X[2] X[3] 2 2 2 - 1015728745869053300 X[1] X[2] X[4] + 1676161373186768938 X[1] X[3] 2 2 2 - 2022012156400660940 X[1] X[3] X[4] + 467077783213805362 X[1] X[4] 3 2 + 837881831336178988 X[1] X[2] + 4417250379056249420 X[1] X[2] X[3] 2 - 2358763422008282148 X[1] X[2] X[4] 2 + 6403431017797320916 X[1] X[2] X[3] - 8448600396744225240 X[1] X[2] X[3] X[4] 2 3 + 2167957955527410916 X[1] X[2] X[4] + 2887032816674918644 X[1] X[3] 2 - 6219926519280438132 X[1] X[3] X[4] 2 3 + 3814770212055275500 X[1] X[3] X[4] - 638166577097877484 X[1] X[4] 4 3 + 554606779414756927 X[2] + 3535775604943713508 X[2] X[3] 3 2 2 - 1914253508844044684 X[2] X[4] + 6988210778242682482 X[2] X[3] 2 2 2 - 9238035915624460316 X[2] X[3] X[4] + 2545414662186970618 X[2] X[4] 3 2 + 5765759206337064180 X[2] X[3] - 12726319584557843860 X[2] X[3] X[4] 2 3 + 8317700192792049388 X[2] X[3] X[4] - 1492658801740088780 X[2] X[4] 4 3 + 1755640826688617191 X[3] - 5481463542251997380 X[3] X[4] 2 2 3 + 5819553102267237554 X[3] X[4] - 2373512322815281972 X[3] X[4] 4 + 311279114828129503 X[4] + 2665529160517466896 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 12 t - 17 t + 14 t - 19 ) a[1, j] t = ----------------------------- / 4 3 2 ----- t - 12 t - 20 t - 20 t + 1 j = 0 infinity ----- 3 2 \ j 4 t + 15 t + t - 6 ) a[2, j] t = ----------------------------- / 4 3 2 ----- t - 12 t - 20 t - 20 t + 1 j = 0 infinity ----- 3 2 \ j -7 t - 13 t - 6 t - 1 ) a[3, j] t = ----------------------------- / 4 3 2 ----- t - 12 t - 20 t - 20 t + 1 j = 0 infinity ----- 3 2 \ j 3 t - t + 15 t - 16 ) a[4, j] t = ----------------------------- / 4 3 2 ----- t - 12 t - 20 t - 20 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 32 Let 4 P(X[1], X[2], X[3], X[4]) = 1584234731538294320 X[1] 3 3 - 4621288482701364176 X[1] X[2] - 3971934056931381776 X[1] X[3] 3 2 2 + 5659227590944409360 X[1] X[4] - 3390311586592227228 X[1] X[2] 2 + 8014387847839534248 X[1] X[2] X[3] 2 2 2 - 12100780196149285320 X[1] X[2] X[4] + 3634540264929761796 X[1] X[3] 2 2 2 - 10368514009160756520 X[1] X[3] X[4] + 7469416302903096420 X[1] X[4] 3 2 - 1039689377031189140 X[1] X[2] + 3517479283739280564 X[1] X[2] X[3] 2 - 6058817496512836068 X[1] X[2] X[4] 2 - 4442045547574965084 X[1] X[2] X[3] + 13168885020958166232 X[1] X[2] X[3] X[4] 2 3 - 10196206128020799996 X[1] X[2] X[4] - 1473892788221942948 X[1] X[3] 2 + 6194053366281838140 X[1] X[3] X[4] 2 3 - 8898978758464821516 X[1] X[3] X[4] + 4322440730803336820 X[1] X[4] 4 3 - 100865266602751753 X[2] + 653840135447325932 X[2] X[3] 3 2 2 - 1032092465433392276 X[2] X[4] - 595580370597877110 X[2] X[3] 2 2 2 + 2728767066659958900 X[2] X[3] X[4] - 2599847627686675278 X[2] X[4] 3 2 + 955307559105085868 X[2] X[3] - 3665444090473552380 X[2] X[3] X[4] 2 3 + 5320670638057228500 X[2] X[3] X[4] - 2792540153609843588 X[2] X[4] 4 3 + 235732705146869783 X[3] - 1249367538210835076 X[3] X[4] 2 2 3 + 2612862254600873778 X[3] X[4] - 2515621990513363508 X[3] X[4] 4 + 926572195954041455 X[4] + 6504805424707616016 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -2 t - 9 t - 9 t + 13 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 14 t - 5 t + 18 t + 1 j = 0 infinity ----- 3 2 \ j -14 t + 4 t - 17 t - 2 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 14 t - 5 t + 18 t + 1 j = 0 infinity ----- 3 2 \ j 16 t + 18 t - 4 t + 1 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 14 t - 5 t + 18 t + 1 j = 0 infinity ----- 3 2 \ j 16 t + 16 t + 9 t - 17 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 14 t - 5 t + 18 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 33 Let 4 P(X[1], X[2], X[3], X[4]) = -22797085457890603 X[1] 3 3 + 92318068659582784 X[1] X[2] - 285181365831643294 X[1] X[3] 3 2 2 - 88340522425682638 X[1] X[4] - 133521873469821758 X[1] X[2] 2 2 + 741620863575334522 X[1] X[2] X[3] + 214962367663950106 X[1] X[2] X[4] 2 2 2 - 792052022159577608 X[1] X[3] - 363500193566383718 X[1] X[3] X[4] 2 2 3 - 29451168065981568 X[1] X[4] + 80985688900336368 X[1] X[2] 2 2 - 582337318016649338 X[1] X[2] X[3] - 150703097214488242 X[1] X[2] X[4] 2 + 835830803803030516 X[1] X[2] X[3] + 171095202388810388 X[1] X[2] X[3] X[4] 2 3 - 41548623804765552 X[1] X[2] X[4] + 563293378127648142 X[1] X[3] 2 2 + 1315968842180411094 X[1] X[3] X[4] + 700456741040100824 X[1] X[3] X[4] 3 4 + 107697653061495840 X[1] X[4] - 17520146015246071 X[2] 3 3 + 143748080942410230 X[2] X[3] + 31738151018547798 X[2] X[4] 2 2 2 - 218247867989304348 X[2] X[3] + 27562781708415018 X[2] X[3] X[4] 2 2 3 + 33383516346468928 X[2] X[4] - 231458329840172422 X[2] X[3] 2 2 - 628713770364221854 X[2] X[3] X[4] - 309746635224479960 X[2] X[3] X[4] 3 4 - 40382785739683408 X[2] X[4] - 567253279731414445 X[3] 3 2 2 - 1098034579863670306 X[3] X[4] - 900781045073247000 X[3] X[4] 3 4 - 331039219530054672 X[3] X[4] - 44192173820071984 X[4] + 263006215396611856 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -4 t - 8 t - t - 19 ) a[1, j] t = --------------------- / 4 2 ----- t + 18 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j -15 t + 9 t - 10 t - 18 ) a[2, j] t = ------------------------- / 4 2 ----- t + 18 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j -9 t + 9 t - 5 t + 7 ) a[3, j] t = ---------------------- / 4 2 ----- t + 18 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j 18 t - 16 t + 20 t - 19 ) a[4, j] t = ------------------------- / 4 2 ----- t + 18 t + 7 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 34 Let 4 P(X[1], X[2], X[3], X[4]) = -3050043567272443030 X[1] 3 3 + 6646135384205006055 X[1] X[2] + 1135270675765885425 X[1] X[3] 3 2 2 + 6611471428016853525 X[1] X[4] - 5187318390493485030 X[1] X[2] 2 - 2008444513216636390 X[1] X[2] X[3] 2 2 2 - 10415273161843279525 X[1] X[2] X[4] - 152126821064581740 X[1] X[3] 2 2 2 - 1962576688166046785 X[1] X[3] X[4] - 5208347285899654125 X[1] X[4] 3 2 + 1749979402598612080 X[1] X[2] + 1113527580893214620 X[1] X[2] X[3] 2 2 + 5288021321021674900 X[1] X[2] X[4] + 217064880885251400 X[1] X[2] X[3] + 2173172722154979680 X[1] X[2] X[3] X[4] 2 3 + 5328120529157277425 X[1] X[2] X[4] + 13458201198562940 X[1] X[3] 2 2 + 183982699326071900 X[1] X[3] X[4] + 1085248244304138675 X[1] X[3] X[4] 3 4 + 1777912295353397375 X[1] X[4] - 218421784399769480 X[2] 3 3 - 202234902060852040 X[2] X[3] - 874583828034904600 X[2] X[4] 2 2 2 - 72132335952093800 X[2] X[3] - 575853287913503860 X[2] X[3] X[4] 2 2 3 - 1327874259329069750 X[2] X[4] - 12110014460345760 X[2] X[3] 2 2 - 119034433258715120 X[2] X[3] X[4] - 572003647060915850 X[2] X[3] X[4] 3 4 - 893089845306532675 X[2] X[4] + 12155056525432 X[3] 3 2 2 - 11078372087100620 X[3] X[4] - 48515335765397240 X[3] X[4] 3 4 - 196165803839079675 X[3] X[4] - 222342135119568425 X[4] + 8241733107575729288 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -9 t + 11 t - 18 t - 3 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 7 t + 19 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j 6 t + 15 t + 8 t - 15 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 7 t + 19 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j -18 t - 8 t - t + 11 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 7 t + 19 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j -17 t - 7 t - 20 t + 3 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 7 t + 19 t - 18 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 35 Let 4 P(X[1], X[2], X[3], X[4]) = -84250162870530829 X[1] 3 3 + 1815513057920951398 X[1] X[2] + 468644746211184015 X[1] X[3] 3 2 2 - 91815890287119272 X[1] X[4] - 13326470110458479691 X[1] X[2] 2 - 6450272130155833260 X[1] X[2] X[3] 2 2 2 + 2071113010658101773 X[1] X[2] X[4] - 767607141823701525 X[1] X[3] 2 2 2 + 578533307514845265 X[1] X[3] X[4] - 13037795264046636 X[1] X[4] 3 2 + 31855453496530676182 X[1] X[2] + 20323997100318565530 X[1] X[2] X[3] 2 - 14138884542117189669 X[1] X[2] X[4] 2 + 4080695737668153525 X[1] X[2] X[3] - 7496564229146840715 X[1] X[2] X[3] X[4] 2 3 + 509490675104735091 X[1] X[2] X[4] + 246802850048433375 X[1] X[3] 2 2 - 992950389310974975 X[1] X[3] X[4] + 150714954105335505 X[1] X[3] X[4] 3 4 + 9837680029206067 X[1] X[4] - 25654589265048480049 X[2] 3 3 - 21317456483928290655 X[2] X[3] + 20064281132127889019 X[2] X[4] 2 2 2 - 6819309033988553100 X[2] X[3] + 14029644893493108510 X[2] X[3] X[4] 2 2 3 - 3692400776403478674 X[2] X[4] - 1052803377325365750 X[2] X[3] 2 + 3134082901849151175 X[2] X[3] X[4] 2 3 - 2120156268494626890 X[2] X[3] X[4] - 88276573100166226 X[2] X[4] 4 3 - 71095935645005625 X[3] + 215306986585608000 X[3] X[4] 2 2 3 - 309476831430365100 X[3] X[4] - 28607366426887305 X[3] X[4] 4 - 1290603591961609 X[4] + 84493151250244140625 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 3 t - 16 t - 20 t + 11 ) a[1, j] t = --------------------------- / 4 3 2 ----- t + 4 t + 18 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j -4 t - 2 t - 15 t + 4 ) a[2, j] t = --------------------------- / 4 3 2 ----- t + 4 t + 18 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j 14 t - 5 t + 14 t - 17 ) a[3, j] t = --------------------------- / 4 3 2 ----- t + 4 t + 18 t + 7 t + 1 j = 0 infinity ----- 3 2 \ j -15 t - 20 t - 20 t - 12 ) a[4, j] t = --------------------------- / 4 3 2 ----- t + 4 t + 18 t + 7 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 36 Let 4 P(X[1], X[2], X[3], X[4]) = 1256575871585988893 X[1] 3 3 - 14163719694744518552 X[1] X[2] + 3463291460087686620 X[1] X[3] 3 2 2 + 15665531684534162991 X[1] X[4] - 9072790065411629346 X[1] X[2] 2 + 17356310465972358021 X[1] X[2] X[3] 2 2 2 + 62066376976503515739 X[1] X[2] X[4] - 4983377654853379506 X[1] X[3] 2 2 2 - 31612713987761356722 X[1] X[3] X[4] - 65170749079640895078 X[1] X[4] 3 2 + 27942011994841069993 X[1] X[2] + 7817053503556221000 X[1] X[2] X[3] 2 - 144630572155625089956 X[1] X[2] X[4] 2 - 9665931805233555738 X[1] X[2] X[3] - 45100674554923554441 X[1] X[2] X[3] X[4] 2 3 + 224704260824803732518 X[1] X[2] X[4] + 1666106592410264769 X[1] X[3] 2 + 23313319105206294558 X[1] X[3] X[4] 2 3 + 48885697531717249347 X[1] X[3] X[4] - 108916621169984377704 X[1] X[4] 4 3 - 10663787914171867933 X[2] - 13163379811334124960 X[2] X[3] 3 2 2 + 63139694696809865943 X[2] X[4] - 560923506724685574 X[2] X[3] 2 2 2 + 66045826567576829379 X[2] X[3] X[4] - 132148969936989243861 X[2] X[4] 3 2 + 2312146442812915287 X[2] X[3] + 4360775579838748413 X[2] X[3] X[4] 2 3 - 102653904394792630809 X[2] X[3] X[4] + 117283336287256162089 X[2] X[4] 4 3 - 102585941221423887 X[3] - 5769368356488482205 X[3] X[4] 2 2 3 - 6409469004156573516 X[3] X[4] + 50596654185467788068 X[3] X[4] 4 - 37401565163865125733 X[4] + 699713323593426404161 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 2 t + 5 t + 14 t + 16 ) a[1, j] t = ----------------------------- / 4 3 2 ----- t - 17 t + 18 t + 10 t + 1 j = 0 infinity ----- 3 2 \ j -15 t + 14 t + 2 t + 20 ) a[2, j] t = ----------------------------- / 4 3 2 ----- t - 17 t + 18 t + 10 t + 1 j = 0 infinity ----- 3 2 \ j 14 t + 18 t - 19 t + 10 ) a[3, j] t = ----------------------------- / 4 3 2 ----- t - 17 t + 18 t + 10 t + 1 j = 0 infinity ----- 3 2 \ j -5 t - 6 t - t + 11 ) a[4, j] t = ----------------------------- / 4 3 2 ----- t - 17 t + 18 t + 10 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 37 Let 4 P(X[1], X[2], X[3], X[4]) = 971673014235172343 X[1] 3 3 + 3131616168151412320 X[1] X[2] + 120832410084066087 X[1] X[3] 3 2 2 + 228991759056380426 X[1] X[4] + 3782730006727183053 X[1] X[2] 2 2 + 289980821335208781 X[1] X[2] X[3] + 550261690236845544 X[1] X[2] X[4] 2 2 2 + 4378093107198306 X[1] X[3] + 20049801199316976 X[1] X[3] X[4] 2 2 3 + 19058871252484026 X[1] X[4] + 2029619459069989918 X[1] X[2] 2 2 + 231913889300925906 X[1] X[2] X[3] + 440454137082423474 X[1] X[2] X[4] 2 + 7020621275416776 X[1] X[2] X[3] + 31950326095191462 X[1] X[2] X[3] X[4] 2 3 + 30292538345523396 X[1] X[2] X[4] + 63873217416015 X[1] X[3] 2 2 + 491027881766964 X[1] X[3] X[4] + 1056094073020302 X[1] X[3] X[4] 3 4 + 644629551497888 X[1] X[4] + 408137932782848618 X[2] 3 3 + 61808109456995424 X[2] X[3] + 117436598616532276 X[2] X[4] 2 2 2 + 2813745234000639 X[2] X[3] + 12722616063982230 X[2] X[3] X[4] 2 2 3 + 12024652253411178 X[2] X[4] + 51386474840595 X[2] X[3] 2 2 + 392119941078258 X[2] X[3] X[4] + 836171603657910 X[2] X[3] X[4] 3 4 + 505818514335904 X[2] X[4] + 362404970661 X[3] 3 2 2 + 3562040561598 X[3] X[4] + 13259307164022 X[3] X[4] 3 4 + 17130428069040 X[3] X[4] + 6917442654848 X[4] + 2448943573163419 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -12 t + 13 t + 7 t - 4 ) a[1, j] t = ------------------------- / 4 3 2 ----- t + t - 6 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j 17 t - 17 t - 10 t + 4 ) a[2, j] t = ------------------------- / 4 3 2 ----- t + t - 6 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j -15 t - 13 t + 13 t - 9 ) a[3, j] t = ------------------------- / 4 3 2 ----- t + t - 6 t - 11 t + 1 j = 0 infinity ----- 3 2 \ j -20 t + 12 t - 11 t + 16 ) a[4, j] t = -------------------------- / 4 3 2 ----- t + t - 6 t - 11 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 38 Let 4 3 P(X[1], X[2], X[3], X[4]) = -291363502799224 X[1] - 32121499040084 X[1] X[2] 3 3 - 3103424144025848 X[1] X[3] - 2754541319138172 X[1] X[4] 2 2 2 - 258143895485376 X[1] X[2] - 485710680567864 X[1] X[2] X[3] 2 2 2 - 641611055898360 X[1] X[2] X[4] - 12426167775497208 X[1] X[3] 2 2 2 - 22160901523545360 X[1] X[3] X[4] - 9922635321928416 X[1] X[4] 3 2 + 71635761468895 X[1] X[2] - 1155923415884472 X[1] X[2] X[3] 2 2 - 955725574960539 X[1] X[2] X[4] - 1735512130887252 X[1] X[2] X[3] 2 - 4044448711126260 X[1] X[2] X[3] X[4] - 2200522370558499 X[1] X[2] X[4] 3 2 - 22129073720206088 X[1] X[3] - 59407523746221852 X[1] X[3] X[4] 2 3 - 53355616652033172 X[1] X[3] X[4] - 16027489601168697 X[1] X[4] 4 3 - 5101221570223 X[2] + 165396987961442 X[2] X[3] 3 2 2 + 141128112665691 X[2] X[4] - 1290684191909460 X[2] X[3] 2 2 2 - 2127081177980334 X[2] X[3] X[4] - 873459743152545 X[2] X[4] 3 2 - 1811916958735024 X[2] X[3] - 5926030015775004 X[2] X[3] X[4] 2 3 - 6185531648310678 X[2] X[3] X[4] - 2089003041928179 X[2] X[4] 4 3 - 14770084406051536 X[3] - 53008924392954792 X[3] X[4] 2 2 3 - 71563905708197976 X[3] X[4] - 43062274469584782 X[3] X[4] 4 - 9742263885436608 X[4] + 47644262326584 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -20 t - 13 t + 15 t + 7 ) a[1, j] t = ------------------------- / 4 2 ----- t + 19 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j -5 t + 2 t - 7 t + 10 ) a[2, j] t = ----------------------- / 4 2 ----- t + 19 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j 9 t - 12 t - 17 t + 15 ) a[3, j] t = ------------------------ / 4 2 ----- t + 19 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j -t + 20 t + 13 t - 20 ) a[4, j] t = ----------------------- / 4 2 ----- t + 19 t - 7 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 39 Let 4 P(X[1], X[2], X[3], X[4]) = 2777335458367632 X[1] 3 3 - 27680402592228224 X[1] X[2] - 63879970984288976 X[1] X[3] 3 2 2 - 4981101238721168 X[1] X[4] + 53358274409810496 X[1] X[2] 2 2 + 233916099267971744 X[1] X[2] X[3] + 52842711264101320 X[1] X[2] X[4] 2 2 2 + 252210017556397184 X[1] X[3] + 121999506724979592 X[1] X[3] X[4] 2 2 3 + 1696744785424564 X[1] X[4] + 96693176739816656 X[1] X[2] 2 2 + 774112411703292896 X[1] X[2] X[3] - 123279640223683864 X[1] X[2] X[4] 2 + 2065821013622569936 X[1] X[2] X[3] - 559167810594507624 X[1] X[2] X[3] X[4] 2 3 - 23078539115762164 X[1] X[2] X[4] + 1834231091715879712 X[1] X[3] 2 2 - 633703650534630776 X[1] X[3] X[4] - 54448124971823092 X[1] X[3] X[4] 3 4 + 398271252004304 X[1] X[4] - 96396243706351376 X[2] 3 3 - 1084528853098566032 X[2] X[3] - 88984089654593992 X[2] X[4] 2 2 2 - 4530602487091038688 X[2] X[3] - 674943501751255200 X[2] X[3] X[4] 2 2 3 + 76370411565889212 X[2] X[4] - 8328806227869433536 X[2] X[3] 2 2 - 1694800604768145336 X[2] X[3] X[4] + 359667732772539752 X[2] X[3] X[4] 3 4 - 2364545436177426 X[2] X[4] - 5687356548784606064 X[3] 3 2 2 - 1410257269880977992 X[3] X[4] + 423338031116372448 X[3] X[4] 3 4 - 5494350978934288 X[3] X[4] + 8282059801375 X[4] + 81457290361159696 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -15 t + 5 t - 18 t + 5 ) a[1, j] t = --------------------------- / 4 3 2 ----- t + 9 t - 15 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j -20 t + 7 t + 7 t - 20 ) a[2, j] t = --------------------------- / 4 3 2 ----- t + 9 t - 15 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j 7 t - 2 t - 4 t + 9 ) a[3, j] t = --------------------------- / 4 3 2 ----- t + 9 t - 15 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j -18 t - 16 t + 20 t + 16 ) a[4, j] t = --------------------------- / 4 3 2 ----- t + 9 t - 15 t - 7 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 40 Let 4 P(X[1], X[2], X[3], X[4]) = -6955163710012385 X[1] 3 3 + 28511784497812175 X[1] X[2] + 28984230111790772 X[1] X[3] 3 2 2 - 2124413711241528 X[1] X[4] - 18105354355133955 X[1] X[2] 2 2 - 30001949990521782 X[1] X[2] X[3] + 15761107960671843 X[1] X[2] X[4] 2 2 2 - 11918509943785158 X[1] X[3] + 16989217152631164 X[1] X[3] X[4] 2 2 3 + 694337763501837 X[1] X[4] + 991610525282720 X[1] X[2] 2 2 - 6216219930310572 X[1] X[2] X[3] - 5916183307008912 X[1] X[2] X[4] 2 - 14683758438131202 X[1] X[2] X[3] 2 - 10828846245053844 X[1] X[2] X[3] X[4] + 3036844687452408 X[1] X[2] X[4] 3 2 - 7417262535782512 X[1] X[3] - 4666446861381648 X[1] X[3] X[4] 2 3 + 3480104738834892 X[1] X[3] X[4] + 317013659457768 X[1] X[4] 4 3 + 5686892861034760 X[2] + 22782934694163448 X[2] X[3] 3 2 2 + 737653179042408 X[2] X[4] + 33777693639963042 X[2] X[3] 2 2 2 + 171416496846672 X[2] X[3] X[4] - 495157628331360 X[2] X[4] 3 2 + 22097185485654412 X[2] X[3] - 1858506824106450 X[2] X[3] X[4] 2 3 - 1037065740062688 X[2] X[3] X[4] + 210331930149864 X[2] X[4] 4 3 + 5429855833198216 X[3] - 1254824935988568 X[3] X[4] 2 2 3 - 499929164903070 X[3] X[4] + 255980176051320 X[3] X[4] 4 + 29584681507656 X[4] + 43102943135415387 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -5 t - 6 t - 8 t - 5 ) a[1, j] t = ----------------------------- / 4 3 2 ----- t + 13 t + 16 t + 11 t + 1 j = 0 infinity ----- 3 2 \ j 3 t + 17 t - 15 t - 5 ) a[2, j] t = ----------------------------- / 4 3 2 ----- t + 13 t + 16 t + 11 t + 1 j = 0 infinity ----- 3 2 \ j -5 t - 11 t + 19 t + 6 ) a[3, j] t = ----------------------------- / 4 3 2 ----- t + 13 t + 16 t + 11 t + 1 j = 0 infinity ----- 3 2 \ j 13 t - t - 2 t + 18 ) a[4, j] t = ----------------------------- / 4 3 2 ----- t + 13 t + 16 t + 11 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 41 Let 4 P(X[1], X[2], X[3], X[4]) = 187960677600356104 X[1] 3 3 - 149219522758822788 X[1] X[2] - 45169035615898304 X[1] X[3] 3 2 2 - 67452884488789068 X[1] X[4] - 162518881861980 X[1] X[2] 2 2 + 53430555564008916 X[1] X[2] X[3] + 53577625773764412 X[1] X[2] X[4] 2 2 2 - 1336886491686456 X[1] X[3] + 10328020419996900 X[1] X[3] X[4] 2 2 3 - 7627539498602424 X[1] X[4] + 10451109923249799 X[1] X[2] 2 2 - 12069916203204234 X[1] X[2] X[3] - 3704421900896193 X[1] X[2] X[4] 2 - 1611371458263264 X[1] X[2] X[3] - 15349300993545840 X[1] X[2] X[3] X[4] 2 3 + 7757692680092769 X[1] X[2] X[4] + 357268065202168 X[1] X[3] 2 2 + 1974865101840000 X[1] X[3] X[4] + 192096612786930 X[1] X[3] X[4] 3 4 + 430926131228769 X[1] X[4] + 732607793829825 X[2] 3 3 - 2018952041011854 X[2] X[3] - 3897610703014428 X[2] X[4] 2 2 2 + 1587719603354544 X[2] X[3] + 5707374780752646 X[2] X[3] X[4] 2 2 3 - 1450104052054734 X[2] X[4] - 264592085802744 X[2] X[3] 2 2 - 1098292929022800 X[2] X[3] X[4] + 160478258553270 X[2] X[3] X[4] 3 4 + 215170902323748 X[2] X[4] + 4115073937888 X[3] 3 2 2 - 8716837154952 X[3] X[4] - 136223444729472 X[3] X[4] 3 4 - 77329572577038 X[3] X[4] + 41048531828613 X[4] + 98382982566599864 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j t - 2 t - 13 t + 3 ) a[1, j] t = --------------------------- / 4 3 2 ----- t + 7 t - 9 t - 20 t + 1 j = 0 infinity ----- 2 \ j -7 t - 17 t + 11 ) a[2, j] t = --------------------------- / 4 3 2 ----- t + 7 t - 9 t - 20 t + 1 j = 0 infinity ----- 3 2 \ j -14 t + 8 t - 17 t + 20 ) a[3, j] t = --------------------------- / 4 3 2 ----- t + 7 t - 9 t - 20 t + 1 j = 0 infinity ----- 3 2 \ j 8 t + 15 t - 5 t + 3 ) a[4, j] t = --------------------------- / 4 3 2 ----- t + 7 t - 9 t - 20 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 42 Let 4 P(X[1], X[2], X[3], X[4]) = -843411212950560046 X[1] 3 3 - 3975568817117645154 X[1] X[2] + 3483488988372750890 X[1] X[3] 3 2 2 + 275730375518018704 X[1] X[4] - 6904404250200363126 X[1] X[2] 2 + 11945309119014235410 X[1] X[2] X[3] 2 2 2 + 1082752199929465512 X[1] X[2] X[4] - 4852298779151211180 X[1] X[3] 2 2 2 - 1178693393500639260 X[1] X[3] X[4] + 38478948887202984 X[1] X[4] 3 2 - 5253785637632996544 X[1] X[2] + 13529108736305431230 X[1] X[2] X[3] 2 + 1346244547275097332 X[1] X[2] X[4] 2 - 10857116148396304710 X[1] X[2] X[3] - 2846875926957702180 X[1] X[2] X[3] X[4] 2 3 + 66670343190840228 X[1] X[2] X[4] + 2639740479853330100 X[1] X[3] 2 2 + 1468735847722782960 X[1] X[3] X[4] - 61770091878762780 X[1] X[3] X[4] 3 4 - 2536162089784616 X[1] X[4] - 1481406670446819561 X[2] 3 3 + 5067795959526275460 X[2] X[3] + 538430546450794914 X[2] X[4] 2 2 2 - 6044082095933028030 X[2] X[3] - 1679979286469986950 X[2] X[3] X[4] 2 2 3 + 26817171209362824 X[2] X[4] + 2905735837800985950 X[2] X[3] 2 2 + 1704220716407776110 X[2] X[3] X[4] - 55627114871907660 X[2] X[3] X[4] 3 4 - 2974948281796176 X[2] X[4] - 462011249354990725 X[3] 3 2 2 - 548202380563932950 X[3] X[4] + 27275128067199420 X[3] X[4] 3 4 + 1785603374706400 X[3] X[4] - 38842502025736 X[4] + 917458616924041201 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -5 t + 8 t - 12 t - 11 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t + 14 t + 6 t - 16 t + 1 j = 0 infinity ----- 3 2 \ j 6 t - 2 t - 5 t + 17 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t + 14 t + 6 t - 16 t + 1 j = 0 infinity ----- 3 2 \ j t - t - 16 t + 10 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t + 14 t + 6 t - 16 t + 1 j = 0 infinity ----- 3 2 \ j -18 t + 4 t - 4 t + 7 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t + 14 t + 6 t - 16 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 43 Let 4 P(X[1], X[2], X[3], X[4]) = -4892804380775991964 X[1] 3 3 + 5958213267457539712 X[1] X[2] + 7056165306984031000 X[1] X[3] 3 2 2 + 5072008688405674546 X[1] X[4] - 2074946742910650264 X[1] X[2] 2 - 5541764949328731888 X[1] X[2] X[3] 2 2 2 - 4515730019304451230 X[1] X[2] X[4] - 3268995249383028168 X[1] X[3] 2 2 2 - 5495490688116690966 X[1] X[3] X[4] - 1943370457493345430 X[1] X[4] 3 2 + 87845038862631568 X[1] X[2] + 659354243102913288 X[1] X[2] X[3] 2 + 1015099318258224942 X[1] X[2] X[4] 2 + 1043750162615114640 X[1] X[2] X[3] + 2764063783884106320 X[1] X[2] X[3] X[4] 2 3 + 1134912517849003872 X[1] X[2] X[4] + 447036470158442704 X[1] X[3] 2 + 1678177664147124498 X[1] X[3] X[4] 2 3 + 1409160280494140202 X[1] X[3] X[4] + 326958545663728756 X[1] X[4] 4 3 + 37332971795224100 X[2] + 154851972474967136 X[2] X[3] 3 2 2 - 17300465410829818 X[2] X[4] + 207925858593975432 X[2] X[3] 2 2 2 - 143789702440502010 X[2] X[3] X[4] - 124691154045584478 X[2] X[4] 3 2 + 107476670695590728 X[2] X[3] - 235903685830327326 X[2] X[3] X[4] 2 3 - 345586446342164586 X[2] X[3] X[4] - 94627174804138054 X[2] X[4] 4 3 + 17601705373636604 X[3] - 104274867114482746 X[3] X[4] 2 2 3 - 214948264036373736 X[3] X[4] - 119187664597136080 X[3] X[4] 4 - 20414278597771333 X[4] + 166859603896599441 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 3 t + 14 t - 5 t + 5 ) a[1, j] t = ----------------------------- / 4 3 2 ----- t - 20 t - 20 t + 14 t + 1 j = 0 infinity ----- 3 2 \ j t + 18 t - 12 ) a[2, j] t = ----------------------------- / 4 3 2 ----- t - 20 t - 20 t + 14 t + 1 j = 0 infinity ----- 3 \ j -t - 16 t + 13 ) a[3, j] t = ----------------------------- / 4 3 2 ----- t - 20 t - 20 t + 14 t + 1 j = 0 infinity ----- 3 2 \ j 15 t + 17 t - t + 19 ) a[4, j] t = ----------------------------- / 4 3 2 ----- t - 20 t - 20 t + 14 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 44 Let 4 3 P(X[1], X[2], X[3], X[4]) = 57317569585664 X[1] - 1026664987382824 X[1] X[2] 3 3 - 271529897545896 X[1] X[3] - 323177564260752 X[1] X[4] 2 2 2 + 5108241126176004 X[1] X[2] + 2599975967356512 X[1] X[2] X[3] 2 2 2 + 3602695191793344 X[1] X[2] X[4] + 434147547813084 X[1] X[3] 2 2 2 + 830657415222576 X[1] X[3] X[4] + 603923516933616 X[1] X[4] 3 2 - 1551133425641164 X[1] X[2] - 724424269042908 X[1] X[2] X[3] 2 2 - 6662122653164976 X[1] X[2] X[4] - 255356794819692 X[1] X[2] X[3] 2 - 3144076466577408 X[1] X[2] X[3] X[4] - 2986434043874928 X[1] X[2] X[4] 3 2 - 49102472259036 X[1] X[3] - 441363057212736 X[1] X[3] X[4] 2 3 - 704743487224272 X[1] X[3] X[4] - 339996024263808 X[1] X[4] 4 3 - 22894786129315861 X[2] - 25341342387380556 X[2] X[3] 3 2 2 - 10201558352278812 X[2] X[4] - 11136526171843986 X[2] X[3] 2 2 2 - 8382127496894604 X[2] X[3] X[4] + 688157209457136 X[2] X[4] 3 2 - 2316542288110956 X[2] X[3] - 2137807468984356 X[2] X[3] X[4] 2 3 + 78597280641888 X[2] X[3] X[4] + 759126134511792 X[2] X[4] 4 3 - 191405110847121 X[3] - 164123612969508 X[3] X[4] 2 2 3 - 17772255922464 X[3] X[4] + 189613752473808 X[3] X[4] 4 + 57716666651664 X[4] + 18281847941427456 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 8 t + 11 t - 4 t + 10 ) a[1, j] t = --------------------------- / 4 3 2 ----- t - 17 t + 9 t - 8 t + 1 j = 0 infinity ----- 3 2 \ j 4 t - 5 t - 2 t - 4 ) a[2, j] t = --------------------------- / 4 3 2 ----- t - 17 t + 9 t - 8 t + 1 j = 0 infinity ----- 3 2 \ j -2 t - 11 t + 20 t + 4 ) a[3, j] t = --------------------------- / 4 3 2 ----- t - 17 t + 9 t - 8 t + 1 j = 0 infinity ----- 3 2 \ j -7 t + 12 t - 11 t + 14 ) a[4, j] t = --------------------------- / 4 3 2 ----- t - 17 t + 9 t - 8 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 45 Let 4 P(X[1], X[2], X[3], X[4]) = 611326640395211969 X[1] 3 3 - 3888182590858703565 X[1] X[2] + 1728137397268044559 X[1] X[3] 3 2 2 + 6098877445526823568 X[1] X[4] + 6454607044342759026 X[1] X[2] 2 - 2660346362102241826 X[1] X[2] X[3] 2 2 2 - 19490270864856651633 X[1] X[2] X[4] - 937098515348544912 X[1] X[3] 2 2 2 + 3363806511613864740 X[1] X[3] X[4] + 14634118070499548537 X[1] X[4] 3 2 - 4141840636441563352 X[1] X[2] + 1198075360713829166 X[1] X[2] X[3] 2 + 18443115123763571135 X[1] X[2] X[4] 2 + 1048183807565900043 X[1] X[2] X[3] - 2746545535316538709 X[1] X[2] X[3] X[4] 2 3 - 27306798699271237979 X[1] X[2] X[4] + 218071079562977600 X[1] X[3] 2 - 1521036010585846349 X[1] X[3] X[4] 2 3 + 1463545481246312878 X[1] X[3] X[4] + 13435174312148004396 X[1] X[4] 4 3 + 928523341929105463 X[2] - 149600405559086049 X[2] X[3] 3 2 2 - 5435864846231719189 X[2] X[4] - 252748740377194210 X[2] X[3] 2 2 2 + 300496870230970481 X[2] X[3] X[4] + 11970714341238886594 X[2] X[4] 3 2 - 140092400373246096 X[2] X[3] + 795009261209634999 X[2] X[3] X[4] 2 3 - 16681564527090855 X[2] X[3] X[4] - 11722539875208049898 X[2] X[4] 4 3 - 17045884618478773 X[3] + 192557092479756379 X[3] X[4] 2 2 3 - 574553595897200509 X[3] X[4] - 172258102245857127 X[3] X[4] 4 + 4288005905111472905 X[4] + 20973273227041080125 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -17 t + 11 t + 10 t - 13 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 16 t - 14 t - 3 t + 1 j = 0 infinity ----- 3 2 \ j -3 t + 20 t - 12 t - 19 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 16 t - 14 t - 3 t + 1 j = 0 infinity ----- 3 2 \ j 5 t + 16 t + 11 t - 16 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 16 t - 14 t - 3 t + 1 j = 0 infinity ----- 3 2 \ j 15 t + 17 t - 11 t - 3 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 16 t - 14 t - 3 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 46 Let 4 P(X[1], X[2], X[3], X[4]) = 1522464015171969 X[1] 3 3 + 43501505354375211 X[1] X[2] + 23268127137471028 X[1] X[3] 3 2 2 - 50004731046717978 X[1] X[4] + 315812754644893488 X[1] X[2] 2 2 + 102744141924135777 X[1] X[2] X[3] - 412506912714444378 X[1] X[2] X[4] 2 2 2 - 67809514846602440 X[1] X[3] + 145671537403337711 X[1] X[3] X[4] 2 2 3 - 16797790764801993 X[1] X[4] + 1618260099165220608 X[1] X[2] 2 2 + 956936416522027136 X[1] X[2] X[3] - 3710198609730958647 X[1] X[2] X[4] 2 - 197346181864204490 X[1] X[2] X[3] - 504418146610526375 X[1] X[2] X[3] X[4] 2 3 + 2149034737045297656 X[1] X[2] X[4] + 62035098548990687 X[1] X[3] 2 2 - 204531572543053501 X[1] X[3] X[4] + 431125043631219241 X[1] X[3] X[4] 3 4 - 521328632147070054 X[1] X[4] - 112869028658525985 X[2] 3 3 - 2076757155154457329 X[2] X[3] + 477045687462575055 X[2] X[4] 2 2 2 - 1636011271978623355 X[2] X[3] + 4822882381430895221 X[2] X[3] X[4] 2 2 3 - 651568234791491805 X[2] X[4] + 10695709385122205 X[2] X[3] 2 + 1250841363207433692 X[2] X[3] X[4] 2 3 - 2829780702639937618 X[2] X[3] X[4] + 327162201190624635 X[2] X[4] 4 3 - 20260276062133721 X[3] + 124187189457459185 X[3] X[4] 2 2 3 - 506815904405545191 X[3] X[4] + 682805128571975977 X[3] X[4] 4 - 68741932879334895 X[4] + 47616331112261655121 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 9 t + 8 t + t - 20 ) a[1, j] t = ------------------------- / 4 3 2 ----- t + 15 t + 9 t - t + 1 j = 0 infinity ----- 3 2 \ j -5 t - 15 t - 14 t - 2 ) a[2, j] t = ------------------------- / 4 3 2 ----- t + 15 t + 9 t - t + 1 j = 0 infinity ----- 3 2 \ j -13 t - 5 t + 2 t - 19 ) a[3, j] t = ------------------------- / 4 3 2 ----- t + 15 t + 9 t - t + 1 j = 0 infinity ----- 3 \ j -9 t - 10 t - 11 ) a[4, j] t = ------------------------- / 4 3 2 ----- t + 15 t + 9 t - t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 47 Let 4 P(X[1], X[2], X[3], X[4]) = -35114338779187343 X[1] 3 3 - 279191342418793349 X[1] X[2] - 221014462112675621 X[1] X[3] 3 2 2 + 72207290526888974 X[1] X[4] - 790384534378754124 X[1] X[2] 2 2 - 1322439350242948539 X[1] X[2] X[3] + 435683069142406035 X[1] X[2] X[4] 2 2 2 - 543699756328763541 X[1] X[3] + 392593394830366761 X[1] X[3] X[4] 2 2 3 - 69129601892081256 X[1] X[4] - 923124817076881556 X[1] X[2] 2 2 - 2528254884988244400 X[1] X[2] X[3] + 815747100679209480 X[1] X[2] X[4] 2 - 2194460554544453931 X[1] X[2] X[3] + 1569334517736899430 X[1] X[2] X[3] X[4] 2 3 - 271383763714064583 X[1] X[2] X[4] - 644391803712730151 X[1] X[3] 2 2 + 746232423543441312 X[1] X[3] X[4] - 269057988331989027 X[1] X[3] X[4] 3 4 + 28718815647696194 X[1] X[4] - 352865212581348764 X[2] 3 3 - 1496581490555310212 X[2] X[3] + 444349462626880268 X[2] X[4] 2 2 2 - 2088463951618939020 X[2] X[3] + 1423959650621552472 X[2] X[3] X[4] 2 2 3 - 226615582280219484 X[2] X[4] - 1294886695750646261 X[2] X[3] 2 2 + 1460577921302734431 X[2] X[3] X[4] - 508746512235381243 X[2] X[3] X[4] 3 4 + 53021166851080697 X[2] X[4] - 316248300099752852 X[3] 3 2 2 + 495354183047705885 X[3] X[4] - 266764131512825907 X[3] X[4] 3 4 + 56625237307679039 X[3] X[4] - 4114347748129241 X[4] + 28881478957466143044 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -15 t - 5 t - 6 t - 15 ) a[1, j] t = ---------------------------- / 4 3 2 ----- t - 8 t + 12 t - 10 t + 1 j = 0 infinity ----- 3 2 \ j 9 t - 10 t + 2 t + 9 ) a[2, j] t = ---------------------------- / 4 3 2 ----- t - 8 t + 12 t - 10 t + 1 j = 0 infinity ----- 3 2 \ j 10 t - 5 t + 17 t - 8 ) a[3, j] t = ---------------------------- / 4 3 2 ----- t - 8 t + 12 t - 10 t + 1 j = 0 infinity ----- 3 2 \ j 19 t - 8 t - 5 t - 5 ) a[4, j] t = ---------------------------- / 4 3 2 ----- t - 8 t + 12 t - 10 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 48 Let 4 P(X[1], X[2], X[3], X[4]) = 6465809671340572081 X[1] 3 3 - 42672573666582749501 X[1] X[2] + 46344382397910832375 X[1] X[3] 3 2 2 + 13017412673431089062 X[1] X[4] + 107371571556029502361 X[1] X[2] 2 - 263873461358689458400 X[1] X[2] X[3] 2 2 2 - 67981169255714053839 X[1] X[2] X[4] + 154685914771469307250 X[1] X[3] 2 2 2 + 89532444894361541650 X[1] X[3] X[4] + 8513378887538574209 X[1] X[4] 3 2 - 184846860267291950326 X[1] X[2] + 663616711768895133825 X[1] X[2] X[3] 2 + 245098357684928816861 X[1] X[2] X[4] 2 - 783795682900661079625 X[1] X[2] X[3] - 573964431756119362975 X[1] X[2] X[3] X[4] 2 3 - 108790034271839251957 X[1] X[2] X[4] + 302244230544215621625 X[1] X[3] 2 + 337955596018576102125 X[1] X[3] X[4] 2 3 + 126091101396836440775 X[1] X[3] X[4] + 17646514558685818453 X[1] X[4] 4 3 + 98164525787960685731 X[2] - 501702414263605499150 X[2] X[3] 3 2 2 - 223307036389162555863 X[2] X[4] + 938003721293606785625 X[2] X[3] 2 + 824542999352548289875 X[2] X[3] X[4] 2 2 3 + 189927603317511238859 X[2] X[4] - 759678269764002647125 X[2] X[3] 2 - 994568085829060606500 X[2] X[3] X[4] 2 3 - 456655119572768133325 X[2] X[3] X[4] - 76949807870155814722 X[2] X[4] 4 3 + 224085091427792811250 X[3] + 391746067723908048625 X[3] X[4] 2 2 3 + 268135588517975128625 X[3] X[4] + 88947501665897287850 X[3] X[4] 4 + 12482344769869133591 X[4] + 2771013736183749600625 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 6 t + 7 t - 14 t + 14 ) a[1, j] t = ----------------------------- / 4 3 2 ----- t - 10 t - 11 t - 15 t + 1 j = 0 infinity ----- 3 2 \ j -19 t - 8 t + 3 t + 13 ) a[2, j] t = ----------------------------- / 4 3 2 ----- t - 10 t - 11 t - 15 t + 1 j = 0 infinity ----- 3 2 \ j -19 t + 9 t + 12 t + 2 ) a[3, j] t = ----------------------------- / 4 3 2 ----- t - 10 t - 11 t - 15 t + 1 j = 0 infinity ----- 3 2 \ j 6 t - 18 t + 17 t + 16 ) a[4, j] t = ----------------------------- / 4 3 2 ----- t - 10 t - 11 t - 15 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 49 Let 4 P(X[1], X[2], X[3], X[4]) = 1931041895381448593 X[1] 3 3 - 8031654666482687119 X[1] X[2] + 1995027747737990788 X[1] X[3] 3 2 2 - 4421761994058594933 X[1] X[4] + 10978567796149936500 X[1] X[2] 2 - 5267177664943646904 X[1] X[2] X[3] 2 2 2 + 16859177467753395618 X[1] X[2] X[4] - 69440919428742918 X[1] X[3] 2 2 2 - 3927335707133626851 X[1] X[3] X[4] + 5720430329656261812 X[1] X[4] 3 2 - 6086342474638670806 X[1] X[2] + 1840152101738713419 X[1] X[2] X[3] 2 - 19926948971889499320 X[1] X[2] X[4] 2 + 1479337034447813784 X[1] X[2] X[3] + 7959592685124749115 X[1] X[2] X[3] X[4] 2 3 - 16268496143403738015 X[1] X[2] X[4] - 461981850404367503 X[1] X[3] 2 2 - 751382384937047640 X[1] X[3] X[4] + 1433465774846209512 X[1] X[3] X[4] 3 4 - 4315464661330745262 X[1] X[4] + 669179429542784999 X[2] 3 3 + 1427170612913768134 X[2] X[3] + 5912731864547235771 X[2] X[4] 2 2 2 - 2103582192455221512 X[2] X[3] - 1540090510715100870 X[2] X[3] X[4] 2 2 3 + 9817748493121667898 X[2] X[4] + 1039915653171134959 X[2] X[3] 2 2 + 786626238313202553 X[2] X[3] X[4] - 122460733631692980 X[2] X[3] X[4] 3 4 + 7179671982378998754 X[2] X[4] - 61573871105990917 X[3] 3 2 2 + 107656302142992513 X[3] X[4] + 168435880518779325 X[3] X[4] 3 4 - 217651858669215657 X[3] X[4] + 1095604855511332599 X[4] + 9859485570530716867041 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j 16 t - 16 t + 19 t + 7 ) a[1, j] t = -------------------------- / 4 3 2 ----- t + 5 t - 6 t + 2 t + 1 j = 0 infinity ----- 3 2 \ j 9 t + 3 t - 16 t + 17 ) a[2, j] t = -------------------------- / 4 3 2 ----- t + 5 t - 6 t + 2 t + 1 j = 0 infinity ----- 3 2 \ j -20 t - 7 t - 9 t + 18 ) a[3, j] t = -------------------------- / 4 3 2 ----- t + 5 t - 6 t + 2 t + 1 j = 0 infinity ----- 3 2 \ j -t + 13 t + 18 t - 8 ) a[4, j] t = -------------------------- / 4 3 2 ----- t + 5 t - 6 t + 2 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- Theorem Number, 50 Let 4 P(X[1], X[2], X[3], X[4]) = -67245329300062208 X[1] 3 3 - 50796134711574528 X[1] X[2] + 23338375960690688 X[1] X[3] 3 2 2 + 165583264584925184 X[1] X[4] - 7124866723736064 X[1] X[2] 2 2 + 4226826133518336 X[1] X[2] X[3] + 75129798288734208 X[1] X[2] X[4] 2 2 2 + 9017671998216192 X[1] X[3] - 105258482327703552 X[1] X[3] X[4] 2 2 3 + 35442768409073664 X[1] X[4] + 672266706627456 X[1] X[2] 2 2 + 743417708062464 X[1] X[2] X[3] - 4750460870001408 X[1] X[2] X[4] 2 - 631129110400512 X[1] X[2] X[3] - 2459731464889344 X[1] X[2] X[3] X[4] 2 3 + 16956817646897664 X[1] X[2] X[4] - 818414026316800 X[1] X[3] 2 2 + 9781438214243328 X[1] X[3] X[4] - 25861949474055168 X[1] X[3] X[4] 3 4 - 17107177788408832 X[1] X[4] + 37448713130373 X[2] 3 3 + 7265204711784 X[2] X[3] + 105653474913960 X[2] X[4] 2 2 2 - 51399939180168 X[2] X[3] + 459197901796464 X[2] X[3] X[4] 2 2 3 - 3367333855664136 X[2] X[4] - 9755184351840 X[2] X[3] 2 2 - 298647313106976 X[2] X[3] X[4] + 2303922992845536 X[2] X[3] X[4] 3 4 - 7232259409899360 X[2] X[4] + 11647078123600 X[3] 3 2 2 - 340687686683840 X[3] X[4] + 1257605621709792 X[3] X[4] 3 4 + 8218125339393856 X[3] X[4] + 805545756651088 X[4] + 941767581661986816 Define , 4, sequences a[i,j] i goes from 1 to, 4, and j from 0 to infinity in terms of the generating functions infinity ----- 3 2 \ j -3 t - 2 t + 11 t - 1 ) a[1, j] t = --------------------------- / 4 3 2 ----- t + 4 t - 5 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j -10 t - 12 t - 8 t + 12 ) a[2, j] t = --------------------------- / 4 3 2 ----- t + 4 t - 5 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j -13 t - 9 t + 3 t - 20 ) a[3, j] t = --------------------------- / 4 3 2 ----- t + 4 t - 5 t - 18 t + 1 j = 0 infinity ----- 3 2 \ j -2 t + 7 t + 17 t - 2 ) a[4, j] t = --------------------------- / 4 3 2 ----- t + 4 t - 5 t - 18 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j]) = 0 ----------------------------- ---------------------------------------- This concludes this article that took, 4.916, seconds to generate