On the Ellenberg-Gijswijt sequence that is an upper bound for the size of Su\ n bsets of, F[5] , with No Three-Term Arithemtical Progressions By Shalosh B. Ekhad In the brilliant paper "On large subsets of F_q^n with no Three-Term Arithmetic Progressions" by Jordan S. Ellenberg and Dion Gijswijt arxiv: 1605.09223v1 the authors prove that if M[d,n] is the sum of the coefficients from x^0 to \ x^d of the polynomial 4 3 2 n (x + x + x + x + 1) n then, for any d, the size of any subset of, F[5] , with no three-term arithmetical progressions is <= n 2 M[d/2, n] + 5 - M[d, n] 8 n Taking d to be , ---, and replacing n by 3m, it is easy to see that the abov\ 3 e is bounded by a constant time the sequence (4 m) 4 3 2 (3 m) A(m):=Coefficient of, x , in the polynomial , (x + x + x + x + 1) Thanks to the Almkvist-Zeilberger algorithm, the sequence A(m) satisfies the\ linear recurrence equation with polynomial coefficients of order, 4 158203125 (3 m + 11) (3 m + 8) (3 m + 5) (3 m + 2) (3 m + 10) (3 m + 7) 12 (3 m + 4) (3 m + 1) (m + 3) (m + 2) (m + 1) (187930257683468 m 11 10 9 + 6398993224079493 m + 99264460806637213 m + 927483728000827218 m 8 7 + 5812461814579871934 m + 25733909474317431189 m 6 5 + 82516715304420652549 m + 193040613885594044964 m 4 3 + 326916290228623876628 m + 390745802542804686768 m 2 + 312789749543928554688 m + 150512206199186544768 m + 32911485589560337920) A(m) - 2531250 (3 m + 11) (3 m + 8) (3 m + 5) 15 (3 m + 10) (3 m + 7) (3 m + 4) (m + 3) (m + 2) (225704239477845068 m 14 13 + 8249451460814083763 m + 138879015473911071104 m 12 11 + 1427371298136535326782 m + 10006956732145152743042 m 10 9 + 50638680251480830286624 m + 190848371101281592927042 m 8 7 + 544754629744293680445886 m + 1185497363658265059469762 m 6 5 + 1963356421241596666248013 m + 2449145350115146878689254 m 4 3 + 2254181600136763952080132 m + 1477407422571797519763528 m 2 + 648606156773964179320800 m + 169799985365935703299200 m + 19872423633375000000000) A(m + 1) + 3375 (3 m + 11) (3 m + 8) (3 m + 10) 18 17 (3 m + 7) (m + 3) (1053660870613335013412 m + 44306259915630986251183 m 16 15 + 870671972457672975223330 m + 10620417651475076601164138 m 14 13 + 90074805541825330447871410 m + 563821127165043790579001504 m 12 11 + 2698050871570250289157572774 m + 10086720477324754134926647518 m 10 9 + 29847220805418208385130348914 m + 70379672162291589540315895873 m 8 7 + 132454976685742635266796798752 m + 198309361471516963062220388920 m 6 5 + 234195778366532286477328293904 m + 214936017023089167325122761840 m 4 3 + 149704832697239442364201009344 m + 76221674460776966820210705024 m 2 + 26656078944116471883368348160 m + 5698470786418918205777664000 m + 558520545354750000000000000) A(m + 2) - 24 (4 m + 11) (3 m + 11) 18 (2 m + 5) (3 m + 10) (4 m + 9) (10832578753447240103044 m 17 16 + 466340667923918447335515 m + 9383100926711463968535873 m 15 14 + 117194169088136916752409702 m + 1017724440493290817798844628 m 13 12 + 6522136649513769339318828150 m + 31948267152753639644683926226 m 11 10 + 122233071139618606548055883364 m + 370033524382969561013319240972 m 9 8 + 892279653739381139152701415455 m + 1716383433609795156912114809829 m 7 6 + 2624879580857188700350850051166 m + 3164005208096294866018084593796 m 5 4 + 2961164497584170167423944842280 m + 2100871735432062890751055863072 m 3 2 + 1088050294424258747867291640768 m + 386368752696271788935912002560 m + 83671873214754529726815513600 m + 8280795518007390000000000000) A(m + 3) + 1024 (8 m + 31) (4 m + 15) (4 m + 11) (8 m + 29) (2 m + 7) (2 m + 5) 12 (8 m + 27) (4 m + 13) (4 m + 9) (8 m + 25) (m + 4) (187930257683468 m 11 10 9 + 4143830131877877 m + 41278932348871678 m + 245439090568464243 m 8 7 6 + 969200594451301872 m + 2674720868772696039 m + 5282496098334175462 m 5 4 3 + 7510359113534427237 m + 7613390541341053040 m + 5352831461029616364 m 2 + 2469451143149083440 m + 668245936591087200 m + 79740049050000000) A(m + 4) = 0 and in Maple notation: 158203125*(3*m+11)*(3*m+8)*(3*m+5)*(3*m+2)*(3*m+10)*(3*m+7)*(3*m+4)*(3*m+1)*(m+ 3)*(m+2)*(m+1)*(187930257683468*m^12+6398993224079493*m^11+99264460806637213*m^ 10+927483728000827218*m^9+5812461814579871934*m^8+25733909474317431189*m^7+ 82516715304420652549*m^6+193040613885594044964*m^5+326916290228623876628*m^4+ 390745802542804686768*m^3+312789749543928554688*m^2+150512206199186544768*m+ 32911485589560337920)*A(m)-2531250*(3*m+11)*(3*m+8)*(3*m+5)*(3*m+10)*(3*m+7)*(3 *m+4)*(m+3)*(m+2)*(225704239477845068*m^15+8249451460814083763*m^14+ 138879015473911071104*m^13+1427371298136535326782*m^12+10006956732145152743042* m^11+50638680251480830286624*m^10+190848371101281592927042*m^9+ 544754629744293680445886*m^8+1185497363658265059469762*m^7+ 1963356421241596666248013*m^6+2449145350115146878689254*m^5+ 2254181600136763952080132*m^4+1477407422571797519763528*m^3+ 648606156773964179320800*m^2+169799985365935703299200*m+19872423633375000000000 )*A(m+1)+3375*(3*m+11)*(3*m+8)*(3*m+10)*(3*m+7)*(m+3)*(1053660870613335013412*m ^18+44306259915630986251183*m^17+870671972457672975223330*m^16+ 10620417651475076601164138*m^15+90074805541825330447871410*m^14+ 563821127165043790579001504*m^13+2698050871570250289157572774*m^12+ 10086720477324754134926647518*m^11+29847220805418208385130348914*m^10+ 70379672162291589540315895873*m^9+132454976685742635266796798752*m^8+ 198309361471516963062220388920*m^7+234195778366532286477328293904*m^6+ 214936017023089167325122761840*m^5+149704832697239442364201009344*m^4+ 76221674460776966820210705024*m^3+26656078944116471883368348160*m^2+ 5698470786418918205777664000*m+558520545354750000000000000)*A(m+2)-24*(4*m+11)* (3*m+11)*(2*m+5)*(3*m+10)*(4*m+9)*(10832578753447240103044*m^18+ 466340667923918447335515*m^17+9383100926711463968535873*m^16+ 117194169088136916752409702*m^15+1017724440493290817798844628*m^14+ 6522136649513769339318828150*m^13+31948267152753639644683926226*m^12+ 122233071139618606548055883364*m^11+370033524382969561013319240972*m^10+ 892279653739381139152701415455*m^9+1716383433609795156912114809829*m^8+ 2624879580857188700350850051166*m^7+3164005208096294866018084593796*m^6+ 2961164497584170167423944842280*m^5+2100871735432062890751055863072*m^4+ 1088050294424258747867291640768*m^3+386368752696271788935912002560*m^2+ 83671873214754529726815513600*m+8280795518007390000000000000)*A(m+3)+1024*(8*m+ 31)*(4*m+15)*(4*m+11)*(8*m+29)*(2*m+7)*(2*m+5)*(8*m+27)*(4*m+13)*(4*m+9)*(8*m+ 25)*(m+4)*(187930257683468*m^12+4143830131877877*m^11+41278932348871678*m^10+ 245439090568464243*m^9+969200594451301872*m^8+2674720868772696039*m^7+ 5282496098334175462*m^6+7510359113534427237*m^5+7613390541341053040*m^4+ 5352831461029616364*m^3+2469451143149083440*m^2+668245936591087200*m+ 79740049050000000)*A(m+4) = 0 The growth-constant is the cubic root of the largest root of the algebraic e\ quation in, M 4 3 2 4294967296 M - 398417930496 M + 1532720404125 M - 2216182781250 M + 1037970703125 = 0 and in Maple notation 4294967296*M^4-398417930496*M^3+1532720404125*M^2-2216182781250*M+1037970703125 = 0 that happens to be equal to (1/3) 27 RootOf( 4 3 2 4294967296 _Z - 14756219648 _Z + 2102497125 _Z - 112593750 _Z + 1953125, (1/3) index = 2) and in Maple notation 27^(1/3)*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+ 1953125,index = 2)^(1/3) that equals 4.4615777657025778113 as you can see, this is less than, 5, . n so indeed the size of the largest such set it is exponentially less than, 5 The asymptotic expansion to order, 3, of the sequence A(m) is / 3 m m 1/2 |18782723397369913978599436713984 %1 %2 27 %1 (1/m) |------------------------------------ | %2 \ 3 36020581823962002742886369675182080 %1 + ----------------------------------- ---- 36941 m %2 2 62394340671541538125860135796992 %1 - ------------------------------------ %2 3 49384676101634233901561043240635943370096640 %1 - -------------------------------------------- ----- 8896071738639 2 m %2 2 131715637858871153376696983355971040 %1 - ------------------------------------ ---- 36941 m %2 2162433079051447972840160583375 %1 11540550305198190008884860000000 m + ---------------------------------- + ---------------------------------- %2 %2 2 888147137441167172511141168586422771607187915 %1 + --------------------------------------------- ----- 47445715939408 2 m %2 340819931508738950777098300282250625 %1 + ------------------------------------ ---- 295528 m %2 1182066228048047304863174890625 - ------------------------------- %2 55615380253892619198384352456121310003712699375 %1 - ----------------------------------------------- ----- 36438309841465344 2 m %2 315774733405193936709221446687421875 1 + ------------------------------------ ---- 5319504 m %2 \ 153071447905761200305089319875622450190496640625 1 | + ------------------------------------------------ -----|/m 2951503097158692864 2 | m %2/ %1 := RootOf( 4 3 2 4294967296 _Z - 14756219648 _Z + 2102497125 _Z - 112593750 _Z + 1953125, index = 2) 3 %2 := 18782723397369913978599436713984 %1 2 - 62394340671541538125860135796992 %1 + 2162433079051447972840160583375 %1 - 1182066228048047304863174890625 and in Maple notation (18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*27^m*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+ 1953125,index = 2)^m/m*(1/m)^(1/2)*(18782723397369913978599436713984/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^3+36020581823962002742886369675182080/36941/m/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^3-62394340671541538125860135796992/(18782723397369913978599436713984 *RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^3-62394340671541538125860135796992*RootOf(4294967296*_Z^4-\ 14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^2-49384676101634233901561043240635943370096640/8896071738639/m^2/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^3-131715637858871153376696983355971040/36941/m/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^2+2162433079051447972840160583375/(18782723397369913978599436713984* RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^3-62394340671541538125860135796992*RootOf(4294967296*_Z^4-\ 14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)+11540550305198190008884860000000*m/(18782723397369913978599436713984 *RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^3-62394340671541538125860135796992*RootOf(4294967296*_Z^4-\ 14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )+888147137441167172511141168586422771607187915/47445715939408/m^2/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^2+340819931508738950777098300282250625/295528/m/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)-1182066228048047304863174890625/(18782723397369913978599436713984* RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)^3-62394340671541538125860135796992*RootOf(4294967296*_Z^4-\ 14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )-55615380253892619198384352456121310003712699375/36438309841465344/m^2/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )*RootOf(4294967296*_Z^4-14756219648*_Z^3+2102497125*_Z^2-112593750*_Z+1953125, index = 2)+315774733405193936709221446687421875/5319504/m/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )+153071447905761200305089319875622450190496640625/2951503097158692864/m^2/( 18782723397369913978599436713984*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^3-\ 62394340671541538125860135796992*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)^2+ 2162433079051447972840160583375*RootOf(4294967296*_Z^4-14756219648*_Z^3+ 2102497125*_Z^2-112593750*_Z+1953125,index = 2)-1182066228048047304863174890625 )) and in floating-point -.697078762e30*27.^m*3.289286006^m/m*(1/m)^(1/2)*(.999999988+.3322000e-1/m-.\ 34328e-2/m^2-16.55559018*m) 1 Note that we even gained a factor of, ---- 1/2 n hence be have an improvement on Ellenberg-Gijswijt! For the sake our beloved OEIS, the first, 30, terms of the upper bound discussed at the beginning are (recall that we doing only n multiple of 3) [90, 6546, 509175, 40870662, 3342804870, 276880835775, 23144061459357, 1948045893535950, 164866452315130890, 14014806094592031846, 1195726139315697010650, 102332786458350447066975, 8780941279035901133744325, 755189083279470616825151685, 65077929078217624100380577700, 5617883784772252682825252313150, 485722029374412023473959613313760, 42054021198143522382135955388794050, 3645620312496436774337593581171285417, 316393646645162334702653741022270417522, 27487284604030233107924119241703068805570, 2390258692234706631820222465258060919219730, 208033388165158336038238013034122626157535450, 18120343165772076728849765998335923890492026975, 1579496447067839502454231320492420971329320883620, 137773693562169830552243765740355491112355474799425, 12025092571571052707906532845763456771864139712810820, 1050186093328252818092751603691773296996824861344091125, 91766233904167328761997001986446951191425416806710929905, 8022747045638173259203212272030001596231492112920046840600] The first, 30, terms of the above-mentioned sequence, A(m) are [15, 951, 69675, 5386887, 429236745, 34871149275, 2871408900522, 238803579061575, 20012426485010115, 1687259064324248721, 142952642274924551700, 12160858387192980968475, 1038050406637202270901825, 88866278127468913677255810, 7626877846943507681184687075, 656006325324370232994899912775, 56533535135217110482595899243995, 4880295302225029056991230388089675, 421937557232602144337647112082230217, 36529596097993940775323599416023616897, 3166492524413488433809102261713268679295, 274788966586411884187182213151395983517780, 23870650264702675392385408444448023320979200, 2075562080151420387035098865576493836398664475, 180626178175381923459862619284196343492556821120, 15731494515220827249577309288732463461865236030425, 1371127128636423559022291089655087766946120102306920, 119586140774070594575310288680448643877829517342229250, 10436592312473122432354528946766577056557301433563246405, 911365898031496034654710479313631566315589412819778037475] Finally, just for fun,, A(1000), equals 1623838841980905261619551168679817647865114147181480351018869133779472117835\ 874620457931340764460018358751395309728549172811407744080806015902475929\ 361881125113166942751453626185033006820220881432131216800168069138625295\ 266924477493104734605969043827519969956216645094722566275980967327089239\ 213205504767727122168476607413646083617185043867766907561817776785451768\ 706557062761165002832800149136549287877726506017794250674187927414974591\ 605001600292858948740984645720063364443728960677598355221882044044089857\ 745822846557737848268519306906430712020084041108027842615753912088668203\ 985635835078275847531234046121367706209823000474057462198222151196004886\ 403199418459215903010791674560966011881588903950060534186153234382324186\ 110438808192853012966318638571719819762727522584470602949337694595051490\ 834915647398355712012100123524315999216580948634241140968993950192971989\ 243797304495046577242718853907259132637446997927266059046914946719107157\ 412409995398347423082769480872591692217761726458711817335172816843862536\ 438849280722973808764620471178548352825661426713823283429933592651109193\ 918924142800878078678984944538487169359374995681780372879818814675054002\ 367088143212107277452506794945203275709221850565457055936570740948142287\ 920225422581594615625344972914538960358159682387365433341144342735488444\ 280035232081224153726467236344327888577751152326515419585267047673659061\ 047522328513054203070665876240251737842174190033365829478082233786835318\ 041562747897343785308244520987605004461109633017360844272899607277962107\ 903150067629112611024998201440243472538294744546505218588991836169407430\ 788044956137708181830289634488038948479513078158859983714426386360500788\ 603022344261731399663138459071422567688264703792681960134161582879047224\ 253540679101372139552522858520787216284089436417280477532604849577521817\ 699049525027793850608981520063624778605728875043753620927504885202189158\ 76209819472353923339260875620042070594356327807160597176213060884113915 --------------------------------------------------- This ends this article that took, 11.330, seconds to generate.