Solutions of a selection of , 100, diophantine equations with, 3, variables of degree, 3 By Shalosh B. Ekhad Theorem Number, 1 Let 3 2 P(X[1], X[2], X[3]) = 42945169 X[1] + 250329976 X[1] X[2] 2 2 - 237828888 X[1] X[3] + 466934053 X[1] X[2] - 864636931 X[1] X[2] X[3] 2 3 2 + 393100037 X[1] X[3] + 282015605 X[2] - 768865951 X[2] X[3] 2 3 + 689194738 X[2] X[3] - 203621149 X[3] + 411744547 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 15 t + 13 t - 15 ) a[1, j] t = --------------------- / 3 2 ----- -t + 11 t + 8 t + 1 j = 0 infinity ----- 2 \ j 6 t - 16 t + 17 ) a[2, j] t = --------------------- / 3 2 ----- -t + 11 t + 8 t + 1 j = 0 infinity ----- 2 \ j 19 t - 4 t + 11 ) a[3, j] t = --------------------- / 3 2 ----- -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]) = 0 ----------------------------- Theorem Number, 2 Let 3 2 P(X[1], X[2], X[3]) = 87315731 X[1] - 452021548 X[1] X[2] 2 2 - 313578013 X[1] X[3] + 749836733 X[1] X[2] + 1040183655 X[1] X[2] X[3] 2 3 2 + 360065524 X[1] X[3] - 402904377 X[2] - 839432384 X[2] X[3] 2 3 - 581393771 X[2] X[3] - 133849919 X[3] + 35611289 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -2 t - 3 t + 13 ) a[1, j] t = --------------------- / 3 2 ----- -t + 5 t + 16 t + 1 j = 0 infinity ----- 2 \ j 8 t + 5 t + 4 ) a[2, j] t = --------------------- / 3 2 ----- -t + 5 t + 16 t + 1 j = 0 infinity ----- 2 \ j -13 t - 7 t + 8 ) a[3, j] t = --------------------- / 3 2 ----- -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]) = 0 ----------------------------- Theorem Number, 3 Let 3 2 2 P(X[1], X[2], X[3]) = 7038328 X[1] - 7390196 X[1] X[2] - 26375580 X[1] X[3] 2 2 - 26959350 X[1] X[2] + 40475820 X[1] X[2] X[3] + 9243850 X[1] X[3] 3 2 2 + 14281425 X[2] + 77215425 X[2] X[3] - 79121525 X[2] X[3] 3 + 14203875 X[3] + 4678731200 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -15 t - 20 t - 5 ) a[1, j] t = --------------------- / 3 2 ----- -t - 8 t - 12 t + 1 j = 0 infinity ----- 2 \ j 7 t + 18 t - 8 ) a[2, j] t = --------------------- / 3 2 ----- -t - 8 t - 12 t + 1 j = 0 infinity ----- 2 \ j -7 t - 18 t - 14 ) a[3, j] t = --------------------- / 3 2 ----- -t - 8 t - 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 4 Let 3 2 P(X[1], X[2], X[3]) = -3157249752 X[1] - 9094390272 X[1] X[2] 2 2 + 4147319664 X[1] X[3] - 8262931846 X[1] X[2] 2 3 + 10551581364 X[1] X[2] X[3] - 681181240 X[1] X[3] - 2131843527 X[2] 2 2 3 + 6488638856 X[2] X[3] - 1198034844 X[2] X[3] + 539419976 X[3] + 568075001768 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -18 t + 5 t + 19 ) a[1, j] t = -------------------- / 3 2 ----- -t - 2 t - 9 t + 1 j = 0 infinity ----- 2 \ j 16 t + 16 t - 18 ) a[2, j] t = -------------------- / 3 2 ----- -t - 2 t - 9 t + 1 j = 0 infinity ----- 2 \ j -11 t + 5 t - 11 ) a[3, j] t = -------------------- / 3 2 ----- -t - 2 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]) = 0 ----------------------------- Theorem Number, 5 Let 3 2 P(X[1], X[2], X[3]) = -3487505944 X[1] - 6484735956 X[1] X[2] 2 2 + 7917149264 X[1] X[3] - 4009145116 X[1] X[2] 2 3 + 9808852300 X[1] X[2] X[3] - 5990896680 X[1] X[3] - 824321023 X[2] 2 2 3 + 3030620452 X[2] X[3] - 3709221884 X[2] X[3] + 1511079848 X[3] + 55742968 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -12 t - 19 t - 13 ) a[1, j] t = --------------------- / 3 2 ----- -t + 13 t + 6 t + 1 j = 0 infinity ----- 2 \ j 18 t + 20 t ) a[2, j] t = --------------------- / 3 2 ----- -t + 13 t + 6 t + 1 j = 0 infinity ----- 2 \ j -2 t - 8 t - 17 ) a[3, j] t = --------------------- / 3 2 ----- -t + 13 t + 6 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 6 Let 3 2 P(X[1], X[2], X[3]) = 315904771 X[1] - 912321475 X[1] X[2] 2 2 + 731764033 X[1] X[3] + 807273950 X[1] X[2] - 1388033525 X[1] X[2] X[3] 2 3 2 + 586113878 X[1] X[3] - 200248375 X[2] + 592733450 X[2] X[3] 2 3 - 533236425 X[2] X[3] + 159209491 X[3] + 7472058875 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 4 t - 9 t - 12 ) a[1, j] t = --------------------- / 3 2 ----- -t - 6 t - 11 t + 1 j = 0 infinity ----- 2 \ j 5 t + 20 t - 13 ) a[2, j] t = --------------------- / 3 2 ----- -t - 6 t - 11 t + 1 j = 0 infinity ----- 2 \ j -4 t + 19 t - 3 ) a[3, j] t = --------------------- / 3 2 ----- -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]) = 0 ----------------------------- Theorem Number, 7 Let 3 2 P(X[1], X[2], X[3]) = -575722381 X[1] + 772091849 X[1] X[2] 2 2 - 662715583 X[1] X[3] + 656640137 X[1] X[2] - 1714012238 X[1] X[2] X[3] 2 3 2 + 1059379641 X[1] X[3] + 112648819 X[2] - 482268415 X[2] X[3] 2 3 + 703279609 X[2] X[3] - 339258581 X[3] + 7856862272 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- \ j 2 + 13 t ) a[1, j] t = --------------------- / 3 2 ----- -t - 4 t - 15 t + 1 j = 0 infinity ----- 2 \ j -17 t + 20 t - 12 ) a[2, j] t = --------------------- / 3 2 ----- -t - 4 t - 15 t + 1 j = 0 infinity ----- 2 \ j -15 t + 15 t - 2 ) a[3, j] t = --------------------- / 3 2 ----- -t - 4 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]) = 0 ----------------------------- Theorem Number, 8 Let 3 2 P(X[1], X[2], X[3]) = 1662737933 X[1] + 12581121447 X[1] X[2] 2 2 + 3984924165 X[1] X[3] + 31725504615 X[1] X[2] 2 3 + 20100536490 X[1] X[2] X[3] + 3178868991 X[1] X[3] + 26662052173 X[2] 2 2 3 + 25342680933 X[2] X[3] + 8016740703 X[2] X[3] + 844024111 X[3] + 24897088 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -19 t + 19 t - 11 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 15 t + 15 t + 1 j = 0 infinity ----- 2 \ j 13 t - 5 t + 2 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 15 t + 15 t + 1 j = 0 infinity ----- 2 \ j -18 t - 10 t + 7 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 15 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]) = 0 ----------------------------- Theorem Number, 9 Let 3 2 P(X[1], X[2], X[3]) = 442592750 X[1] + 430773225 X[1] X[2] 2 2 + 480335925 X[1] X[3] - 336861540 X[1] X[2] + 36923160 X[1] X[2] X[3] 2 3 2 + 100730340 X[1] X[3] - 93980412 X[2] - 318511908 X[2] X[3] 2 3 - 149622024 X[2] X[3] - 15907024 X[3] + 11212884000 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -10 t + 16 t - 10 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 18 t - 16 t + 1 j = 0 infinity ----- 2 \ j t + 6 t - 17 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 18 t - 16 t + 1 j = 0 infinity ----- 2 \ j 13 t + 8 t + 19 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 18 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]) = 0 ----------------------------- Theorem Number, 10 Let 3 2 2 P(X[1], X[2], X[3]) = 8558753 X[1] - 10760409 X[1] X[2] - 22191330 X[1] X[3] 2 2 - 12153201 X[1] X[2] + 37151778 X[1] X[2] X[3] + 13887783 X[1] X[3] 3 2 2 3 - 2151631 X[2] + 17663748 X[2] X[3] - 30379527 X[2] X[3] + 631071 X[3] + 84027672 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 19 t + 11 t - 15 ) a[1, j] t = --------------------- / 3 2 ----- -t + 9 t - 10 t + 1 j = 0 infinity ----- 2 \ j 11 t - 5 t + 3 ) a[2, j] t = --------------------- / 3 2 ----- -t + 9 t - 10 t + 1 j = 0 infinity ----- 2 \ j 20 t + 2 t - 10 ) a[3, j] t = --------------------- / 3 2 ----- -t + 9 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]) = 0 ----------------------------- Theorem Number, 11 Let 3 2 P(X[1], X[2], X[3]) = -697042114 X[1] - 4220036543 X[1] X[2] 2 2 - 2081103830 X[1] X[3] - 7872217147 X[1] X[2] 2 3 - 5753725315 X[1] X[2] X[3] + 296028850 X[1] X[3] - 4714415821 X[2] 2 2 3 - 3906211855 X[2] X[3] + 580003775 X[2] X[3] - 231783500 X[3] + 40885346125 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -19 t - 16 t - 20 ) a[1, j] t = --------------------- / 3 2 ----- -t - 4 t - 18 t + 1 j = 0 infinity ----- 2 \ j 11 t - t + 15 ) a[2, j] t = --------------------- / 3 2 ----- -t - 4 t - 18 t + 1 j = 0 infinity ----- 2 \ j -4 t + 10 t + 9 ) a[3, j] t = --------------------- / 3 2 ----- -t - 4 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]) = 0 ----------------------------- Theorem Number, 12 Let 3 2 P(X[1], X[2], X[3]) = -9649090121 X[1] + 12676665456 X[1] X[2] 2 2 - 39963396621 X[1] X[3] - 4776808896 X[1] X[2] 2 3 + 34403986656 X[1] X[2] X[3] - 49888720203 X[1] X[3] + 520915968 X[2] 2 2 3 - 5861588544 X[2] X[3] + 19640165040 X[2] X[3] - 18811690591 X[3] + 145199817216 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -9 t - 19 t + 18 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 18 t + 10 t + 1 j = 0 infinity ----- 2 \ j 14 t + 5 t + 16 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 18 t + 10 t + 1 j = 0 infinity ----- 2 \ j 15 t + 5 t - 6 ) a[3, j] t = ---------------------- / 3 2 ----- -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]) = 0 ----------------------------- Theorem Number, 13 Let 3 2 P(X[1], X[2], X[3]) = 122880451 X[1] + 297730040 X[1] X[2] 2 2 - 248138447 X[1] X[3] - 83829050 X[1] X[2] - 380599090 X[1] X[2] X[3] 2 3 2 + 126074441 X[1] X[3] - 266506250 X[2] - 123121500 X[2] X[3] 2 3 + 126692300 X[2] X[3] - 2465195 X[3] + 16405872725 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -12 t + 2 t - 7 ) a[1, j] t = --------------------- / 3 2 ----- -t - 13 t + 3 t + 1 j = 0 infinity ----- 2 \ j 13 t + 11 t + 1 ) a[2, j] t = --------------------- / 3 2 ----- -t - 13 t + 3 t + 1 j = 0 infinity ----- 2 \ j 13 t - 18 t - 12 ) a[3, j] t = --------------------- / 3 2 ----- -t - 13 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]) = 0 ----------------------------- Theorem Number, 14 Let 2 P(X[1], X[2], X[3]) = 100959504 X[1] c[4] - 1449880800 X[1] X[2] c[4] 2 - 565990332 X[1] X[3] c[4] - 11674850052 X[2] c[4] 2 - 9198987198 X[2] X[3] c[4] - 1811997399 X[3] c[4] - 202542240 X[1] X[2] 2 - 91808524 X[1] X[3] + 1269280548 X[1] c[4] - 1766862852 X[2] 2 - 1611946558 X[2] X[3] + 22825259640 X[2] c[4] - 370095403 X[3] + 10919626002 X[3] c[4] + 50597348 X[1] + 909883640 X[2] + 435289202 X[3] + 100959504 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 16 t + 3 t - 12 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 18 t - 16 t + 1 j = 0 infinity ----- 2 \ j 8 t - 5 t + 9 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 18 t - 16 t + 1 j = 0 infinity ----- 2 \ j -18 t + 20 t - 18 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 18 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]) = 0 ----------------------------- Theorem Number, 15 Let 3 2 2 P(X[1], X[2], X[3]) = -3050927 X[1] + 6430729 X[1] X[2] - 4124042 X[1] X[3] 2 2 + 7337307 X[1] X[2] + 25234300 X[1] X[2] X[3] + 7341360 X[1] X[3] 3 2 2 - 16197109 X[2] - 31224466 X[2] X[3] - 9845264 X[2] X[3] 3 + 1915784 X[3] + 118251328 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -16 t - 11 t + 20 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 14 t + 19 t + 1 j = 0 infinity ----- 2 \ j -20 t + 9 t + 12 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 14 t + 19 t + 1 j = 0 infinity ----- 2 \ j 20 t - 12 t - 2 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 14 t + 19 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 16 Let 3 2 P(X[1], X[2], X[3]) = -2068355393 X[1] + 7342445097 X[1] X[2] 2 2 + 2092132692 X[1] X[3] - 8361106455 X[1] X[2] 2 3 - 3987410388 X[1] X[2] X[3] + 269863620 X[1] X[3] + 3005577847 X[2] 2 2 3 + 1428569040 X[2] X[3] - 1074928956 X[2] X[3] - 423981368 X[3] + 25776946664 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 20 t + 3 t + 16 ) a[1, j] t = --------------------- / 3 2 ----- -t - 12 t - 9 t + 1 j = 0 infinity ----- 2 \ j 10 t - 9 t + 16 ) a[2, j] t = --------------------- / 3 2 ----- -t - 12 t - 9 t + 1 j = 0 infinity ----- 2 \ j 7 t + 7 t - 13 ) a[3, j] t = --------------------- / 3 2 ----- -t - 12 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]) = 0 ----------------------------- Theorem Number, 17 Let 3 2 2 P(X[1], X[2], X[3]) = -778176 X[1] + 571632 X[1] X[2] + 3170592 X[1] X[3] 2 2 + 24956 X[1] X[2] - 1861296 X[1] X[2] X[3] - 4112784 X[1] X[3] 3 2 2 3 - 17239 X[2] + 52522 X[2] X[3] + 1364652 X[2] X[3] + 1721016 X[3] + 557568 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -7 t + 9 t - 12 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 12 t - 19 t + 1 j = 0 infinity ----- \ j -6 - 12 t ) a[2, j] t = ---------------------- / 3 2 ----- -t + 12 t - 19 t + 1 j = 0 infinity ----- 2 \ j -6 t + 8 t - 7 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 12 t - 19 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 18 Let 3 2 P(X[1], X[2], X[3]) = 595150645 X[1] + 3136217776 X[1] X[2] 2 2 - 4083463776 X[1] X[3] - 5174991104 X[1] X[2] 2 3 - 3613326336 X[1] X[2] X[3] + 7026990080 X[1] X[3] + 2552434688 X[2] 2 2 3 + 6195552256 X[2] X[3] + 652787712 X[2] X[3] - 4369186816 X[3] + 476013658112 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 16 t ) a[1, j] t = --------------------- / 3 2 ----- -t - 4 t - 16 t + 1 j = 0 infinity ----- 2 \ j t - 18 t - 16 ) a[2, j] t = --------------------- / 3 2 ----- -t - 4 t - 16 t + 1 j = 0 infinity ----- 2 \ j 7 t - 17 t + 12 ) a[3, j] t = --------------------- / 3 2 ----- -t - 4 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]) = 0 ----------------------------- Theorem Number, 19 Let 3 2 P(X[1], X[2], X[3]) = -645492544 X[1] + 1991776496 X[1] X[2] 2 2 + 481448000 X[1] X[3] - 2195156892 X[1] X[2] - 901639456 X[1] X[2] X[3] 2 3 2 + 162049600 X[1] X[3] + 764521493 X[2] + 140885572 X[2] X[3] 2 3 - 509654544 X[2] X[3] - 196350272 X[3] + 46200848384 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j t - 17 ) a[1, j] t = --------------------- / 3 2 ----- -t + 13 t + 3 t + 1 j = 0 infinity ----- 2 \ j -8 t + 12 t - 16 ) a[2, j] t = --------------------- / 3 2 ----- -t + 13 t + 3 t + 1 j = 0 infinity ----- 2 \ j 13 t + 19 t + 13 ) a[3, j] t = --------------------- / 3 2 ----- -t + 13 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]) = 0 ----------------------------- Theorem Number, 20 Let 3 2 P(X[1], X[2], X[3]) = 86677183 X[1] + 213579874 X[1] X[2] 2 2 - 1086571734 X[1] X[3] + 156240724 X[1] X[2] - 1847150968 X[1] X[2] X[3] 2 3 2 + 4119680244 X[1] X[3] + 34416408 X[2] - 729551384 X[2] X[3] 2 3 + 3419649544 X[2] X[3] - 4621498568 X[3] + 3950784000 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 14 t + 8 t - 20 ) a[1, j] t = --------------------- / 3 2 ----- -t + 8 t + 16 t + 1 j = 0 infinity ----- 2 \ j -19 t + 13 t + 8 ) a[2, j] t = --------------------- / 3 2 ----- -t + 8 t + 16 t + 1 j = 0 infinity ----- \ j -2 + 11 t ) a[3, j] t = --------------------- / 3 2 ----- -t + 8 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]) = 0 ----------------------------- Theorem Number, 21 Let 3 2 P(X[1], X[2], X[3]) = -124504923 X[1] + 235204406 X[1] X[2] 2 2 - 1077415061 X[1] X[3] - 111388228 X[1] X[2] + 1368036804 X[1] X[2] X[3] 2 3 2 - 2732199905 X[1] X[3] + 999736 X[2] - 300556404 X[2] X[3] 2 3 + 1974737710 X[2] X[3] - 1976873975 X[3] + 1427628376 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 7 t - 9 t + 19 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 17 t + 15 t + 1 j = 0 infinity ----- 2 \ j -7 t - 4 t + 18 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 17 t + 15 t + 1 j = 0 infinity ----- 2 \ j -3 t - 3 t - 1 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 17 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]) = 0 ----------------------------- Theorem Number, 22 Let 3 2 P(X[1], X[2], X[3]) = 323186688 X[1] + 895189248 X[1] X[2] 2 2 + 696611328 X[1] X[3] + 542287728 X[1] X[2] + 698793408 X[1] X[2] X[3] 2 3 2 + 53030336 X[1] X[3] - 30989205 X[2] - 437379630 X[2] X[3] 2 3 - 853020860 X[2] X[3] - 382961000 X[3] + 11000295424 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -8 t + 15 t - 11 ) a[1, j] t = --------------------- / 3 2 ----- -t - 7 t + 13 t + 1 j = 0 infinity ----- 2 \ j 16 t - 8 t + 2 ) a[2, j] t = --------------------- / 3 2 ----- -t - 7 t + 13 t + 1 j = 0 infinity ----- 2 \ j -16 t + 12 t + 7 ) a[3, j] t = --------------------- / 3 2 ----- -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]) = 0 ----------------------------- Theorem Number, 23 Let 3 2 P(X[1], X[2], X[3]) = 7597235 X[1] - 145273430 X[1] X[2] 2 2 - 463555140 X[1] X[3] - 4345483180 X[1] X[2] + 3522386370 X[1] X[2] X[3] 2 3 2 + 7124666590 X[1] X[3] + 3747164522 X[2] - 294805666 X[2] X[3] 2 3 - 7351841094 X[2] X[3] - 3519625502 X[3] + 449983383247 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -13 t - 18 t - 5 ) a[1, j] t = --------------------- / 3 2 ----- -t - 7 t + 17 t + 1 j = 0 infinity ----- 2 \ j -7 t + 20 t - 12 ) a[2, j] t = --------------------- / 3 2 ----- -t - 7 t + 17 t + 1 j = 0 infinity ----- 2 \ j -6 t + 3 t + 14 ) a[3, j] t = --------------------- / 3 2 ----- -t - 7 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]) = 0 ----------------------------- Theorem Number, 24 Let 3 2 P(X[1], X[2], X[3]) = -66747061 X[1] + 226667419 X[1] X[2] 2 2 - 205852886 X[1] X[3] - 254987870 X[1] X[2] + 461544869 X[1] X[2] X[3] 2 3 2 - 208529313 X[1] X[3] + 94907339 X[2] - 256581759 X[2] X[3] 2 3 + 230720522 X[2] X[3] - 68972699 X[3] + 101847563 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 14 t - 2 t - 9 ) a[1, j] t = -------------------- / 3 2 ----- -t + 6 t + 5 t + 1 j = 0 infinity ----- 2 \ j 15 t + 10 t - 13 ) a[2, j] t = -------------------- / 3 2 ----- -t + 6 t + 5 t + 1 j = 0 infinity ----- 2 \ j 3 t + 17 t - 4 ) a[3, j] t = -------------------- / 3 2 ----- -t + 6 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]) = 0 ----------------------------- Theorem Number, 25 Let 3 2 P(X[1], X[2], X[3]) = 185600732 X[1] - 404575384 X[1] X[2] 2 2 - 442533028 X[1] X[3] + 99279270 X[1] X[2] + 107336570 X[1] X[2] X[3] 2 3 2 - 133951052 X[1] X[3] + 81555419 X[2] + 180392135 X[2] X[3] 2 3 + 80906837 X[2] X[3] - 4749393 X[3] + 132729045301 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -6 t - 17 t - 4 ) a[1, j] t = ------------------ / 3 2 ----- -t - 5 t - t + 1 j = 0 infinity ----- 2 \ j t + 13 t - 18 ) a[2, j] t = ------------------ / 3 2 ----- -t - 5 t - t + 1 j = 0 infinity ----- 2 \ j 12 t - 11 t + 7 ) a[3, j] t = ------------------ / 3 2 ----- -t - 5 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]) = 0 ----------------------------- Theorem Number, 26 Let 3 2 2 P(X[1], X[2], X[3]) = 653265 X[1] + 2994938 X[1] X[2] - 2601523 X[1] X[3] 2 2 + 4540331 X[1] X[2] - 7903915 X[1] X[2] X[3] + 3441294 X[1] X[3] 3 2 2 3 + 2282227 X[2] - 5972625 X[2] X[3] + 5212792 X[2] X[3] - 1517239 X[3] + 117649 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 3 t + 9 t + 4 ) a[1, j] t = -------------------- / 3 2 ----- -t - 3 t - 4 t + 1 j = 0 infinity ----- 2 \ j -7 t - 4 t - 11 ) a[2, j] t = -------------------- / 3 2 ----- -t - 3 t - 4 t + 1 j = 0 infinity ----- 2 \ j -6 t + 2 t - 9 ) a[3, j] t = -------------------- / 3 2 ----- -t - 3 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]) = 0 ----------------------------- Theorem Number, 27 Let 2 P(X[1], X[2], X[3]) = 604841094 X[1] c[9] + 1127346012 X[1] X[2] c[9] 2 - 225287244 X[1] X[3] c[9] + 515169720 X[2] c[9] 2 2 - 194505948 X[2] X[3] c[9] + 15093702 X[3] c[9] - 387534144 X[1] - 703118601 X[1] X[2] + 18499658 X[1] X[3] + 1784443284 X[1] c[9] 2 - 319151202 X[2] + 17032153 X[2] X[3] + 1620339552 X[2] c[9] - 52730028 X[3] c[9] + 41456829 X[1] + 37644312 X[2] - 1225043 X[3] + 15093702 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -18 t - 2 t + 5 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 16 t - 14 t + 1 j = 0 infinity ----- 2 \ j 20 t - t - 5 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 16 t - 14 t + 1 j = 0 infinity ----- 2 \ j 12 t + 9 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 16 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]) = 0 ----------------------------- Theorem Number, 28 Let 3 2 P(X[1], X[2], X[3]) = 19599704 X[1] + 163735456 X[1] X[2] 2 2 - 51322452 X[1] X[3] + 411889120 X[1] X[2] - 308703928 X[1] X[2] X[3] 2 3 2 + 36364688 X[1] X[3] + 287662272 X[2] - 410066160 X[2] X[3] 2 3 + 135439464 X[2] X[3] - 4052817 X[3] + 601211584 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 16 t - 18 t - 20 ) a[1, j] t = --------------------- / 3 2 ----- -t + 4 t + 17 t + 1 j = 0 infinity ----- \ j 5 + 14 t ) a[2, j] t = --------------------- / 3 2 ----- -t + 4 t + 17 t + 1 j = 0 infinity ----- 2 \ j 12 t + 16 t - 12 ) a[3, j] t = --------------------- / 3 2 ----- -t + 4 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]) = 0 ----------------------------- Theorem Number, 29 Let 3 2 P(X[1], X[2], X[3]) = 48833653 X[1] + 197722722 X[1] X[2] 2 2 - 105610986 X[1] X[3] + 260050395 X[1] X[2] - 285040863 X[1] X[2] X[3] 2 3 2 + 75828447 X[1] X[3] + 111586333 X[2] - 187852500 X[2] X[3] 2 3 + 102223431 X[2] X[3] - 18067995 X[3] + 26609229 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -19 t - 7 t + 3 ) a[1, j] t = --------------------- / 3 2 ----- -t - 3 t + 14 t + 1 j = 0 infinity ----- 2 \ j 4 t + 16 t + 3 ) a[2, j] t = --------------------- / 3 2 ----- -t - 3 t + 14 t + 1 j = 0 infinity ----- 2 \ j -19 t + 12 t + 10 ) a[3, j] t = --------------------- / 3 2 ----- -t - 3 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]) = 0 ----------------------------- Theorem Number, 30 Let 3 2 P(X[1], X[2], X[3]) = -8379410729 X[1] + 25796161717 X[1] X[2] 2 2 - 15007564711 X[1] X[3] - 25672348011 X[1] X[2] 2 3 + 32217022754 X[1] X[2] X[3] - 9006323955 X[1] X[3] + 7979320823 X[2] 2 2 3 - 16805400967 X[2] X[3] + 9997364237 X[2] X[3] - 1864696373 X[3] + 972653141312 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 15 t - 12 t - 11 ) a[1, j] t = --------------------- / 3 2 ----- -t + 11 t - 6 t + 1 j = 0 infinity ----- 2 \ j 18 t + 19 t - 12 ) a[2, j] t = --------------------- / 3 2 ----- -t + 11 t - 6 t + 1 j = 0 infinity ----- 2 \ j 19 t + 17 t + 7 ) a[3, j] t = --------------------- / 3 2 ----- -t + 11 t - 6 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 31 Let 3 2 P(X[1], X[2], X[3]) = 153128963 X[1] - 377585184 X[1] X[2] 2 2 + 1268248958 X[1] X[3] + 207365704 X[1] X[2] - 1773937746 X[1] X[2] X[3] 2 3 2 + 3290634876 X[1] X[3] - 34283108 X[2] + 456539488 X[2] X[3] 2 3 - 2059736356 X[2] X[3] + 2759960374 X[3] + 1501123625 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 15 t - 4 t + 9 ) a[1, j] t = --------------------- / 3 2 ----- -t + 19 t - 5 t + 1 j = 0 infinity ----- 2 \ j 15 t + 14 t - 8 ) a[2, j] t = --------------------- / 3 2 ----- -t + 19 t - 5 t + 1 j = 0 infinity ----- 2 \ j -5 t + 9 t - 6 ) a[3, j] t = --------------------- / 3 2 ----- -t + 19 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]) = 0 ----------------------------- Theorem Number, 32 Let 3 2 P(X[1], X[2], X[3]) = -1273630968 X[1] - 56086452 X[1] X[2] 2 2 + 3570267492 X[1] X[3] + 208155636 X[1] X[2] + 175617684 X[1] X[2] X[3] 2 3 2 - 707379552 X[1] X[3] + 20960847 X[2] - 473726385 X[2] X[3] 2 3 - 846875007 X[2] X[3] - 329061799 X[3] + 925719092936 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -6 t + 13 t + 13 ) a[1, j] t = -------------------- / 3 2 ----- -t + 3 t + 6 t + 1 j = 0 infinity ----- 2 \ j 17 t + 9 t + 20 ) a[2, j] t = -------------------- / 3 2 ----- -t + 3 t + 6 t + 1 j = 0 infinity ----- 2 \ j -19 t - 19 t + 2 ) a[3, j] t = -------------------- / 3 2 ----- -t + 3 t + 6 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 33 Let 3 2 2 P(X[1], X[2], X[3]) = 9739048 X[1] + 19205300 X[1] X[2] + 27537636 X[1] X[3] 2 2 - 957650 X[1] X[2] + 41166700 X[1] X[2] X[3] + 25333534 X[1] X[3] 3 2 2 3 - 5211625 X[2] - 192975 X[2] X[3] + 22134125 X[2] X[3] + 7509907 X[3] + 12800000 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -20 t - 4 t - 18 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 19 t + 18 t + 1 j = 0 infinity ----- 2 \ j 4 t - 7 t - 1 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 19 t + 18 t + 1 j = 0 infinity ----- 2 \ j 20 t + 11 t + 17 ) a[3, j] t = ---------------------- / 3 2 ----- -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]) = 0 ----------------------------- Theorem Number, 34 Let 3 2 P(X[1], X[2], X[3]) = -672872832 X[1] - 2188404864 X[1] X[2] 2 2 + 1614130416 X[1] X[3] - 2290352256 X[1] X[2] 2 3 + 3600430992 X[1] X[2] X[3] - 1111332636 X[1] X[3] - 770015744 X[2] 2 2 3 + 1893372912 X[2] X[3] - 1331907948 X[2] X[3] + 215879121 X[3] + 1389928896 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -2 t - 10 t + 19 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 14 t + 20 t + 1 j = 0 infinity ----- 2 \ j 6 t + 9 t - 12 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 14 t + 20 t + 1 j = 0 infinity ----- 2 \ j 4 t - 12 t + 4 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 14 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]) = 0 ----------------------------- Theorem Number, 35 Let 3 2 P(X[1], X[2], X[3]) = 116537071 X[1] - 125003816 X[1] X[2] 2 2 - 186669560 X[1] X[3] + 23612176 X[1] X[2] + 132257328 X[1] X[2] X[3] 2 3 2 - 123969552 X[1] X[3] + 1067968 X[2] - 44086592 X[2] X[3] 2 3 + 79504128 X[2] X[3] + 215202240 X[3] + 3321287488 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 12 t - 8 t - 16 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 10 t + 11 t + 1 j = 0 infinity ----- 2 \ j 20 t + 7 t - 3 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 10 t + 11 t + 1 j = 0 infinity ----- 2 \ j -3 t - 17 t - 12 ) a[3, j] t = ---------------------- / 3 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]) = 0 ----------------------------- Theorem Number, 36 Let 3 2 P(X[1], X[2], X[3]) = -11449499 X[1] - 12160941 X[1] X[2] 2 2 - 39596796 X[1] X[3] - 5735184 X[1] X[2] - 12201345 X[1] X[2] X[3] 2 3 2 - 53315625 X[1] X[3] - 3491033 X[2] + 2651103 X[2] X[3] 2 3 - 1258842 X[2] X[3] - 30556637 X[3] + 721734273 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -7 t + 18 t - 9 ) a[1, j] t = ------------------ / 3 2 ----- -t + 6 t + t + 1 j = 0 infinity ----- 2 \ j 9 t - 17 t - 1 ) a[2, j] t = ------------------ / 3 2 ----- -t + 6 t + t + 1 j = 0 infinity ----- 2 \ j 4 t - t + 7 ) a[3, j] t = ------------------ / 3 2 ----- -t + 6 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]) = 0 ----------------------------- Theorem Number, 37 Let 3 2 P(X[1], X[2], X[3]) = 1567991725 X[1] - 7704079985 X[1] X[2] 2 2 - 4613104900 X[1] X[3] + 12540143042 X[1] X[2] 2 3 + 14980163569 X[1] X[2] X[3] + 2795537753 X[1] X[3] - 6759161759 X[2] 2 2 3 - 11991897428 X[2] X[3] - 4447245411 X[2] X[3] - 470150815 X[3] + 8353070389 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -3 t + 20 t + 16 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 20 t - 19 t + 1 j = 0 infinity ----- 2 \ j -6 t + 11 t + 12 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 20 t - 19 t + 1 j = 0 infinity ----- 2 \ j 19 t - 15 t - 1 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 20 t - 19 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 38 Let 2 P(X[1], X[2], X[3]) = 12822012 X[1] c[4] - 35075712 X[1] X[2] c[4] 2 - 4269384 X[1] X[3] c[4] + 23961728 X[2] c[4] + 5803392 X[2] X[3] c[4] 2 + 342972 X[3] c[4] - 17873772 X[1] X[2] + 130251204 X[1] X[3] 2 + 2744784 X[1] c[4] + 19314904 X[2] - 181705452 X[2] X[3] 2 - 3744384 X[2] c[4] - 25797132 X[3] - 529200 X[3] c[4] + 529774821 X[1] - 722709096 X[2] - 102141675 X[3] + 12822012 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -2 t - 18 t + 11 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 15 t - 13 t + 1 j = 0 infinity ----- \ j 9 - 15 t ) a[2, j] t = ---------------------- / 3 2 ----- -t - 15 t - 13 t + 1 j = 0 infinity ----- 2 \ j -10 t + 18 t - 7 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 15 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]) = 0 ----------------------------- Theorem Number, 39 Let 3 2 2 P(X[1], X[2], X[3]) = -37376 X[1] + 69364 X[1] X[2] - 211372 X[1] X[3] 2 2 - 20080 X[1] X[2] + 231952 X[1] X[2] X[3] - 388864 X[1] X[3] 3 2 2 3 - 4001 X[2] - 22659 X[2] X[3] + 187229 X[2] X[3] - 232897 X[3] + 256 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -10 t - 6 t - 18 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 20 t - 12 t + 1 j = 0 infinity ----- 2 \ j 5 t + 8 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 20 t - 12 t + 1 j = 0 infinity ----- 2 \ j 7 t + 4 t + 12 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 20 t - 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 40 Let 3 2 P(X[1], X[2], X[3]) = -15121233 X[1] + 78289113 X[1] X[2] 2 2 + 47660889 X[1] X[3] - 130171531 X[1] X[2] - 173806486 X[1] X[2] X[3] 2 3 2 - 38688779 X[1] X[3] + 69833099 X[2] + 147867041 X[2] X[3] 2 3 + 81466273 X[2] X[3] + 6282123 X[3] + 140608000 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 15 t + 13 t - 13 ) a[1, j] t = --------------------- / 3 2 ----- -t + 3 t + 11 t + 1 j = 0 infinity ----- 2 \ j 7 t - 2 t - 6 ) a[2, j] t = --------------------- / 3 2 ----- -t + 3 t + 11 t + 1 j = 0 infinity ----- 2 \ j 6 t + 17 t - 1 ) a[3, j] t = --------------------- / 3 2 ----- -t + 3 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]) = 0 ----------------------------- Theorem Number, 41 Let 3 2 P(X[1], X[2], X[3]) = -4779755 X[1] + 117285047 X[1] X[2] 2 2 - 114622505 X[1] X[3] - 908432209 X[1] X[2] + 1778126942 X[1] X[2] X[3] 2 3 2 - 870108305 X[1] X[3] + 2263678733 X[2] - 6653200121 X[2] X[3] 2 3 + 6518356647 X[2] X[3] - 2128808275 X[3] + 12487168 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -5 t - 11 t - 19 ) a[1, j] t = ------------------- / 3 2 ----- -t - t - 12 t + 1 j = 0 infinity ----- 2 \ j 14 t - 3 t - 8 ) a[2, j] t = ------------------- / 3 2 ----- -t - t - 12 t + 1 j = 0 infinity ----- 2 \ j 15 t - 2 t - 5 ) a[3, j] t = ------------------- / 3 2 ----- -t - t - 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 42 Let 3 2 P(X[1], X[2], X[3]) = 64296431 X[1] - 89219791 X[1] X[2] 2 2 + 171351279 X[1] X[3] - 282817083 X[1] X[2] - 1481607426 X[1] X[2] X[3] 2 3 2 - 1222199003 X[1] X[3] - 69035933 X[2] - 792118849 X[2] X[3] 2 3 - 2612117759 X[2] X[3] - 2270666963 X[3] + 1990865512 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -11 t + 18 ) a[1, j] t = --------------------- / 3 2 ----- -t - 7 t + 17 t + 1 j = 0 infinity ----- 2 \ j -18 t - 19 t + 16 ) a[2, j] t = --------------------- / 3 2 ----- -t - 7 t + 17 t + 1 j = 0 infinity ----- 2 \ j 9 t + t - 8 ) a[3, j] t = --------------------- / 3 2 ----- -t - 7 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]) = 0 ----------------------------- Theorem Number, 43 Let 3 2 P(X[1], X[2], X[3]) = 9968233745 X[1] - 2920219305 X[1] X[2] 2 2 + 12444799495 X[1] X[3] - 591586325 X[1] X[2] 2 3 - 1056201950 X[1] X[2] X[3] + 2844924971 X[1] X[3] + 74517485 X[2] 2 2 3 - 805701605 X[2] X[3] + 975270443 X[2] X[3] - 231383635 X[3] + 134375075848 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -2 t + 9 t - 11 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 10 t + 12 t + 1 j = 0 infinity ----- 2 \ j 18 t + 18 t - 17 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 10 t + 12 t + 1 j = 0 infinity ----- 2 \ j 6 t + 13 t + 12 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 10 t + 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 44 Let 3 2 P(X[1], X[2], X[3]) = 27861424 X[1] - 97334769 X[1] X[2] 2 2 + 201879807 X[1] X[3] - 2516769 X[1] X[2] - 157322901 X[1] X[2] X[3] 2 3 2 + 277316508 X[1] X[3] - 71199 X[2] - 2677680 X[2] X[3] 2 3 - 64790616 X[2] X[3] + 109996697 X[3] + 2749884201 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 2 t + t - 12 ) a[1, j] t = -------------------- / 3 2 ----- -t + 4 t + 2 t + 1 j = 0 infinity ----- 2 \ j 14 t - 5 t + 19 ) a[2, j] t = -------------------- / 3 2 ----- -t + 4 t + 2 t + 1 j = 0 infinity ----- 2 \ j -t - 8 t + 12 ) a[3, j] t = -------------------- / 3 2 ----- -t + 4 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]) = 0 ----------------------------- Theorem Number, 45 Let 3 2 P(X[1], X[2], X[3]) = 8507200 X[1] + 1122304672 X[1] X[2] 2 2 - 1135092080 X[1] X[3] - 1670543264 X[1] X[2] + 815040144 X[1] X[2] X[3] 2 3 2 + 844563040 X[1] X[3] + 604113128 X[2] + 299902020 X[2] X[3] 2 3 - 2014691594 X[2] X[3] + 1112784335 X[3] + 162056905408 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -2 t - 5 t + 19 ) a[1, j] t = --------------------- / 3 2 ----- -t - 11 t - 2 t + 1 j = 0 infinity ----- 2 \ j -7 t + 19 t + 19 ) a[2, j] t = --------------------- / 3 2 ----- -t - 11 t - 2 t + 1 j = 0 infinity ----- 2 \ j -10 t - 14 t + 6 ) a[3, j] t = --------------------- / 3 2 ----- -t - 11 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]) = 0 ----------------------------- Theorem Number, 46 Let 3 2 P(X[1], X[2], X[3]) = -523114875 X[1] + 698107500 X[1] X[2] 2 2 + 2397991500 X[1] X[3] - 246324780 X[1] X[2] - 1552043700 X[1] X[2] X[3] 2 3 2 - 2969797500 X[1] X[3] - 49382632 X[2] - 171472080 X[2] X[3] 2 3 - 16929000 X[2] X[3] + 490887000 X[3] + 33076161000 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -4 t + 2 t + 8 ) a[1, j] t = --------------------- / 3 2 ----- -t - 7 t - 16 t + 1 j = 0 infinity ----- 2 \ j 15 t - 15 t ) a[2, j] t = --------------------- / 3 2 ----- -t - 7 t - 16 t + 1 j = 0 infinity ----- 2 \ j -12 t - 16 t + 5 ) a[3, j] t = --------------------- / 3 2 ----- -t - 7 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]) = 0 ----------------------------- Theorem Number, 47 Let 3 2 P(X[1], X[2], X[3]) = -132269138 X[1] + 183642192 X[1] X[2] 2 2 - 333337503 X[1] X[3] + 58716288 X[1] X[2] - 168703740 X[1] X[2] X[3] 2 3 2 + 100122120 X[1] X[3] - 111936384 X[2] + 571991472 X[2] X[3] 2 3 - 984939624 X[2] X[3] + 570035884 X[3] + 10106279712 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 16 t + 16 t + 4 ) a[1, j] t = --------------------- / 3 2 ----- -t + 16 t + 6 t + 1 j = 0 infinity ----- 2 \ j 20 t + 15 t - 19 ) a[2, j] t = --------------------- / 3 2 ----- -t + 16 t + 6 t + 1 j = 0 infinity ----- 2 \ j 2 t + 14 t - 10 ) a[3, j] t = --------------------- / 3 2 ----- -t + 16 t + 6 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 48 Let 3 2 P(X[1], X[2], X[3]) = 18934128 X[1] + 55977480 X[1] X[2] 2 2 + 38732580 X[1] X[3] + 55129500 X[1] X[2] + 76366350 X[1] X[2] X[3] 2 3 2 + 26411625 X[1] X[3] + 18086000 X[2] + 37618500 X[2] X[3] 2 3 + 26045250 X[2] X[3] + 6003625 X[3] + 729000 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -10 t + 20 ) a[1, j] t = -------------------- / 3 2 ----- -t + 6 t - 4 t + 1 j = 0 infinity ----- 2 \ j 7 t + 8 t - 11 ) a[2, j] t = -------------------- / 3 2 ----- -t + 6 t - 4 t + 1 j = 0 infinity ----- 2 \ j 4 t - 10 t - 14 ) a[3, j] t = -------------------- / 3 2 ----- -t + 6 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]) = 0 ----------------------------- Theorem Number, 49 Let 3 2 P(X[1], X[2], X[3]) = 61140325 X[1] - 881410205 X[1] X[2] 2 2 - 874295080 X[1] X[3] + 964576084 X[1] X[2] + 1797077882 X[1] X[2] X[3] 2 3 2 + 833299892 X[1] X[3] - 275409469 X[2] - 747009642 X[2] X[3] 2 3 - 683652272 X[2] X[3] - 212104360 X[3] + 8972978552 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 14 t + 10 t ) a[1, j] t = ------------------- / 3 2 ----- -t + 12 t + t + 1 j = 0 infinity ----- 2 \ j 20 t + 6 t - 18 ) a[2, j] t = ------------------- / 3 2 ----- -t + 12 t + t + 1 j = 0 infinity ----- 2 \ j 6 t - 3 t + 20 ) a[3, j] t = ------------------- / 3 2 ----- -t + 12 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]) = 0 ----------------------------- Theorem Number, 50 Let 3 2 P(X[1], X[2], X[3]) = 605688905 X[1] - 978262717 X[1] X[2] 2 2 + 2086660890 X[1] X[3] - 11586722961 X[1] X[2] 2 3 + 22634841896 X[1] X[2] X[3] - 10826468120 X[1] X[3] - 9568921811 X[2] 2 2 3 + 30279316750 X[2] X[3] - 30257520048 X[2] X[3] + 9271233320 X[3] + 380447722936 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 14 t - 13 t - 18 ) a[1, j] t = --------------------- / 3 2 ----- -t - 8 t + 13 t + 1 j = 0 infinity ----- 2 \ j -14 t - 11 t - 4 ) a[2, j] t = --------------------- / 3 2 ----- -t - 8 t + 13 t + 1 j = 0 infinity ----- 2 \ j -13 t + t - 15 ) a[3, j] t = --------------------- / 3 2 ----- -t - 8 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]) = 0 ----------------------------- Theorem Number, 51 Let 3 2 P(X[1], X[2], X[3]) = 110597211 X[1] + 2680930278 X[1] X[2] 2 2 + 498296412 X[1] X[3] + 16951769748 X[1] X[2] 2 3 + 5178942684 X[1] X[2] X[3] - 3647045832 X[1] X[3] + 11587601556 X[2] 2 2 3 + 16349193072 X[2] X[3] - 836189712 X[2] X[3] - 3915230416 X[3] + 583229338875 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -11 t - 20 t + 19 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 17 t - 11 t + 1 j = 0 infinity ----- 2 \ j -11 t - 13 t - 6 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 17 t - 11 t + 1 j = 0 infinity ----- 2 \ j 18 t - 3 t - 12 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 17 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]) = 0 ----------------------------- Theorem Number, 52 Let 3 2 2 P(X[1], X[2], X[3]) = 2433507 X[1] - 27383535 X[1] X[2] + 6701960 X[1] X[3] 2 2 - 55856811 X[1] X[2] - 276859980 X[1] X[2] X[3] - 106202780 X[1] X[3] 3 2 2 - 19664721 X[2] - 135354220 X[2] X[3] - 111094860 X[2] X[3] 3 - 24251800 X[3] + 25776946664 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 10 t + 11 t + 2 ) a[1, j] t = -------------------- / 3 2 ----- -t - 5 t + 2 t + 1 j = 0 infinity ----- 2 \ j -16 t + 13 t ) a[2, j] t = -------------------- / 3 2 ----- -t - 5 t + 2 t + 1 j = 0 infinity ----- 2 \ j -t - 15 t + 8 ) a[3, j] t = -------------------- / 3 2 ----- -t - 5 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]) = 0 ----------------------------- Theorem Number, 53 Let P(X[1], X[2], X[3]) = 362 X[1] + 3223 X[2] + 2877 X[3] + 6158 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 10 t + 11 t + 18 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 10 t - 10 t + 1 j = 0 infinity ----- 2 \ j 5 t - t - 20 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 10 t - 10 t + 1 j = 0 infinity ----- 2 \ j -9 t + 19 t + 18 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 10 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]) = 0 ----------------------------- Theorem Number, 54 Let 3 2 2 P(X[1], X[2], X[3]) = -938215 X[1] + 15527886 X[1] X[2] - 4374174 X[1] X[3] 2 2 - 52248768 X[1] X[2] + 38799948 X[1] X[2] X[3] - 10231164 X[1] X[3] 3 2 2 3 - 103621012 X[2] + 834900 X[2] X[3] + 9616068 X[2] X[3] - 4803444 X[3] + 10905591092 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 20 t + 18 t + 10 ) a[1, j] t = -------------------- / 3 2 ----- -t - 3 t - 3 t + 1 j = 0 infinity ----- 2 \ j -6 t - 7 t + 4 ) a[2, j] t = -------------------- / 3 2 ----- -t - 3 t - 3 t + 1 j = 0 infinity ----- 2 \ j -13 t + 7 t - 1 ) a[3, j] t = -------------------- / 3 2 ----- -t - 3 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]) = 0 ----------------------------- Theorem Number, 55 Let 3 2 P(X[1], X[2], X[3]) = -6130701 X[1] - 19131408 X[1] X[2] 2 2 - 13818798 X[1] X[3] - 32153991 X[1] X[2] - 29212641 X[1] X[2] X[3] 2 3 2 - 8200089 X[1] X[3] - 22552271 X[2] - 34025107 X[2] X[3] 2 3 - 10786662 X[2] X[3] - 1541673 X[3] + 122867271 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -4 t + t + 6 ) a[1, j] t = --------------------- / 3 2 ----- -t + 16 t + 5 t + 1 j = 0 infinity ----- \ j -3 - 15 t ) a[2, j] t = --------------------- / 3 2 ----- -t + 16 t + 5 t + 1 j = 0 infinity ----- 2 \ j 11 t + 15 t - 1 ) a[3, j] t = --------------------- / 3 2 ----- -t + 16 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]) = 0 ----------------------------- Theorem Number, 56 Let P(X[1], X[2], X[3]) = -1926 X[1] - 2758 X[2] - 281 X[3] + 4861 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -16 t - 19 t + 1 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 18 t - 18 t + 1 j = 0 infinity ----- 2 \ j 11 t - 17 t + 3 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 18 t - 18 t + 1 j = 0 infinity ----- 2 \ j 19 t + 3 t - 19 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 18 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]) = 0 ----------------------------- Theorem Number, 57 Let 3 2 P(X[1], X[2], X[3]) = -39488951080 X[1] - 752533552 X[1] X[2] 2 2 + 67964212800 X[1] X[3] + 20608804678 X[1] X[2] 2 3 - 4079791988 X[1] X[2] X[3] - 35984900120 X[1] X[3] - 1952754883 X[2] 2 2 3 - 5296488984 X[2] X[3] + 1946249020 X[2] X[3] + 5465840200 X[3] + 671761055512 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -7 t - 19 t - 5 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 19 t + 14 t + 1 j = 0 infinity ----- 2 \ j -12 t - 12 t + 14 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 19 t + 14 t + 1 j = 0 infinity ----- 2 \ j -19 t - 2 t - 20 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 19 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]) = 0 ----------------------------- Theorem Number, 58 Let 3 2 P(X[1], X[2], X[3]) = -20346323 X[1] - 50497128 X[1] X[2] 2 2 - 122981898 X[1] X[3] + 142295496 X[1] X[2] + 296983500 X[1] X[2] X[3] 2 3 2 + 112005108 X[1] X[3] + 88498792 X[2] + 285284160 X[2] X[3] 2 3 + 277913832 X[2] X[3] + 83212432 X[3] + 27211180824 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -20 t + 2 t + 6 ) a[1, j] t = --------------------- / 3 2 ----- -t - 12 t - 6 t + 1 j = 0 infinity ----- 2 \ j -19 t - 12 t - 20 ) a[2, j] t = --------------------- / 3 2 ----- -t - 12 t - 6 t + 1 j = 0 infinity ----- 2 \ j 12 t - 17 t + 14 ) a[3, j] t = --------------------- / 3 2 ----- -t - 12 t - 6 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 59 Let 3 2 2 P(X[1], X[2], X[3]) = -7225000 X[1] + 658700 X[1] X[2] - 451100 X[1] X[3] 2 2 + 4184650 X[1] X[2] + 20996364 X[1] X[2] X[3] + 14419498 X[1] X[3] 3 2 2 3 + 3680325 X[2] + 9897349 X[2] X[3] + 12898567 X[2] X[3] + 9794543 X[3] + 2210782784 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -t + 4 t + 15 ) a[1, j] t = ------------------ / 3 2 ----- -t + t - 6 t + 1 j = 0 infinity ----- 2 \ j -7 t - 13 t - 6 ) a[2, j] t = ------------------ / 3 2 ----- -t + t - 6 t + 1 j = 0 infinity ----- 2 \ j -t + 15 t - 16 ) a[3, j] t = ------------------ / 3 2 ----- -t + t - 6 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 60 Let 3 2 P(X[1], X[2], X[3]) = -59799409 X[1] + 187511988 X[1] X[2] 2 2 - 171952086 X[1] X[3] - 173327016 X[1] X[2] + 354281010 X[1] X[2] X[3] 2 3 2 - 131433456 X[1] X[3] + 39558094 X[2] - 152235804 X[2] X[3] 2 3 + 152731452 X[2] X[3] - 26374732 X[3] + 3534344658 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 18 t + 8 t + 4 ) a[1, j] t = -------------------- / 3 2 ----- -t - 9 t + 3 t + 1 j = 0 infinity ----- 2 \ j 13 t + 14 t - 5 ) a[2, j] t = -------------------- / 3 2 ----- -t - 9 t + 3 t + 1 j = 0 infinity ----- 2 \ j -2 t - 9 t - 9 ) a[3, j] t = -------------------- / 3 2 ----- -t - 9 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]) = 0 ----------------------------- Theorem Number, 61 Let 3 2 P(X[1], X[2], X[3]) = -2202791608 X[1] + 1076624696 X[1] X[2] 2 2 - 2510691152 X[1] X[3] + 1916471882 X[1] X[2] - 970094076 X[1] X[2] X[3] 2 3 2 - 411742264 X[1] X[3] - 450841157 X[2] + 529365460 X[2] X[3] 2 3 - 100407252 X[2] X[3] - 18349336 X[3] + 53072595512 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j t - 14 t + 4 ) a[1, j] t = --------------------- / 3 2 ----- -t - 17 t - 2 t + 1 j = 0 infinity ----- 2 \ j 16 t + 18 t - 4 ) a[2, j] t = --------------------- / 3 2 ----- -t - 17 t - 2 t + 1 j = 0 infinity ----- 2 \ j 16 t + 9 t - 17 ) a[3, j] t = --------------------- / 3 2 ----- -t - 17 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]) = 0 ----------------------------- Theorem Number, 62 Let 3 2 P(X[1], X[2], X[3]) = 220591489 X[1] + 2909324799 X[1] X[2] 2 2 + 1076077794 X[1] X[3] - 628712346 X[1] X[2] + 4008458697 X[1] X[2] X[3] 2 3 2 + 999339399 X[1] X[3] - 5846820139 X[2] + 3582290286 X[2] X[3] 2 3 + 1367656365 X[2] X[3] - 13315093 X[3] + 131949968571 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 13 t - 10 t - 9 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 13 t + 16 t + 1 j = 0 infinity ----- \ j 7 + 18 t ) a[2, j] t = ---------------------- / 3 2 ----- -t + 13 t + 16 t + 1 j = 0 infinity ----- 2 \ j -8 t - t - 19 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 13 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]) = 0 ----------------------------- Theorem Number, 63 Let 3 2 P(X[1], X[2], X[3]) = -3677000994 X[1] - 16751167425 X[1] X[2] 2 2 - 760852332 X[1] X[3] - 25437496706 X[1] X[2] 2 3 - 2310793357 X[1] X[2] X[3] - 52444492 X[1] X[3] - 12876019276 X[2] 2 2 3 - 1754531050 X[2] X[3] - 79640145 X[2] X[3] - 1204186 X[3] + 24389 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -15 t + 9 t - 10 ) a[1, j] t = --------------------- / 3 2 ----- -t - 18 t - 4 t + 1 j = 0 infinity ----- 2 \ j 9 t - 5 t + 7 ) a[2, j] t = --------------------- / 3 2 ----- -t - 18 t - 4 t + 1 j = 0 infinity ----- 2 \ j 20 t - 19 t - 9 ) a[3, j] t = --------------------- / 3 2 ----- -t - 18 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]) = 0 ----------------------------- Theorem Number, 64 Let 3 2 P(X[1], X[2], X[3]) = -1126886044 X[1] + 8655096735 X[1] X[2] 2 2 + 6873409488 X[1] X[3] - 20597062644 X[1] X[2] 2 3 - 32739010641 X[1] X[2] X[3] - 12874135656 X[1] X[3] + 15438660669 X[2] 2 2 3 + 37250286399 X[2] X[3] + 29463874209 X[2] X[3] + 7627441396 X[3] + 19486825371 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -18 t - 13 t + 18 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 18 t - 16 t + 1 j = 0 infinity ----- 2 \ j -3 t - 7 t + 19 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 18 t - 16 t + 1 j = 0 infinity ----- 2 \ j -9 t + 11 t - 18 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 18 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]) = 0 ----------------------------- Theorem Number, 65 Let 3 2 P(X[1], X[2], X[3]) = -157277257 X[1] - 1317509334 X[1] X[2] 2 2 + 1658270879 X[1] X[3] - 2940260216 X[1] X[2] 2 3 + 6833085272 X[1] X[2] X[3] - 3789906927 X[1] X[3] - 767888968 X[2] 2 2 3 + 507627024 X[2] X[3] + 3048919342 X[2] X[3] - 3119582975 X[3] + 87646718888 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 11 t + 6 t + 15 ) a[1, j] t = --------------------- / 3 2 ----- -t + 8 t - 15 t + 1 j = 0 infinity ----- 2 \ j -18 t - 8 t - 1 ) a[2, j] t = --------------------- / 3 2 ----- -t + 8 t - 15 t + 1 j = 0 infinity ----- 2 \ j -7 t - 20 t + 3 ) a[3, j] t = --------------------- / 3 2 ----- -t + 8 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]) = 0 ----------------------------- Theorem Number, 66 Let 3 2 P(X[1], X[2], X[3]) = -2423528344 X[1] + 4099431596 X[1] X[2] 2 2 + 1887095260 X[1] X[3] - 2311419842 X[1] X[2] 2 3 - 2128038164 X[1] X[2] X[3] - 489780082 X[1] X[3] + 434423873 X[2] 2 2 3 + 599937391 X[2] X[3] + 276158283 X[2] X[3] + 42371165 X[3] + 16384 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 7 t - 2 t + 5 ) a[1, j] t = --------------------- / 3 2 ----- -t + 7 t - 17 t + 1 j = 0 infinity ----- 2 \ j 11 t + 4 t + 18 ) a[2, j] t = --------------------- / 3 2 ----- -t + 7 t - 17 t + 1 j = 0 infinity ----- 2 \ j 3 t - 16 t - 20 ) a[3, j] t = --------------------- / 3 2 ----- -t + 7 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]) = 0 ----------------------------- Theorem Number, 67 Let 3 2 P(X[1], X[2], X[3]) = -985049408 X[1] - 2797805184 X[1] X[2] 2 2 - 127691280 X[1] X[3] + 3283142592 X[1] X[2] + 8226782352 X[1] X[2] X[3] 2 3 2 + 3129937848 X[1] X[3] - 256673216 X[2] - 1610345952 X[2] X[3] 2 3 - 2631382524 X[2] X[3] - 979721109 X[3] + 95569357248 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -17 t - 4 t - 2 ) a[1, j] t = --------------------- / 3 2 ----- -t - 15 t + 4 t + 1 j = 0 infinity ----- 2 \ j 14 t - 5 t + 14 ) a[2, j] t = --------------------- / 3 2 ----- -t - 15 t + 4 t + 1 j = 0 infinity ----- 2 \ j -20 t - 20 t - 12 ) a[3, j] t = --------------------- / 3 2 ----- -t - 15 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]) = 0 ----------------------------- Theorem Number, 68 Let 3 2 P(X[1], X[2], X[3]) = 110296481 X[1] - 194391171 X[1] X[2] 2 2 - 52667265 X[1] X[3] + 119114931 X[1] X[2] + 65171490 X[1] X[2] X[3] 2 3 2 - 246758949 X[1] X[3] - 24977105 X[2] - 31087569 X[2] X[3] 2 3 + 180194661 X[2] X[3] - 128319363 X[3] + 3061257408 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 10 t - 2 t ) a[1, j] t = ------------------- / 3 2 ----- -t - t - 15 t + 1 j = 0 infinity ----- 2 \ j 16 t - 17 t + 18 ) a[2, j] t = ------------------- / 3 2 ----- -t - t - 15 t + 1 j = 0 infinity ----- 2 \ j 2 t + 5 t + 14 ) a[3, j] t = ------------------- / 3 2 ----- -t - 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]) = 0 ----------------------------- Theorem Number, 69 Let 3 2 P(X[1], X[2], X[3]) = 31166989 X[1] + 486982943 X[1] X[2] 2 2 + 880339377 X[1] X[3] - 715772605 X[1] X[2] - 2049561002 X[1] X[2] X[3] 2 3 2 - 1546901425 X[1] X[3] - 1851550543 X[2] - 14545609551 X[2] X[3] 2 3 - 35952811541 X[2] X[3] - 28230384949 X[3] + 50899819816 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 10 t - 15 t + 14 ) a[1, j] t = --------------------- / 3 2 ----- -t + 2 t + 20 t + 1 j = 0 infinity ----- 2 \ j 14 t + 18 t - 19 ) a[2, j] t = --------------------- / 3 2 ----- -t + 2 t + 20 t + 1 j = 0 infinity ----- 2 \ j -6 t - t + 11 ) a[3, j] t = --------------------- / 3 2 ----- -t + 2 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]) = 0 ----------------------------- Theorem Number, 70 Let 3 2 P(X[1], X[2], X[3]) = -616371928 X[1] + 582500064 X[1] X[2] 2 2 + 1856750916 X[1] X[3] + 52564248 X[1] X[2] - 1338452844 X[1] X[2] X[3] 2 3 2 - 1877855232 X[1] X[3] - 127966456 X[2] + 18700380 X[2] X[3] 2 3 + 735333456 X[2] X[3] + 643966477 X[3] + 17293606056 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j t - 6 t - 11 ) a[1, j] t = --------------------- / 3 2 ----- -t + 12 t - 5 t + 1 j = 0 infinity ----- 2 \ j 13 t + 7 t - 4 ) a[2, j] t = --------------------- / 3 2 ----- -t + 12 t - 5 t + 1 j = 0 infinity ----- 2 \ j -10 t + 4 t - 12 ) a[3, j] t = --------------------- / 3 2 ----- -t + 12 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]) = 0 ----------------------------- Theorem Number, 71 Let P(X[1], X[2], X[3]) = 2568 X[1] - 2047 X[2] - 710 X[3] + 927 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -13 t + 13 t - 9 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 17 t - 17 t + 1 j = 0 infinity ----- 2 \ j -11 t + 16 t - 15 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 17 t - 17 t + 1 j = 0 infinity ----- 2 \ j -14 t - 20 t + 12 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 17 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]) = 0 ----------------------------- Theorem Number, 72 Let 3 2 P(X[1], X[2], X[3]) = -7520754287 X[1] - 8617922025 X[1] X[2] 2 2 + 18617567619 X[1] X[3] - 16457685 X[1] X[2] + 7638284622 X[1] X[2] X[3] 2 3 2 - 10882857645 X[1] X[3] + 603010661 X[2] - 376483461 X[2] X[3] 2 3 - 2056024521 X[2] X[3] + 1793167713 X[3] + 184803393536 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- \ j -7 + 19 t ) a[1, j] t = --------------------- / 3 2 ----- -t + 9 t + 18 t + 1 j = 0 infinity ----- 2 \ j -13 t + 15 t + 7 ) a[2, j] t = --------------------- / 3 2 ----- -t + 9 t + 18 t + 1 j = 0 infinity ----- 2 \ j -7 t + 10 t - 20 ) a[3, j] t = --------------------- / 3 2 ----- -t + 9 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]) = 0 ----------------------------- Theorem Number, 73 Let 3 2 P(X[1], X[2], X[3]) = 773670509 X[1] - 833558807 X[1] X[2] 2 2 - 530311835 X[1] X[3] - 428576053 X[1] X[2] + 701283210 X[1] X[2] X[3] 2 3 2 - 130591677 X[1] X[3] + 71799351 X[2] + 451005845 X[2] X[3] 2 3 - 72767623 X[2] X[3] + 1142083 X[3] + 343882756216 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -12 t - 17 t + 15 ) a[1, j] t = -------------------- / 3 2 ----- -t - 5 t + 2 t + 1 j = 0 infinity ----- 2 \ j 13 t - 20 t + 9 ) a[2, j] t = -------------------- / 3 2 ----- -t - 5 t + 2 t + 1 j = 0 infinity ----- 2 \ j -13 t - t + 20 ) a[3, j] t = -------------------- / 3 2 ----- -t - 5 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]) = 0 ----------------------------- Theorem Number, 74 Let 3 2 P(X[1], X[2], X[3]) = -217629736 X[1] + 641606788 X[1] X[2] 2 2 - 188934180 X[1] X[3] - 48358910 X[1] X[2] - 870072548 X[1] X[2] X[3] 2 3 2 + 608326306 X[1] X[3] + 708187 X[2] + 32702311 X[2] X[3] 2 3 + 293449217 X[2] X[3] - 256886723 X[3] + 4126796864 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 5 t + 9 t - 15 ) a[1, j] t = ------------------ / 3 2 ----- -t - 7 t - t + 1 j = 0 infinity ----- 2 \ j -15 t + 5 t - 18 ) a[2, j] t = ------------------ / 3 2 ----- -t - 7 t - t + 1 j = 0 infinity ----- 2 \ j 7 t + 7 t - 20 ) a[3, j] t = ------------------ / 3 2 ----- -t - 7 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]) = 0 ----------------------------- Theorem Number, 75 Let 3 2 P(X[1], X[2], X[3]) = -8563720 X[1] - 25433880 X[1] X[2] 2 2 - 11902584 X[1] X[3] - 6632910 X[1] X[2] + 20774280 X[1] X[2] X[3] 2 3 2 + 21067848 X[1] X[3] + 17936435 X[2] + 70462338 X[2] X[3] 2 3 + 78356016 X[2] X[3] + 26755704 X[3] + 179406144 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 7 t - 2 t - 4 ) a[1, j] t = --------------------- / 3 2 ----- -t + 9 t - 20 t + 1 j = 0 infinity ----- 2 \ j -16 t + 20 t + 16 ) a[2, j] t = --------------------- / 3 2 ----- -t + 9 t - 20 t + 1 j = 0 infinity ----- 2 \ j 11 t - 6 t - 18 ) a[3, j] t = --------------------- / 3 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]) = 0 ----------------------------- Theorem Number, 76 Let 3 2 P(X[1], X[2], X[3]) = 1656186125 X[1] - 1167539325 X[1] X[2] 2 2 + 340861500 X[1] X[3] + 24810270 X[1] X[2] - 504615825 X[1] X[2] X[3] 2 3 2 - 141064875 X[1] X[3] + 65228773 X[2] + 109310985 X[2] X[3] 2 3 - 9175950 X[2] X[3] - 25207875 X[3] + 2222447625 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -6 t - 8 t - 5 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 13 t + 16 t + 1 j = 0 infinity ----- 2 \ j -15 t - 5 t - 5 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 13 t + 16 t + 1 j = 0 infinity ----- 2 \ j 6 t + 3 t + 17 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 13 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]) = 0 ----------------------------- Theorem Number, 77 Let 3 2 P(X[1], X[2], X[3]) = 99376264 X[1] - 757437204 X[1] X[2] 2 2 + 77199348 X[1] X[3] + 1716308622 X[1] X[2] - 454751244 X[1] X[2] X[3] 2 3 2 + 9425358 X[1] X[3] - 1163939891 X[2] + 525786297 X[2] X[3] 2 3 - 53789937 X[2] X[3] - 1068277 X[3] + 279726264 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -2 t + 18 t - 5 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 11 t + 19 t + 1 j = 0 infinity ----- 2 \ j -6 t + 13 t - 1 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 11 t + 19 t + 1 j = 0 infinity ----- 2 \ j -20 t + 13 t + 15 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 11 t + 19 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 78 Let 3 2 2 P(X[1], X[2], X[3]) = 17139287 X[1] + 2382127 X[1] X[2] + 23605043 X[1] X[3] 2 2 - 38902655 X[1] X[2] - 41198652 X[1] X[2] X[3] - 1851773 X[1] X[3] 3 2 2 3 + 1178283 X[2] - 6355885 X[2] X[3] - 7886635 X[2] X[3] - 1140119 X[3] + 476379541 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -2 t - 13 t + 3 ) a[1, j] t = -------------------- / 3 2 ----- -t + 7 t - 9 t + 1 j = 0 infinity ----- 2 \ j -17 t + 11 t + 1 ) a[2, j] t = -------------------- / 3 2 ----- -t + 7 t - 9 t + 1 j = 0 infinity ----- 2 \ j 20 t - 7 ) a[3, j] t = -------------------- / 3 2 ----- -t + 7 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]) = 0 ----------------------------- Theorem Number, 79 Let 3 2 P(X[1], X[2], X[3]) = 738937656 X[1] + 1309111464 X[1] X[2] 2 2 - 201818784 X[1] X[3] + 775022272 X[1] X[2] - 221843564 X[1] X[2] X[3] 2 3 2 - 23323958 X[1] X[3] + 153207352 X[2] - 59970236 X[2] X[3] 2 3 - 15759584 X[2] X[3] + 7827843 X[3] + 3221199000 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -5 t + 3 t - 14 ) a[1, j] t = --------------------- / 3 2 ----- -t + 8 t - 17 t + 1 j = 0 infinity ----- 2 \ j 6 t + 8 t + 15 ) a[2, j] t = --------------------- / 3 2 ----- -t + 8 t - 17 t + 1 j = 0 infinity ----- 2 \ j 6 t - 16 t - 18 ) a[3, j] t = --------------------- / 3 2 ----- -t + 8 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]) = 0 ----------------------------- Theorem Number, 80 Let 3 2 P(X[1], X[2], X[3]) = -191896000 X[1] - 507946400 X[1] X[2] 2 2 + 255618800 X[1] X[3] - 447930000 X[1] X[2] + 451151600 X[1] X[2] X[3] 2 3 2 - 113161700 X[1] X[3] - 131597784 X[2] + 198943884 X[2] X[3] 2 3 - 99872978 X[2] X[3] + 16653817 X[3] + 576000 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -5 t + 8 t - 12 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 11 t + 14 t + 1 j = 0 infinity ----- 2 \ j -2 t - 5 t + 17 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 11 t + 14 t + 1 j = 0 infinity ----- 2 \ j -16 t + 10 t + 6 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 11 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]) = 0 ----------------------------- Theorem Number, 81 Let P(X[1], X[2], X[3]) = -66 X[1] + 38 X[2] + 17 X[3] + 908 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 4 t - 4 t + 7 ) a[1, j] t = ---------------- / 3 2 ----- -t + t - t + 1 j = 0 infinity ----- 2 \ j -8 t + 2 t - 18 ) a[2, j] t = ---------------- / 3 2 ----- -t + t - t + 1 j = 0 infinity ----- 2 \ j -20 t - 20 t + 14 ) a[3, j] t = ------------------ / 3 2 ----- -t + 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]) = 0 ----------------------------- Theorem Number, 82 Let P(X[1], X[2], X[3]) = 140 X[1] - 1391 X[2] - 387 X[3] + 3071 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -12 t + 3 t + 14 ) a[1, j] t = -------------------- / 3 2 ----- -t - 5 t + 5 t + 1 j = 0 infinity ----- 2 \ j t + 18 t ) a[2, j] t = -------------------- / 3 2 ----- -t - 5 t + 5 t + 1 j = 0 infinity ----- \ j 13 - 16 t ) a[3, j] t = -------------------- / 3 2 ----- -t - 5 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]) = 0 ----------------------------- Theorem Number, 83 Let 3 2 2 P(X[1], X[2], X[3]) = -263197 X[1] - 222503 X[1] X[2] - 245931 X[1] X[3] 2 2 + 1965001 X[1] X[2] + 2481604 X[1] X[2] X[3] + 625029 X[1] X[3] 3 2 2 3 + 3753461 X[2] + 9404891 X[2] X[3] + 8091007 X[2] X[3] + 2339263 X[3] + 26335125 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 15 t + 17 t - 1 ) a[1, j] t = ------------------- / 3 2 ----- -t + 19 t - t + 1 j = 0 infinity ----- 2 \ j -17 t + 9 t - 8 ) a[2, j] t = ------------------- / 3 2 ----- -t + 19 t - t + 1 j = 0 infinity ----- 2 \ j 11 t - 4 t + 10 ) a[3, j] t = ------------------- / 3 2 ----- -t + 19 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]) = 0 ----------------------------- Theorem Number, 84 Let 3 2 2 P(X[1], X[2], X[3]) = -5076000 X[1] - 1262988 X[1] X[2] + 2493972 X[1] X[3] 2 2 + 920154 X[1] X[2] + 1398864 X[1] X[2] X[3] + 205590 X[1] X[3] 3 2 2 3 + 159091 X[2] + 332643 X[2] X[3] + 103401 X[2] X[3] - 79111 X[3] + 6434856 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 4 t - 5 t - 2 ) a[1, j] t = -------------------- / 3 2 ----- -t - 4 t + 8 t + 1 j = 0 infinity ----- 2 \ j -11 t + 20 t + 4 ) a[2, j] t = -------------------- / 3 2 ----- -t - 4 t + 8 t + 1 j = 0 infinity ----- 2 \ j -11 t + 14 t - 2 ) a[3, j] t = -------------------- / 3 2 ----- -t - 4 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]) = 0 ----------------------------- Theorem Number, 85 Let 3 2 P(X[1], X[2], X[3]) = 750895669 X[1] - 1447657104 X[1] X[2] 2 2 - 2343214492 X[1] X[3] + 310330313 X[1] X[2] + 2109075623 X[1] X[2] X[3] 2 3 2 + 2017020227 X[1] X[3] + 109432313 X[2] - 166618703 X[2] X[3] 2 3 - 791934394 X[2] X[3] - 522958879 X[3] + 35976742225 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -14 t - 3 t - 1 ) a[1, j] t = --------------------- / 3 2 ----- -t - 7 t + 12 t + 1 j = 0 infinity ----- 2 \ j 10 t - 13 t - 16 ) a[2, j] t = --------------------- / 3 2 ----- -t - 7 t + 12 t + 1 j = 0 infinity ----- 2 \ j -19 t - 17 t + 11 ) a[3, j] t = --------------------- / 3 2 ----- -t - 7 t + 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 86 Let 3 2 P(X[1], X[2], X[3]) = -247291974 X[1] - 215028072 X[1] X[2] 2 2 + 225277569 X[1] X[3] - 34723428 X[1] X[2] + 171803919 X[1] X[2] X[3] 2 3 2 - 446910 X[1] X[3] + 5268982 X[2] + 35367903 X[2] X[3] 2 3 - 17720220 X[2] X[3] - 26809012 X[3] + 5685163659 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 11 t - 16 t - 3 ) a[1, j] t = ---------------------- / 3 2 ----- -t + 20 t - 12 t + 1 j = 0 infinity ----- 2 \ j -3 t + 5 t + 16 ) a[2, j] t = ---------------------- / 3 2 ----- -t + 20 t - 12 t + 1 j = 0 infinity ----- 2 \ j 15 t + 17 t - 11 ) a[3, j] t = ---------------------- / 3 2 ----- -t + 20 t - 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 87 Let 3 2 P(X[1], X[2], X[3]) = -501179305 X[1] - 459577913 X[1] X[2] 2 2 - 1611307659 X[1] X[3] - 116922081 X[1] X[2] - 897072695 X[1] X[2] X[3] 2 3 2 - 1603998641 X[1] X[3] - 8508994 X[2] - 100779374 X[2] X[3] 2 3 - 383012598 X[2] X[3] - 464958136 X[3] + 2797260929 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 15 t + 9 t - 1 ) a[1, j] t = -------------------- / 3 2 ----- -t + 8 t + 8 t + 1 j = 0 infinity ----- 2 \ j 8 t + t - 20 ) a[2, j] t = -------------------- / 3 2 ----- -t + 8 t + 8 t + 1 j = 0 infinity ----- 2 \ j -14 t - 2 t + 9 ) a[3, j] t = -------------------- / 3 2 ----- -t + 8 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]) = 0 ----------------------------- Theorem Number, 88 Let 3 2 P(X[1], X[2], X[3]) = -6469146 X[1] + 50108544 X[1] X[2] 2 2 - 41178780 X[1] X[3] - 102875022 X[1] X[2] + 156834630 X[1] X[2] X[3] 2 3 2 - 58407300 X[1] X[3] + 63570987 X[2] - 139554495 X[2] X[3] 2 3 + 102323025 X[2] X[3] - 25608625 X[3] + 121287375 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -5 t + 2 t - 19 ) a[1, j] t = --------------------- / 3 2 ----- -t - 5 t - 15 t + 1 j = 0 infinity ----- 2 \ j -10 t - 11 t - 13 ) a[2, j] t = --------------------- / 3 2 ----- -t - 5 t - 15 t + 1 j = 0 infinity ----- 2 \ j -9 t - 9 t ) a[3, j] t = --------------------- / 3 2 ----- -t - 5 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]) = 0 ----------------------------- Theorem Number, 89 Let 3 2 P(X[1], X[2], X[3]) = -337135277 X[1] + 434261697 X[1] X[2] 2 2 + 1649239731 X[1] X[3] - 56943564 X[1] X[2] - 1197047619 X[1] X[2] X[3] 2 3 2 - 2536833006 X[1] X[3] - 8899819 X[2] + 97512039 X[2] X[3] 2 3 + 828242172 X[2] X[3] + 1228775427 X[3] + 8303765625 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -15 t - 8 t + 12 ) a[1, j] t = --------------------- / 3 2 ----- -t - 10 t + 3 t + 1 j = 0 infinity ----- 2 \ j -15 t - 5 t - 6 ) a[2, j] t = --------------------- / 3 2 ----- -t - 10 t + 3 t + 1 j = 0 infinity ----- 2 \ j -10 t + 2 t + 9 ) a[3, j] t = --------------------- / 3 2 ----- -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]) = 0 ----------------------------- Theorem Number, 90 Let 3 2 P(X[1], X[2], X[3]) = 28526581 X[1] + 105854947 X[1] X[2] 2 2 - 51160198 X[1] X[3] + 130924592 X[1] X[2] - 126581040 X[1] X[2] X[3] 2 3 2 + 30519035 X[1] X[3] + 53973471 X[2] - 78291153 X[2] X[3] 2 3 + 37760790 X[2] X[3] - 6056793 X[3] + 54872 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 10 t - 5 t + 17 ) a[1, j] t = -------------------- / 3 2 ----- -t - 8 t + 9 t + 1 j = 0 infinity ----- 2 \ j -8 t - 5 t - 5 ) a[2, j] t = -------------------- / 3 2 ----- -t - 8 t + 9 t + 1 j = 0 infinity ----- \ j 19 - 19 t ) a[3, j] t = -------------------- / 3 2 ----- -t - 8 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]) = 0 ----------------------------- Theorem Number, 91 Let 3 2 P(X[1], X[2], X[3]) = -356664520 X[1] - 2923498828 X[1] X[2] 2 2 - 3799708752 X[1] X[3] - 4583871638 X[1] X[2] 2 3 - 12069995152 X[1] X[2] X[3] - 7622294240 X[1] X[3] - 2346193 X[2] 2 2 3 - 3516163572 X[2] X[3] - 7533188720 X[2] X[3] - 3770027200 X[3] + 73244501504 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -11 t - 15 t + 11 ) a[1, j] t = ---------------------- / 3 2 ----- -t - 10 t + 15 t + 1 j = 0 infinity ----- 2 \ j -14 t + 14 t - 10 ) a[2, j] t = ---------------------- / 3 2 ----- -t - 10 t + 15 t + 1 j = 0 infinity ----- 2 \ j 13 t + 6 t + 7 ) a[3, j] t = ---------------------- / 3 2 ----- -t - 10 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]) = 0 ----------------------------- Theorem Number, 92 Let 3 2 P(X[1], X[2], X[3]) = 158755151 X[1] - 912422844 X[1] X[2] 2 2 + 1063904717 X[1] X[3] + 1685520380 X[1] X[2] 2 3 - 4006309236 X[1] X[2] X[3] + 2353692193 X[1] X[3] - 1010358200 X[2] 2 2 3 + 3649924420 X[2] X[3] - 4358746424 X[2] X[3] + 1718473083 X[3] + 825293672 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 12 t + 2 t - 19 ) a[1, j] t = -------------------- / 3 2 ----- -t - 8 t + 3 t + 1 j = 0 infinity ----- 2 \ j 16 t - 19 t + 9 ) a[2, j] t = -------------------- / 3 2 ----- -t - 8 t + 3 t + 1 j = 0 infinity ----- 2 \ j 6 t - 18 t + 17 ) a[3, j] t = -------------------- / 3 2 ----- -t - 8 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]) = 0 ----------------------------- Theorem Number, 93 Let 3 2 P(X[1], X[2], X[3]) = -93109797 X[1] + 724130996 X[1] X[2] 2 2 - 816119545 X[1] X[3] - 1565685593 X[1] X[2] + 3445487973 X[1] X[2] X[3] 2 3 2 - 1838295829 X[1] X[3] + 930287326 X[2] - 3200212686 X[2] X[3] 2 3 + 3533374114 X[2] X[3] - 1233679108 X[3] + 43651389761 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 19 t + 7 t + 5 ) a[1, j] t = -------------------- / 3 2 ----- -t - 6 t + 2 t + 1 j = 0 infinity ----- 2 \ j 17 t + 16 t - 16 ) a[2, j] t = -------------------- / 3 2 ----- -t - 6 t + 2 t + 1 j = 0 infinity ----- 2 \ j 9 t + 3 t - 16 ) a[3, j] t = -------------------- / 3 2 ----- -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]) = 0 ----------------------------- Theorem Number, 94 Let 3 2 P(X[1], X[2], X[3]) = 113514515 X[1] + 1023032529 X[1] X[2] 2 2 + 924692817 X[1] X[3] + 3046081146 X[1] X[2] + 5509186389 X[1] X[2] X[3] 2 3 2 + 2486551698 X[1] X[3] + 2999674053 X[2] + 8137381065 X[2] X[3] 2 3 + 7343388846 X[2] X[3] + 2204771009 X[3] + 64481201 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -8 t - 20 t - 7 ) a[1, j] t = --------------------- / 3 2 ----- -t - 9 t + 18 t + 1 j = 0 infinity ----- 2 \ j -t + 13 t + 18 ) a[2, j] t = --------------------- / 3 2 ----- -t - 9 t + 18 t + 1 j = 0 infinity ----- 2 \ j 4 t - 5 t - 18 ) a[3, j] t = --------------------- / 3 2 ----- -t - 9 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]) = 0 ----------------------------- Theorem Number, 95 Let 3 2 P(X[1], X[2], X[3]) = -718845944 X[1] - 2182669068 X[1] X[2] 2 2 + 1074244680 X[1] X[3] - 1593030330 X[1] X[2] 2 3 + 2017495896 X[1] X[2] X[3] - 671380392 X[1] X[3] + 344901375 X[2] 2 2 3 - 145718910 X[2] X[3] - 92775756 X[2] X[3] + 55997144 X[3] + 56093919552 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 12 t - 3 t - 2 ) a[1, j] t = ------------------- / 3 2 ----- -t + 11 t - t + 1 j = 0 infinity ----- 2 \ j -10 t - 12 t - 8 ) a[2, j] t = ------------------- / 3 2 ----- -t + 11 t - t + 1 j = 0 infinity ----- 2 \ j -9 t + 3 t - 20 ) a[3, j] t = ------------------- / 3 2 ----- -t + 11 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]) = 0 ----------------------------- Theorem Number, 96 Let 3 2 P(X[1], X[2], X[3]) = -2168201853 X[1] - 1251879915 X[1] X[2] 2 2 + 4783712826 X[1] X[3] - 263137660 X[1] X[2] + 1795807655 X[1] X[2] X[3] 2 3 2 - 3472240769 X[1] X[3] - 19555915 X[2] + 180617670 X[2] X[3] 2 3 - 642185725 X[2] X[3] + 820796087 X[3] + 1775956931 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -2 t + 7 t + 17 ) a[1, j] t = --------------------- / 3 2 ----- -t - 2 t - 13 t + 1 j = 0 infinity ----- 2 \ j 13 t + 16 t - 17 ) a[2, j] t = --------------------- / 3 2 ----- -t - 2 t - 13 t + 1 j = 0 infinity ----- \ j 16 + 17 t ) a[3, j] t = --------------------- / 3 2 ----- -t - 2 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]) = 0 ----------------------------- Theorem Number, 97 Let 3 2 P(X[1], X[2], X[3]) = 1765592 X[1] - 277874976 X[1] X[2] 2 2 + 277927344 X[1] X[3] + 447768198 X[1] X[2] + 502830120 X[1] X[2] X[3] 2 3 2 - 993039360 X[1] X[3] + 11477379 X[2] - 2149747812 X[2] X[3] 2 3 + 3781210560 X[2] X[3] - 1624754368 X[3] + 62429032063 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -12 t + 11 t + 5 ) a[1, j] t = --------------------- / 3 2 ----- -t - 9 t - 15 t + 1 j = 0 infinity ----- 2 \ j -13 t - 16 t + 13 ) a[2, j] t = --------------------- / 3 2 ----- -t - 9 t - 15 t + 1 j = 0 infinity ----- 2 \ j -2 t + 8 t + 11 ) a[3, j] t = --------------------- / 3 2 ----- -t - 9 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]) = 0 ----------------------------- Theorem Number, 98 Let 3 2 P(X[1], X[2], X[3]) = -18568808 X[1] + 46820016 X[1] X[2] 2 2 - 34111224 X[1] X[3] - 16339194 X[1] X[2] + 720216 X[1] X[2] X[3] 2 3 2 + 16598754 X[1] X[3] - 11945667 X[2] + 41625801 X[2] X[3] 2 3 - 40834485 X[2] X[3] + 10109127 X[3] + 219103896 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 15 t - 9 t + 3 ) a[1, j] t = --------------------- / 3 2 ----- -t - 15 t + 6 t + 1 j = 0 infinity ----- 2 \ j 19 t - t + 8 ) a[2, j] t = --------------------- / 3 2 ----- -t - 15 t + 6 t + 1 j = 0 infinity ----- 2 \ j 9 t - 5 t + 10 ) a[3, j] t = --------------------- / 3 2 ----- -t - 15 t + 6 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j]) = 0 ----------------------------- Theorem Number, 99 Let 3 2 P(X[1], X[2], X[3]) = -38604352 X[1] - 20248848 X[1] X[2] 2 2 - 12235440 X[1] X[3] - 3356100 X[1] X[2] - 4722264 X[1] X[2] X[3] 2 3 2 - 1902756 X[1] X[3] - 180171 X[2] - 433539 X[2] X[3] 2 3 - 436329 X[2] X[3] - 65929 X[3] + 68024448 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j -5 t - t - 1 ) a[1, j] t = ------------------ / 3 2 ----- -t - t - 3 t + 1 j = 0 infinity ----- 2 \ j 20 t + 3 t + 12 ) a[2, j] t = ------------------ / 3 2 ----- -t - t - 3 t + 1 j = 0 infinity ----- 2 \ j 8 t + 19 t - 8 ) a[3, j] t = ------------------ / 3 2 ----- -t - 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]) = 0 ----------------------------- Theorem Number, 100 Let 3 2 2 P(X[1], X[2], X[3]) = 1561800 X[1] + 53844120 X[1] X[2] + 69750760 X[1] X[3] 2 2 + 90344592 X[1] X[2] + 178097212 X[1] X[2] X[3] + 76527058 X[1] X[3] 3 2 2 - 28740504 X[2] - 74634560 X[2] X[3] - 59923402 X[2] X[3] 3 - 15315899 X[3] + 1529074600 Define , 3, sequences a[i,j] i goes from 1 to, 3, and j from 0 to infinity in terms of the generating functions infinity ----- 2 \ j 16 t - 20 t - 1 ) a[1, j] t = --------------------- / 3 2 ----- -t - 7 t + 10 t + 1 j = 0 infinity ----- 2 \ j -17 t - 3 t + 7 ) a[2, j] t = --------------------- / 3 2 ----- -t - 7 t + 10 t + 1 j = 0 infinity ----- 2 \ j 14 t + 16 t - 4 ) a[3, j] t = --------------------- / 3 2 ----- -t - 7 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]) = 0 ----------------------------- ---------------------------------------- This concludes this article that took, 1.565, seconds to generate