Polynomial solutions to Pell's equation ======================================= The values given in any line below satisfy D(k) = n and P^2 - DQ^2 = 1. n = 1: perfect square n = 2: k = 1, D = x^2+1, P = 2x^2+1, Q = 2x n = 3: k = 1, D = x^2+2, P = x^2+1, Q = x n = 4: perfect square n = 5: k = 2, D = x^2+1, P = 2x^2+1, Q = 2x n = 6: k = 2, D = x^2+2, P = x^2+1, Q = x n = 7: k = 1, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 8: k = 2, D = x^2+2x, P = x+1, Q = 1 n = 9: perfect square n = 10: k = 3, D = x^2+1, P = 2x^2+1, Q = 2x n = 11: k = 3, D = x^2+2, P = x^2+1, Q = x n = 12: k = 2, D = x^2+3x+2, P = 2x+3, Q = 2 n = 13: k = 1, D = 4x^2+4x+5, P = 32x^6+96x^5+168x^4+176x^3+120x^2+48x+9, Q = 16x^5+40x^4+56x^3+44x^2+20x+4 n = 14: k = 2, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 15: k = 3, D = x^2+2x, P = x+1, Q = 1 n = 16: perfect square n = 17: k = 4, D = x^2+1, P = 2x^2+1, Q = 2x n = 18: k = 4, D = x^2+2, P = x^2+1, Q = x n = 19: k = 1, D = 9x^2+8x+2, P = 81x^2+72x+17, Q = 27x+12 n = 20: k = 3, D = x^2+3x+2, P = 2x+3, Q = 2 n = 21: k = 1, D = 4x^2+12x+5, P = 4x^3+18x^2+24x+9, Q = 2x^2+6x+4 n = 22: k = 1, D = 9x^2+10x+3, P = 81x^2+90x+26, Q = 27x+15 n = 23: k = 3, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 24: k = 4, D = x^2+2x, P = x+1, Q = 1 n = 25: perfect square n = 26: k = 5, D = x^2+1, P = 2x^2+1, Q = 2x n = 27: k = 5, D = x^2+2, P = x^2+1, Q = x n = 28: k = 1, D = 36x^2-8x, P = 162x^2-36x+1, Q = 27x-3 n = 29: k = 2, D = 4x^2+4x+5, P = 32x^6+96x^5+168x^4+176x^3+120x^2+48x+9, Q = 16x^5+40x^4+56x^3+44x^2+20x+4 n = 30: k = 4, D = x^2+3x+2, P = 2x+3, Q = 2 n = 31: k = 1, D = 49x^2-20x+2, P = 2401x^2-980x+99, Q = 343x-70 n = 32: k = 2, D = 9x^2-2x, P = 9x-1, Q = 3 n = 33: k = 2, D = 4x^2+7x+3, P = 8x+7, Q = 4 n = 34: k = 4, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 35: k = 5, D = x^2+2x, P = x+1, Q = 1 n = 36: perfect square n = 37: k = 6, D = x^2+1, P = 2x^2+1, Q = 2x n = 38: k = 6, D = x^2+2, P = x^2+1, Q = x n = 39: k = 2, D = 9x^2+3, P = 6x^2+1, Q = 2x n = 40: k = 3, D = 4x^2+4, P = 2x^2+1, Q = x n = 41: k = 1, D = 25x^2+14x+2, P = 1250x^2+700x+99, Q = 250x+70 n = 42: k = 5, D = x^2+3x+2, P = 2x+3, Q = 2 n = 43: k = 1, D = 81x^2-44x+6, P = 6561x^2-3564x+485, Q = 729x-198 n = 44: k = 1, D = 16x^2+16x+12, P = 32x^4+64x^3+64x^2+32x+7, Q = 8x^3+12x^2+8x+2 n = 45: k = 2, D = 4x^2+12x+5, P = 4x^3+18x^2+24x+9, Q = 2x^2+6x+4 n = 46: k = 1, D = 529x^2-746x+263, P = 279841x^2-394634x+139128, Q = 12167x-8579 n = 47: k = 5, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 48: k = 6, D = x^2+2x, P = x+1, Q = 1 n = 49: perfect square n = 50: k = 7, D = x^2+1, P = 2x^2+1, Q = 2x n = 51: k = 7, D = x^2+2, P = x^2+1, Q = x n = 52: k = 1, D = 100x^2-56x+8, P = 1250x^2-700x+99, Q = 125x-35 n = 53: k = 3, D = 4x^2+4x+5, P = 32x^6+96x^5+168x^4+176x^3+120x^2+48x+9, Q = 16x^5+40x^4+56x^3+44x^2+20x+4 n = 54: k = 2, D = 9x^2+8x+2, P = 81x^2+72x+17, Q = 27x+12 n = 55: k = 1, D = 25x^2+25x+5, P = 40x^2+40x+9, Q = 8x+4 n = 56: k = 6, D = x^2+3x+2, P = 2x+3, Q = 2 n = 57: k = 2, D = 9x^2+9x+3, P = 24x^2+24x+7, Q = 8x+4 n = 58: k = 1, D = 169x^2-140x+29, P = 57122x^2-47320x+9801, Q = 4394x-1820 n = 59: k = 2, D = 9x^2+10x+3, P = 81x^2+90x+26, Q = 27x+15 n = 60: k = 3, D = 4x^2+7x+3, P = 8x+7, Q = 4 n = 61: k = 1, D = 100x^2-44x+5, P = 7812500000x^6-10312500000x^5+5690625000x^4-1680250000x^3+279975000x^2-24961200x+930249, Q = 781250000x^5-859375000x^4+379375000x^3-84012500x^2+9332500x-416020 n = 62: k = 6, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 63: k = 7, D = x^2+2x, P = x+1, Q = 1 n = 64: perfect square n = 65: k = 8, D = x^2+1, P = 2x^2+1, Q = 2x n = 66: k = 8, D = x^2+2, P = x^2+1, Q = x n = 67: k = 1, D = 729x^2-1016x+354, P = 531441x^2-740664x+258065, Q = 19683x-13716 n = 68: k = 4, D = 4x^2+4, P = 2x^2+1, Q = x n = 69: k = 1, D = 36x^2+28x+5, P = 2916x^3+3402x^2+1296x+161, Q = 486x^2+378x+72 n = 70: k = 1, D = 225x^2-200x+45, P = 810x^2-720x+161, Q = 54x-24 n = 71: k = 1, D = 49x^2+20x+2, P = 2401x^2+980x+99, Q = 343x+70 n = 72: k = 7, D = x^2+3x+2, P = 2x+3, Q = 2 n = 73: k = 1, D = 15625x^2-29114x+13562, P = 488281250x^2-909812500x+423812499, Q = 3906250x-3639250 n = 74: k = 2, D = 25x^2-14x+2, P = 1250x^2-700x+99, Q = 250x-70 n = 75: k = 3, D = 9x^2-2x, P = 9x-1, Q = 3 n = 76: k = 1, D = 144x^2-80x+12, P = 209952x^4-233280x^3+98496x^2-18720x+1351, Q = 17496x^3-14580x^2+4104x-390 n = 77: k = 3, D = 4x^2+12x+5, P = 4x^3+18x^2+24x+9, Q = 2x^2+6x+4 n = 78: k = 2, D = 9x^2+18x+6, P = 6x^2+12x+5, Q = 2x+2 n = 79: k = 7, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 80: k = 8, D = x^2+2x, P = x+1, Q = 1 n = 81: perfect square n = 82: k = 9, D = x^2+1, P = 2x^2+1, Q = 2x n = 83: k = 9, D = x^2+2, P = x^2+1, Q = x n = 84: k = 3, D = 9x^2+3, P = 6x^2+1, Q = 2x n = 85: k = 4, D = 4x^2+4x+5, P = 32x^6+96x^5+168x^4+176x^3+120x^2+48x+9, Q = 16x^5+40x^4+56x^3+44x^2+20x+4 n = 86: k = 1, D = 121x^2-38x+3, P = 14641x^2-4598x+362, Q = 1331x-209 n = 87: k = 3, D = 9x^2+2x, P = 9x+1, Q = 3 n = 88: k = 1, D = 144x^2-64x+8, P = 324x^2-144x+17, Q = 27x-6 n = 89: k = 1, D = 2809x^2-4618x+1898, P = 15780962x^2-25943924x+10662963, Q = 297754x-244754 n = 90: k = 8, D = x^2+3x+2, P = 2x+3, Q = 2 n = 91: k = 1, D = 5929x^2-10388x+4550, P = 102487x^2-179564x+78651, Q = 1331x-1166 n = 92: k = 1, D = 16x^2+48x+28, P = 32x^4+192x^3+416x^2+384x+127, Q = 8x^3+36x^2+52x+24 n = 93: k = 1, D = 36x^2+36x+21, P = 864x^6+2592x^5+3672x^4+3024x^3+1512x^2+432x+55, Q = 144x^5+360x^4+408x^3+252x^2+84x+12 n = 94: k = 1, D = 22801x^2-42674x+19967, P = 519885601x^2-973009874x+455267568, Q = 3442951x-3221887 n = 95: k = 4, D = 4x^2+7x+3, P = 8x+7, Q = 4 n = 96: k = 2, D = 25x^2-2x, P = 25x-1, Q = 5 n = 97: k = 1, D = 323761x^2-636314x+312650, P = 209642370242x^2-412027313908x+202447753299, Q = 368440018x-362062666 n = 98: k = 8, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 99: k = 9, D = x^2+2x, P = x+1, Q = 1 n = 100: perfect square n = 101: k = 10, D = x^2+1, P = 2x^2+1, Q = 2x n = 102: k = 10, D = x^2+2, P = x^2+1, Q = x n = 103: k = 1, D = 2209x^2-3464x+1358, P = 4879681x^2-7651976x+2999823, Q = 103823x-81404 n = 104: k = 5, D = 4x^2+4, P = 2x^2+1, Q = x n = 105: k = 4, D = 4x^2+9x+5, P = 8x+9, Q = 4 n = 106: k = 1, D = 151321x^2-294632x+143417, P = 45796090082x^2-89168017744x+43404007713, Q = 117727738x-114611848 n = 107: k = 3, D = 9x^2+8x+2, P = 81x^2+72x+17, Q = 27x+12 n = 108: k = 2, D = 16x^2+16x+12, P = 32x^4+64x^3+64x^2+32x+7, Q = 8x^3+12x^2+8x+2 n = 109: k = 1, D = 2500x^2-3956x+1565, P = 1907348632812500000x^6-9054565429687500000x^5+17909937744140625000x^4-18893805777343750000x^3+11211598103109375000x^2-3548252371850970000x+467897169491706249, Q = 38146972656250000x^5-150909423828125000x^4+238799169921875000x^3-188938057773437500x^2+74743987354062500x-11827507906169900 n = 110: k = 9, D = x^2+3x+2, P = 2x+3, Q = 2 n = 111: k = 3, D = 9x^2+9x+3, P = 24x^2+24x+7, Q = 8x+4 n = 112: k = 2, D = 36x^2-17x+2, P = 72x-17, Q = 12 n = 113: k = 1, D = 5329x^2-9106x+3890, P = 56796482x^2-97051748x+41459619, Q = 778034x-664738 n = 114: k = 3, D = 9x^2+10x+3, P = 81x^2+90x+26, Q = 27x+15 n = 115: k = 1, D = 1225x^2-1700x+590, P = 12005x^2-16660x+5781, Q = 343x-238 n = 116: k = 1, D = 676x^2-792x+232, P = 57122x^2-66924x+19603, Q = 2197x-1287 n = 117: k = 4, D = 4x^2+12x+5, P = 4x^3+18x^2+24x+9, Q = 2x^2+6x+4 n = 118: k = 1, D = 2601x^2-4094x+1611, P = 6765201x^2-10648494x+4190210, Q = 132651x-104397 n = 119: k = 9, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 120: k = 10, D = x^2+2x, P = x+1, Q = 1 n = 121: perfect square n = 122: k = 11, D = x^2+1, P = 2x^2+1, Q = 2x n = 123: k = 11, D = x^2+2, P = x^2+1, Q = x n = 124: k = 1, D = 784x^2-944x+284, P = 184473632x^4-444242624x^3+401139872x^2-160970880x+24220799, Q = 6588344x^3-11899356x^2+7163212x-1437240 n = 125: k = 5, D = 4x^2+4x+5, P = 32x^6+96x^5+168x^4+176x^3+120x^2+48x+9, Q = 16x^5+40x^4+56x^3+44x^2+20x+4 n = 126: k = 1, D = 324x^2-243x+45, P = 1152x^2-864x+161, Q = 64x-24 n = 127: k = 1, D = 37249x^2-70148x+33026, P = 1387488001x^2-2612942852x+1230185475, Q = 7189057x-6769282 n = 128: k = 2, D = 36x^2-8x, P = 162x^2-36x+1, Q = 27x-3 n = 129: k = 1, D = 441x^2-405x+93, P = 57624x^2-52920x+12151, Q = 2744x-1260 n = 130: k = 2, D = 25x^2+14x+2, P = 1250x^2+700x+99, Q = 250x+70 n = 131: k = 1, D = 81x^2+44x+6, P = 6561x^2+3564x+485, Q = 729x+198 n = 132: k = 10, D = x^2+3x+2, P = 2x+3, Q = 2 n = 133: k = 1, D = 900x^2-1108x+341, P = 45562500x^3-84138750x^2+51791400x-10626551, Q = 1518750x^2-1869750x+575460 n = 134: k = 1, D = 1089x^2-1414x+459, P = 1185921x^2-1539846x+499850, Q = 35937x-23331 n = 135: k = 1, D = 81x^2+54x, P = 108x^3+108x^2+27x+1, Q = 12x^2+8x+1 n = 136: k = 4, D = 9x^2-2x, P = 9x-1, Q = 3 n = 137: k = 1, D = 22201x^2-40914x+18850, P = 985768802x^2-1816663428x+836977699, Q = 6615898x-6096186 n = 138: k = 5, D = 4x^2+7x+3, P = 8x+7, Q = 4 n = 139: k = 1, D = 558009x^2-1098404x+540534, P = 311374044081x^2-612919317636x+301622836805, Q = 416832723x-410253894 n = 140: k = 4, D = 4x^2+16x+12, P = 2x^2+8x+7, Q = x+2 n = 141: k = 3, D = 9x^2+18x+6, P = 6x^2+12x+5, Q = 2x+2 n = 142: k = 10, D = x^2+4x+2, P = x^2+4x+3, Q = x+2 n = 143: k = 11, D = x^2+2x, P = x+1, Q = 1 n = 144: perfect square n = 145: k = 12, D = x^2+1, P = 2x^2+1, Q = 2x n = 146: k = 12, D = x^2+2, P = x^2+1, Q = x n = 147: k = 4, D = 9x^2+3, P = 6x^2+1, Q = 2x n = 148: k = 6, D = 4x^2+4, P = 2x^2+1, Q = x n = 149: k = 1, D = 100x^2+44x+5, P = 7812500000x^6+10312500000x^5+5690625000x^4+1680250000x^3+279975000x^2+24961200x+930249, Q = 781250000x^5+859375000x^4+379375000x^3+84012500x^2+9332500x+416020 n = 150: k = 4, D = 9x^2+6, P = 3x^2+1, Q = x