From: kesslert@surfree.com (Troy Kessler) Subject: Re: High Rank curve y^2=x^3-n^2*x+1 Date: 9 Sep 00 16:24:08 GMT Newsgroups: sci.math.numberthy Summary: [missing] Ranks of y^2=x^3-n^2*x+1 The following are points on the curve: (2-n,+-(2n-3)) (-n,+-1) (-1,+-n) (0,+-1) (n,+-1) (n+2,+-(2n+3)) Rank 6 (n<2500)Rank 7 Rank 8 Rank 9 61 347 4412 11416 107 443 4972 16228 124 614 5476 20529 128 757 5846 21456 146 778 6569 22226 148 784 7324 37072 178 857 7609 59317 199 877 8617 253 888 8627 262 1134 8951 264 1163 12755 280 1202 14137 295 1263 14401 308 1309 16667 359 1474 23878 365 1532 25371 376 1535 25823 393 1538 27428 394 1562 30814 398 1606 30841 399 1618 40447 418 1658 41517 431 1756 41582 447 1821 42818 452 1832 43552 474 1856 46027 479 1947 46174 496 1953 46318 512 1964 46682 514 2074 47966 522 2239 48018 526 2314 48152 533 2341 48650 547 2386 49659 553 2392 50329 569 2417 51332 577 2423 56756 583 2442 58511 585 2457 59815 586 2497 592 2522 596 2524 597 2547 598 2587 612 2609 629 2744 643 2788 644 2928 652 2965 671 2977 701 2998 707 3104 711 3176 713 3187 717 3228 721 3256 735 3281 739 3283 755 3544 763 3595 767 3628 769 3669 820 3734 838 3778 842 3786 876 3830 878 3929 907 4015 920 4054 922 4073 942 4088 947 4208 950 4248 951 4298 952 4354 962 4361 996 4367 1004 4494 1010 4648 1011 4739 1018 4757 1039 4831 1041 4846 1054 4918 1063 5098 1073 5186 1078 5234 1081 5273 1082 5311 1092 5333 1094 5348 1096 5371 1114 5378 1120 5431 1121 6124 1144 6133 1154 6542 1174 6584 1179 6638 1192 6691 1217 7159 1226 8167 1229 8401 1232 8431 1251 8464 1264 9892 1277 10958 1312 17551 1327 25009 1328 40122 1337 41684 1338 44051 1347 45063 1348 45433 1371 45592 1386 49109 1411 50157 1423 51529 1424 1435 1447 1468 1487 1511 1516 1545 1554 1565 1583 1589 1598 1619 1620 1634 1663 1674 1675 1687 1714 1715 1719 1726 1727 1732 1733 1762 1771 1777 1784 1792 1804 1831 1837 1844 1853 1863 1868 1871 1873 1889 1896 1910 1917 1923 1929 1942 1967 1969 2011 2020 2027 2049 2050 2060 2062 2065 2066 2083 2094 2103 2104 2106 2113 2117 2123 2140 2157 2162 2176 2179 2185 2194 2200 2215 2218 2230 2248 2249 2251 2263 2293 2302 2308 2312 2338 2359 2371 2390 2398 2401 2412 2422 2435 2439 ============================================================================== From: kesslert@surfree.com (Troy Kesslert) Subject: Re y^=x^3-n^2*x+1 (Rank 10 Curve Found) Date: 16 Sep 00 19:00:16 GMT Newsgroups: sci.math.numberthy Summary: [missing] Can someone verify that n=216493 makes a rank 10 curve? This is the first rank 10 curve I have found in this family. Ranks of y^2=x^3-n^2*x+1 The following are points on the curve: (2-n,+-(2n-3)) (-n,+-1) (-1,+-n) (0,+-1) (n,+-1) (n+2,+-(2n+3)) Rank 6 (n<2500)Rank 7 Rank 8 Rank 9 Rank 10 61 347 4412 11416 216493 107 443 4972 16228 124 614 5476 20529 128 757 5846 21456 146 778 6569 22226 148 784 7324 37072 178 857 7609 59317 199 877 8617 87519 253 888 8627 91441 262 1134 8951 113857 264 1163 12755 165560 280 1202 14137 185353 295 1263 14401 239732 308 1309 16667 359 1474 23878 365 1532 25371 376 1535 25823 393 1538 27428 394 1562 30814 398 1606 30841 399 1618 40447 418 1658 41517 431 1756 41582 447 1821 42818 452 1832 43552 474 1856 46027 479 1947 46174 496 1953 46318 512 1964 46682 514 2074 47966 522 2239 48018 526 2314 48152 533 2341 48650 547 2386 49659 553 2392 50329 569 2417 51332 577 2423 56756 583 2442 58511 585 2457 59815 586 2497 67359 592 2522 69878 596 2524 76621 597 2547 77336 598 2587 77393 612 2609 78125 629 2744 80111 643 2788 84067 644 2928 97238 652 2965 101482 671 2977 107872 701 2998 120644 707 3104 124561 711 3176 128023 713 3187 145598 717 3228 147164 721 3256 149993 735 3281 150067 739 3283 153481 755 3544 153761 763 3595 158198 767 3628 159398 769 3669 160779 820 3734 170294 838 3778 173918 842 3786 175114 876 3830 179845 878 3929 186551 907 4015 194992 920 4054 198734 922 4073 942 4088 947 4208 950 4248 951 4298 952 4354 962 4361 996 4367 1004 4494 1010 4648 1011 4739 1018 4757 1039 4831 1041 4846 1054 4918 1063 5098 1073 5186 1078 5234 1081 5273 1082 5311 1092 5333 1094 5348 1096 5371 1114 5378 1120 5431 1121 6124 1144 6133 1154 6542 1174 6584 1179 6638 1192 6691 1217 7159 1226 8167 1229 8324 1232 8401 1251 8431 1264 8464 1277 9892 1312 10958 1327 17551 1328 25009 1337 40122 1338 41684 1347 44051 1348 45063 1371 45433 1386 45592 1411 49109 1423 51529 1424 51529 1435 1447 1468 1487 1511 1516 1545 1554 1565 1583 1589 1598 1619 1620 1634 1663 1674 1675 1687 1714 1715 1719 1726 1727 1732 1733 1762 1771 1777 1784 1792 1804 1831 1837 1844 1853 1863 1868 1871 1873 1889 1896 1910 1917 1923 1929 1942 1967 1969 2011 2020 2027 2049 2050 2060 2062 2065 2066 2083 2094 2103 2104 2106 2113 2117 2123 2140 2157 2162 2176 2179 2185 2194 2200 2215 2218 2230 2248 2249 2251 2263 2293 2302 2308 2312 2338 2359 2371 2390 2398 2401 2412 2422 2435 2439 ============================================================================== From: kesslert@surfree.com (Troy Kessler) Subject: y^2=x^3-n^2*x+1 Date: 5 Oct 00 16:06:55 GMT Newsgroups: sci.math.numberthy Summary: [missing] Here is the the most recent data. If you would like to search for high ranks and have some spare computer time, please email me. Ranks of y^2=x^3-n^2*x+1 The following are points on the curve: (2-n,+-(2n-3)) (-n,+-1) (-1,+-n) (0,+-1) (n,+-1) (n+2,+-(2n+3)) Rank 6 (n<2500)Rank 7 Rank 8 Rank 9 Rank 10 61 347 4412 11416 216493 107 443 4972 16228 333974 124 614 5476 20529 128 757 5846 21456 146 778 6569 22226 148 784 7324 37072 178 857 7609 59317 199 877 8617 87519 253 888 8627 91441 262 1134 8951 113857 264 1163 12755 165560 280 1202 14137 185353 295 1263 14401 239732 308 1309 16667 279662 359 1474 23878 310192 365 1532 25371 327616 376 1535 25823 349794 393 1538 27428 367958 394 1562 30814 377748 398 1606 30841 417712 399 1618 40447 426137 418 1658 41517 467899 431 1756 41582 468454 447 1821 42818 491132 452 1832 43552 500216 474 1856 46027 528452 479 1947 46174 553853 496 1953 46318 591704 512 1964 46682 594602 514 2074 47966 522 2239 48018 526 2314 48152 533 2341 48650 547 2386 49659 553 2392 50329 569 2417 51332 577 2423 56756 583 2442 58511 585 2457 59815 586 2497 67359 592 2522 69878 596 2524 76621 597 2547 77336 598 2587 77393 612 2609 78125 629 2744 80111 643 2788 84067 644 2928 97238 652 2965 101482 671 2977 107872 701 2998 120644 707 3104 124561 711 3176 128023 713 3187 145598 717 3228 147164 721 3256 149993 735 3281 150067 739 3283 153481 755 3544 153761 763 3595 158198 767 3628 159398 769 3669 160779 820 3734 170294 838 3778 173918 842 3786 175114 876 3830 179845 878 3929 186551 907 4015 194992 920 4054 198734 922 4073 209756 942 4088 226552 947 4208 269474 950 4248 270428 951 4298 282227 952 4354 297762 962 4361 316633 996 4367 345942 1004 4494 385322 1010 4648 392664 1011 4739 397237 1018 4757 402251 1039 4831 405333 1041 4846 428159 1054 4918 434437 1063 5098 443977 1073 5186 462544 1078 5234 464632 1081 5273 471034 1082 5311 505997 1092 5333 514874 1094 5348 541832 1096 5371 543164 1114 5378 553947 1120 5431 574417 1121 6124 586847 1144 6133 600736 1154 6542 608423 1174 6584 615776 1179 6638 1192 6691 1217 7159 1226 8167 1229 8324 1232 8401 1251 8431 1264 8464 1277 9892 1312 10958 1327 17551 1328 25009 1337 40122 1338 41684 1347 44051 1348 45063 1371 45433 1386 45592 1411 49109 1423 51529 1424 51529 1435 1447 1468 1487 1511 1516 1545 1554 1565 1583 1589 1598 1619 1620 1634 1663 1674 1675 1687 1714 1715 1719 1726 1727 1732 1733 1762 1771 1777 1784 1792 1804 1831 1837 1844 1853 1863 1868 1871 1873 1889 1896 1910 1917 1923 1929 1942 1967 1969 2011 2020 2027 2049 2050 2060 2062 2065 2066 2083 2094 2103 2104 2106 2113 2117 2123 2140 2157 2162 2176 2179 2185 2194 2200 2215 2218 2230 2248 2249 2251 2263 2293 2302 2308 2312 2338 2359 2371 2390 2398 2401 2412 2422 2435 2439