## ``Congruent number curves'' ny2=x3-x with |n| congruent to 5 mod 8 and less than a million: Positivity of rank, and list of curves of conjectural rank 3

For which numbers n is there a rational Pythagorean triangle of area n? This problem is known to be equivalent to: for which n does the elliptic curve

E(n): n y2 = x3 - x

have a rational point other than the point at infinity and the 2-torsion points with y=0? Such n are said to be ``congruent'' because they are also known to be the numbers that occur as the common difference (``congruum'' in Latin) of a three-term arithmetic progression of squares.

We may, and henceforth do, assume that n is a positive squarefree integer. It is conjectured that if n is congruent mod 8 to 5, 6, or 7, then n is congruent. See this page, concerning the 8k+7 case, for more background on this problem and the relevant conjectures.

Theorem. If n=8k+5 and n<106 then n is a congruent number.

The proof, combining computation and theory, proceeds as described in the 8k+7 page, except that the Heegner-point construction is replaced by a variant, called ``mock Heegner points'' by Monsky [see his Math. Zeitschrift paper ``Mock Heegner points and congruent numbers'' (Vol.204 (1990) #1, 45-67) -- though Bryan Birch notes that this variant is if anything closer to Heegner's original construction...]. As a result, we cannot use Gross-Zagier to deduce that the analytic rank of E(n) is at least 3 if the resulting point P(n) is torsion, and 1 if not. Nor can we use Kolyvagin to deduce that the arithmetic rank is 1 in the non-torsion case. But it seems a reasonable conjecture that both Gross-Zagier and Kolyvagin extend to the ``mock Heegner'' context. At any rate, to complete the proof of the theorem it suffices to exhibit a single rational non-torsion point on each of the 381 curves E(n) on which P(n) is numerically indistinguishable from a torsion point. We do this in the table at the end of this page, giving for each n the numerator and denominator of the x-coordinate of a nontrivial rational solution of ny2=x3-x. The points with large numerator and denominator were obtained with Cremona's MWRANK.

In all 381 cases, 2-descent on E(n) and/or on a curve 2-isogenous with E(n) shows that the arithmetic rank is at most 3. In all but 18 cases, the 2-part of the Tate-Shafarevich group SHA of E(n) is trivial if E(n) has arithmetic rank 3 as expected. The 18 exceptional cases are:
 n = 80189 = 17*53*89 n = 82205 = 5*41*401 n = 83845 = 5*41*409 n = 92045 = 5*41*449 n = 116645 = 5*41*569 n = 125749 = 13*17*569 n = 182005 = 5*89*409 n = 199805 = 5*89*449 n = 270805 = 5*41*1321 n = 282613 = 41*61*113 n = 342205 = 5*89*769 n = 440453 = 13*17*1993 n = 477581 = 13*17*2161 n = 517789 = 41*73*173 n = 689005 = 5*41*3361 n = 749989 = 17*157*281 n = 758693 = 13*17*3433 n = 982085 = 5*13*29*521
In each of these cases, MWRANK confirms arithmetic rank 3 (by 2-descent on an isogenous curve), but finds a subgroup of SHA isomorphic with (Z/2Z)2.

 n n, factored numerator and denominator of x 2605 5*521 441, 80 4669 7*23*29 1856, 7 8005 5*1601 1681, 1521 10549 7*11*137 2601, 137 11005 5*31*71 2205, 4 11029 41*269 369, 169 13317 3*23*193 768, 193 14965 5*41*73 369, 361 18221 7*19*137 17161, 137 20005 5*4001 7921, 81 20909 7*29*103 725, 4 23405 5*31*151 6125, 1444 23709 3*7*1129 1156, 27 26013 3*13*23*29 52, 23 26245 5*29*181 1369, 441 26565 3*5*7*11*23 28, 5 27613 53*521 1590121, 339200 28221 3*23*409 9408, 1 32893 7*37*127 841, 175 33117 3*7*19*83 108, 25 35269 13*2713 10201, 7488 40397 7*29*199 3509, 2116 40885 5*13*17*37 117, 68 41181 3*7*37*53 37, 16 41309 101*409 60025, 58176 42029 13*53*61 44521, 52 43405 5*8681 4761, 3920 45037 29*1553 1089, 464 47957 7*13*17*31 425, 9 50061 3*11*37*41 2209, 1025 50629 197*257 15957, 3364 51933 3*7*2473 2500, 27 53605 5*71*151 80, 71 55549 13*4273 2401, 1872 60229 13*41*113 533, 484 63005 5*12601 116281, 103680 68605 5*13721 57696805, 22638564 69085 5*41*337 1849, 1521 69509 11*71*89 2500, 979 70013 53*1321 190969, 190800 70189 7*37*271 181447, 961 70941 3*13*17*107 43681, 25 76245 3*5*13*17*23 2209, 1 80189 17*53*89 53, 36 81469 257*317 143641, 135953 81669 3*7*3889 19663, 11449 82205 5*41*401 205, 196 82309 53*1553 2401, 848 83845 5*41*409 245, 164 90597 3*13*23*101 126025, 23 91749 3*7*17*257 2057, 1 92045 5*41*449 162409, 320 92157 3*13*17*139 625, 208 99309 3*7*4729 8836, 4107 99645 3*5*7*13*73 125, 21 103309 17*59*103 961, 927 108717 3*7*31*167 192, 25 109389 3*7*5209 41503, 169 110589 3*191*193 192, 1 110805 3*5*83*89 169, 80 115221 3*193*199 196, 3 116645 5*41*569 5041, 80 125749 13*17*569 361, 208 132405 3*5*7*13*97 36261232, 13 134805 3*5*11*19*43 4096, 11 136445 5*29*941 1521, 361 137613 3*7*6553 6889, 336 139893 3*13*17*211 841, 425 140005 5*28001 43681, 12321 141141 3*7*11*13*47 6724, 3 141661 13*17*641 12769, 1872 143021 17*47*179 5687, 5329 148733 13*17*673 16641, 6241 150805 5*30161 271445, 4 155309 7*11*2017 11264, 6889 160701 3*17*23*137 1225, 1104 160805 5*29*1109 58081, 6241 161605 5*32321 16641, 15680 162285 3*5*31*349 605, 93 166957 7*17*23*61 3388, 1525 170877 3*7*79*103 625, 7 174709 17*43*239 196, 43 180277 41*4397 1412251601, 930405200 181589 41*43*103 2116, 2107 182005 5*89*409 729, 89 187013 23*47*173 8649, 1 190805 5*31*1231 2511, 49 198061 37*53*101 25921, 25012 199805 5*89*449 449, 441 205933 7*13*31*73 1444, 819 207021 3*151*457 18438436, 9093243 209005 5*41801 35628961, 34680321 212749 41*5189 8256575941, 7563780900 217141 17*53*241 3249, 848 218285 5*149*293 149, 144 224005 5*71*631 5751, 4489 225301 17*29*457 529, 457 226005 3*5*13*19*61 1369, 95 230061 3*13*17*347 833, 208 232981 7*83*401 484, 83 234069 3*11*41*173 2809, 2768 234821 17*19*727 3332, 3211 239669 61*3929 48734361, 7051600 243165 3*5*13*29*43 245, 13 246205 5*41*1201 35721, 13520 248469 3*13*23*277 1681, 1127 251669 11*137*167 340736, 136161 258621 3*11*17*461 561, 361 265109 13*20393 10201, 10192 266541 3*11*41*197 361, 33 268933 7*103*373 235225, 98983 269717 7*53*727 1575, 121 270805 5*41*1321 2601, 41 271765 5*13*37*113 1089, 185 276261 3*71*1297 124848, 32761 276501 3*37*47*53 61009, 1225 278621 7*53*751 3481, 2527 280165 5*137*409 299209, 261121 280261 47*67*89 423, 289 282341 349*809 335241, 229441 282405 3*5*67*281 281, 121 282613 41*61*113 17629, 12996 284005 5*79*719 638401, 57591 289445 5*13*61*73 8649, 841 289765 5*7*17*487 20160, 11881 290789 23*47*269 13181, 12100 293645 5*11*19*281 245, 36 296429 7*17*47*53 529, 423 305949 3*7*17*857 2057, 343 307365 3*5*31*661 6125, 837 310405 5*62081 330245, 228484 318237 3*37*47*61 218700, 8281 318773 7*13*31*113 10923025, 113 319349 41*7789 373321, 16129 330941 13*25457 50625, 289 331813 41*8093 1225684987841, 809300000000 335901 3*19*71*83 151321, 71 338261 7*11*23*191 99, 92 342205 5*89*769 445, 324 343181 113*3037 14100025, 2961841 343493 53*6481 41899729, 19080000 344757 3*7*16417 114244, 675 348909 3*11*97*109 139129, 24025 353381 7*19*2657 10647, 10609 354173 149*2377 29300569, 1051344 355589 13*17*1609 42237, 14884 356741 7*11*41*113 77, 36 362645 5*29*41*61 208849, 64980 363613 17*73*293 7325, 3844 367285 5*17*29*149 3279721, 2313441 369421 13*157*181 1026493, 46656 370069 7*29*1823 13225, 464 372701 7*37*1439 47961, 37888 375173 17*29*761 1830609, 253009 377405 5*7*41*263 346921, 214369 388245 3*5*11*13*181 3721, 1911 390949 13*17*29*61 60025, 121 391629 3*7*17*1097 1029, 68 395637 3*11*19*631 20667, 475 397421 113*3517 11789177, 7969329 397797 3*97*1367 29058096, 25775929 399189 3*7*19009 10609, 8400 404549 17*53*449 18395521, 68688 410437 29*14153 13689, 464 410821 109*3769 93224432725, 2931995556 411677 7*23*2557 13689, 8575 416005 5*19*29*151 1539, 29 419765 5*37*2269 4205, 333 420045 3*5*41*683 18605, 164 421405 5*271*311 144653564, 5335445 421429 23*73*251 324, 251 424461 3*151*937 3775, 3721 425405 5*85081 88445, 3364 425901 3*7*17*1193 768, 425 429709 7*17*23*157 426409, 28577 434701 19*137*167 441, 167 436805 5*199*439 700569, 1681 438605 5*87721 52441, 35280 440453 13*17*1993 41209, 10609 443541 3*7*21121 120127, 48841 443685 3*5*11*2689 13456, 11 449837 17*47*563 987377, 140625 452341 23*71*277 214775, 104329 453277 11*89*463 4656964, 4547939 454205 5*90841 512886609, 457929481 461029 349*1321 94300378921, 67017301129 462685 5*37*41*61 19321, 3249 465861 3*11*19*743 1521691, 27 468741 3*7*13*17*101 625, 404 470373 3*17*23*401 7921, 1104 473421 3*13*61*199 1555009, 2809 477581 13*17*2161 32761, 6137 485485 5*7*11*13*97 97, 20 490845 3*5*43*761 279841, 5120 496509 3*13*29*439 1369, 52 497005 5*99401 1202693035383946129, 1166247624159514121 498885 3*5*79*421 229441, 1684 504341 41*12301 14801, 9801 505421 7*103*701 168921, 721 507085 5*37*2741 5445, 37 508485 3*5*109*311 1369, 436 513949 31*59*281 11471769, 281 516581 13*79*503 21261321, 240448 517789 41*73*173 23409, 11072 522805 5*104561 10936696805, 401932484 523981 113*4637 1381878017, 178134992 525941 13*23*1759 48668, 4693 527845 5*229*461 90601, 24649 529581 3*13*37*367 4624, 147 535381 7*11*17*409 2209, 409 535501 37*41*353 39698496, 2096704 536781 3*7*25561 91204, 87723 537205 5*107441 253374309769, 173564596481 539437 41*59*223 81796, 80771 543205 5*13*61*137 17689, 121 544357 11*17*41*71 34496, 34425 544685 5*41*2657 26569, 1 545669 41*13309 398325061, 305568900 547205 5*109441 129605, 20164 550981 19*47*617 112847, 89401 551733 3*7*13*43*47 700, 141 551973 3*17*79*137 60025, 4913 553213 41*103*131 116738816, 87036809 569837 37*15401 70804125925, 62966742084 571909 13*29*37*41 3025, 9 572101 17*73*461 37341, 36100 573013 7*109*751 1666617020581, 1652553195300 573781 13*19*23*101 1863, 1369 576821 7*19*4337 4356, 19 581269 13*61*733 247009, 229441 581509 23*131*193 2411809, 2783 582981 3*7*17*23*71 71, 48 584885 5*7*17*983 8836, 7875 585741 3*13*23*653 637, 16 585821 37*71*223 64447, 17161 587221 29*20249 7395725, 7365796 592997 733*809 77423125, 27988164 594741 3*7*127*223 175, 48 597365 5*37*3229 345845, 320013 599461 41*14621 16118289, 2157961 600061 7*11*7793 6561, 1232 600773 41*14653 1952145925, 580258404 605829 3*7*17*1697 34225, 5376 611749 353*1733 570025, 431649 619989 3*19*73*149 1125721, 38617 622973 13*173*277 225, 52 623733 3*11*41*461 625, 297 626381 7*43*2081 24768, 10201 628845 3*5*7*53*113 60, 53 629333 37*73*233 848241, 14800 630597 3*11*97*197 197, 100 631085 5*7*13*19*73 567, 73 632613 3*433*487 9597604, 735075 633085 5*53*2389 117077, 117045 635789 7*11*23*359 103499, 252 637941 3*337*631 484, 147 640101 3*7*11*17*163 175, 12 647309 13*17*29*101 4693, 256 648669 3*7*17*23*79 51, 28 649605 3*5*11*31*127 361, 20 650301 3*17*41*311 1313828, 147 653029 13*191*263 14161, 4775 655221 3*7*41*761 861, 100 660005 5*132001 167281, 96721 671573 7*197*487 3721, 175 672605 5*17*41*193 6561, 1 673293 3*29*71*109 17689, 436 675853 113*5981 1109664744821, 849604523300 679205 5*135841 6390639845, 1971182404 681245 5*19*71*101 56169, 2209 686933 13*53*997 49729, 121 687797 11*31*2017 49729, 49104 688541 7*19*31*167 13689, 8512 689005 5*41*3361 5125, 1764 692189 17*19*2143 49098049, 41694208 697885 5*29*4813 199809, 194996 697965 3*5*19*31*79 2401, 31 702149 7*37*2711 2368, 343 702429 3*7*13*31*83 52, 31 704685 3*5*109*431 867, 5 713765 5*13*79*139 256, 139 716885 5*13*41*269 3721, 1076 719693 13*23*29*83 1472, 725 720853 7*29*53*67 31329, 19604 730229 313*2333 2817, 1849 731645 5*41*43*83 1805, 1764 732669 3*7*139*251 139, 112 735005 5*29*37*137 24505, 14641 739381 17*23*31*61 153, 31 740485 5*23*47*137 92, 45 744765 3*5*7*41*173 189, 16 749405 5*71*2111 174845, 24964 749989 17*157*281 5041, 17 751485 3*5*7*17*421 961, 119 753445 5*7*11*19*103 1369, 721 758693 13*17*3433 27889, 425 759829 7*19*29*197 197, 64 762013 7*23*4733 196249, 40401 764949 3*17*53*283 8957, 5476 765669 3*31*8233 37975, 27889 770109 3*23*11161 284592, 27889 774701 47*53*311 15625, 2799 779605 5*155921 271441, 40401 780845 5*13*41*293 3645, 164 785437 41*19157 5968026009, 191570000 786445 5*11*79*181 89401, 711 789965 5*11*53*271 15884, 1521 790621 13*61*997 529, 468 791245 5*7*13*37*47 3388, 333 791997 3*41*47*137 3364, 3075 800101 23*43*809 985527, 108241 801413 53*15121 3348238378325, 93450743716 801933 3*11*19*1279 24300, 1 802405 5*160481 17341582643878249, 14458796624428001 802869 3*31*89*97 265225, 361 804221 11*113*647 4056196, 188771 808917 3*47*5737 3481, 2256 811941 3*13*109*191 2401, 2292 812181 3*97*2791 2500, 291 813101 569*1429 46541924, 1758276 813365 5*7*17*1367 1372, 5 818845 5*389*421 405, 16 821541 3*7*19*29*71 961, 464 821845 5*19*41*211 2009, 2000 824261 101*8161 161264601, 41139601 831405 3*5*43*1289 734449, 196209 839613 3*17*101*163 10609, 1616 840885 3*5*61*919 7381, 4624 844781 7*17*31*229 4761, 3025 845189 17*83*599 1328, 729 848309 7*11*23*479 22253, 11236 859677 3*7*13*47*67 343, 268 861445 5*13*29*457 261, 196 868749 3*7*41*1009 525, 484 868965 3*5*19*3049 9196, 49 871653 3*29*43*233 2785561, 313664 873717 3*13*43*521 325, 196 876509 61*14369 116488849, 76572401 877533 3*73*4007 480000, 187489 880157 113*7789 76454401621, 62033033700 880469 29*97*313 2817, 2809 883277 337*2621 261942763589, 260857390564 887565 3*5*7*79*107 107, 28 889413 3*7*41*1033 1029, 4 889477 257*3461 491908469, 482329444 891429 3*7*11*17*227 30899, 27 897005 5*17*61*173 78125, 10309 898885 5*13*13829 26569, 1089 901901 7*11*13*17*53 1849, 153 903901 29*71*439 375769, 320031 906373 13*113*617 697225, 602177 917165 5*53*3461 96603801985, 92580310224 930741 3*7*23*41*47 5329, 329 931917 3*7*199*223 34969, 33631 933069 3*31*79*127 79, 48 933317 7*11*17*23*31 275, 252 935085 3*5*17*19*193 125, 68 939605 5*187921 421201, 233280 950629 353*2693 3177, 2209 951405 3*5*7*13*17*41 80, 39 955245 3*5*43*1481 1161, 320 956685 3*5*23*47*59 34347, 245 957885 3*5*19*3361 502445, 8427 961653 3*7*11*23*181 756, 575 961989 3*7*19*2411 2299, 112 966485 5*13*14869 42661205, 42542253 970669 7*23*6029 17161, 5103 971741 41*137*173 29237, 21316 975605 5*195121 1673951058961, 1447069363920 975973 19*31*1657 1216, 441 978437 457*2141 173289420582269, 6925497509956 979685 5*7*23*1217 11881, 289 982085 5*13*29*521 637, 405 982181 229*4289 78074977561, 55801815761 984885 3*5*11*47*127 80, 47 987861 3*7*47041 1297321, 1054729 989485 5*7*17*1663 2268, 605 989989 7*11*13*23*43 343, 44 992165 5*61*3253 1896129, 1230004 992381 13*23*3319 6647, 9 996149 7*11*17*761 761, 625