``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*521441, 80
4669 7*23*291856, 7
8005 5*16011681, 1521
10549 7*11*1372601, 137
11005 5*31*712205, 4
11029 41*269369, 169
13317 3*23*193768, 193
14965 5*41*73369, 361
18221 7*19*13717161, 137
20005 5*40017921, 81
20909 7*29*103725, 4
23405 5*31*1516125, 1444
23709 3*7*11291156, 27
26013 3*13*23*2952, 23
26245 5*29*1811369, 441
26565 3*5*7*11*2328, 5
27613 53*5211590121, 339200
28221 3*23*4099408, 1
32893 7*37*127841, 175
33117 3*7*19*83108, 25
35269 13*271310201, 7488
40397 7*29*1993509, 2116
40885 5*13*17*37117, 68
41181 3*7*37*5337, 16
41309 101*40960025, 58176
42029 13*53*6144521, 52
43405 5*86814761, 3920
45037 29*15531089, 464
47957 7*13*17*31425, 9
50061 3*11*37*412209, 1025
50629 197*25715957, 3364
51933 3*7*24732500, 27
53605 5*71*15180, 71
55549 13*42732401, 1872
60229 13*41*113533, 484
63005 5*12601116281, 103680
68605 5*1372157696805, 22638564
69085 5*41*3371849, 1521
69509 11*71*892500, 979
70013 53*1321190969, 190800
70189 7*37*271181447, 961
70941 3*13*17*10743681, 25
76245 3*5*13*17*232209, 1
80189 17*53*8953, 36
81469 257*317143641, 135953
81669 3*7*388919663, 11449
82205 5*41*401205, 196
82309 53*15532401, 848
83845 5*41*409245, 164
90597 3*13*23*101126025, 23
91749 3*7*17*2572057, 1
92045 5*41*449162409, 320
92157 3*13*17*139625, 208
99309 3*7*47298836, 4107
99645 3*5*7*13*73125, 21
103309 17*59*103961, 927
108717 3*7*31*167192, 25
109389 3*7*520941503, 169
110589 3*191*193192, 1
110805 3*5*83*89169, 80
115221 3*193*199196, 3
116645 5*41*5695041, 80
125749 13*17*569361, 208
132405 3*5*7*13*9736261232, 13
134805 3*5*11*19*434096, 11
136445 5*29*9411521, 361
137613 3*7*65536889, 336
139893 3*13*17*211841, 425
140005 5*2800143681, 12321
141141 3*7*11*13*476724, 3
141661 13*17*64112769, 1872
143021 17*47*1795687, 5329
148733 13*17*67316641, 6241
150805 5*30161271445, 4
155309 7*11*201711264, 6889
160701 3*17*23*1371225, 1104
160805 5*29*110958081, 6241
161605 5*3232116641, 15680
162285 3*5*31*349605, 93
166957 7*17*23*613388, 1525
170877 3*7*79*103625, 7
174709 17*43*239196, 43
180277 41*43971412251601, 930405200
181589 41*43*1032116, 2107
182005 5*89*409729, 89
187013 23*47*1738649, 1
190805 5*31*12312511, 49
198061 37*53*10125921, 25012
199805 5*89*449449, 441
205933 7*13*31*731444, 819
207021 3*151*45718438436, 9093243
209005 5*4180135628961, 34680321
212749 41*51898256575941, 7563780900
217141 17*53*2413249, 848
218285 5*149*293149, 144
224005 5*71*6315751, 4489
225301 17*29*457529, 457
226005 3*5*13*19*611369, 95
230061 3*13*17*347833, 208
232981 7*83*401484, 83
234069 3*11*41*1732809, 2768
234821 17*19*7273332, 3211
239669 61*392948734361, 7051600
243165 3*5*13*29*43245, 13
246205 5*41*120135721, 13520
248469 3*13*23*2771681, 1127
251669 11*137*167340736, 136161
258621 3*11*17*461561, 361
265109 13*2039310201, 10192
266541 3*11*41*197361, 33
268933 7*103*373235225, 98983
269717 7*53*7271575, 121
270805 5*41*13212601, 41
271765 5*13*37*1131089, 185
276261 3*71*1297124848, 32761
276501 3*37*47*5361009, 1225
278621 7*53*7513481, 2527
280165 5*137*409299209, 261121
280261 47*67*89423, 289
282341 349*809335241, 229441
282405 3*5*67*281281, 121
282613 41*61*11317629, 12996
284005 5*79*719638401, 57591
289445 5*13*61*738649, 841
289765 5*7*17*48720160, 11881
290789 23*47*26913181, 12100
293645 5*11*19*281245, 36
296429 7*17*47*53529, 423
305949 3*7*17*8572057, 343
307365 3*5*31*6616125, 837
310405 5*62081330245, 228484
318237 3*37*47*61218700, 8281
318773 7*13*31*11310923025, 113
319349 41*7789373321, 16129
330941 13*2545750625, 289
331813 41*80931225684987841, 809300000000
335901 3*19*71*83151321, 71
338261 7*11*23*19199, 92
342205 5*89*769445, 324
343181 113*303714100025, 2961841
343493 53*648141899729, 19080000
344757 3*7*16417114244, 675
348909 3*11*97*109139129, 24025
353381 7*19*265710647, 10609
354173 149*237729300569, 1051344
355589 13*17*160942237, 14884
356741 7*11*41*11377, 36
362645 5*29*41*61208849, 64980
363613 17*73*2937325, 3844
367285 5*17*29*1493279721, 2313441
369421 13*157*1811026493, 46656
370069 7*29*182313225, 464
372701 7*37*143947961, 37888
375173 17*29*7611830609, 253009
377405 5*7*41*263346921, 214369
388245 3*5*11*13*1813721, 1911
390949 13*17*29*6160025, 121
391629 3*7*17*10971029, 68
395637 3*11*19*63120667, 475
397421 113*351711789177, 7969329
397797 3*97*136729058096, 25775929
399189 3*7*1900910609, 8400
404549 17*53*44918395521, 68688
410437 29*1415313689, 464
410821 109*376993224432725, 2931995556
411677 7*23*255713689, 8575
416005 5*19*29*1511539, 29
419765 5*37*22694205, 333
420045 3*5*41*68318605, 164
421405 5*271*311144653564, 5335445
421429 23*73*251324, 251
424461 3*151*9373775, 3721
425405 5*8508188445, 3364
425901 3*7*17*1193768, 425
429709 7*17*23*157426409, 28577
434701 19*137*167441, 167
436805 5*199*439700569, 1681
438605 5*8772152441, 35280
440453 13*17*199341209, 10609
443541 3*7*21121120127, 48841
443685 3*5*11*268913456, 11
449837 17*47*563987377, 140625
452341 23*71*277214775, 104329
453277 11*89*4634656964, 4547939
454205 5*90841512886609, 457929481
461029 349*132194300378921, 67017301129
462685 5*37*41*6119321, 3249
465861 3*11*19*7431521691, 27
468741 3*7*13*17*101625, 404
470373 3*17*23*4017921, 1104
473421 3*13*61*1991555009, 2809
477581 13*17*216132761, 6137
485485 5*7*11*13*9797, 20
490845 3*5*43*761279841, 5120
496509 3*13*29*4391369, 52
497005 5*994011202693035383946129, 1166247624159514121
498885 3*5*79*421229441, 1684
504341 41*1230114801, 9801
505421 7*103*701168921, 721
507085 5*37*27415445, 37
508485 3*5*109*3111369, 436
513949 31*59*28111471769, 281
516581 13*79*50321261321, 240448
517789 41*73*17323409, 11072
522805 5*10456110936696805, 401932484
523981 113*46371381878017, 178134992
525941 13*23*175948668, 4693
527845 5*229*46190601, 24649
529581 3*13*37*3674624, 147
535381 7*11*17*4092209, 409
535501 37*41*35339698496, 2096704
536781 3*7*2556191204, 87723
537205 5*107441253374309769, 173564596481
539437 41*59*22381796, 80771
543205 5*13*61*13717689, 121
544357 11*17*41*7134496, 34425
544685 5*41*265726569, 1
545669 41*13309398325061, 305568900
547205 5*109441129605, 20164
550981 19*47*617112847, 89401
551733 3*7*13*43*47700, 141
551973 3*17*79*13760025, 4913
553213 41*103*131116738816, 87036809
569837 37*1540170804125925, 62966742084
571909 13*29*37*413025, 9
572101 17*73*46137341, 36100
573013 7*109*7511666617020581, 1652553195300
573781 13*19*23*1011863, 1369
576821 7*19*43374356, 19
581269 13*61*733247009, 229441
581509 23*131*1932411809, 2783
582981 3*7*17*23*7171, 48
584885 5*7*17*9838836, 7875
585741 3*13*23*653637, 16
585821 37*71*22364447, 17161
587221 29*202497395725, 7365796
592997 733*80977423125, 27988164
594741 3*7*127*223175, 48
597365 5*37*3229345845, 320013
599461 41*1462116118289, 2157961
600061 7*11*77936561, 1232
600773 41*146531952145925, 580258404
605829 3*7*17*169734225, 5376
611749 353*1733570025, 431649
619989 3*19*73*1491125721, 38617
622973 13*173*277225, 52
623733 3*11*41*461625, 297
626381 7*43*208124768, 10201
628845 3*5*7*53*11360, 53
629333 37*73*233848241, 14800
630597 3*11*97*197197, 100
631085 5*7*13*19*73567, 73
632613 3*433*4879597604, 735075
633085 5*53*2389117077, 117045
635789 7*11*23*359103499, 252
637941 3*337*631484, 147
640101 3*7*11*17*163175, 12
647309 13*17*29*1014693, 256
648669 3*7*17*23*7951, 28
649605 3*5*11*31*127361, 20
650301 3*17*41*3111313828, 147
653029 13*191*26314161, 4775
655221 3*7*41*761861, 100
660005 5*132001167281, 96721
671573 7*197*4873721, 175
672605 5*17*41*1936561, 1
673293 3*29*71*10917689, 436
675853 113*59811109664744821, 849604523300
679205 5*1358416390639845, 1971182404
681245 5*19*71*10156169, 2209
686933 13*53*99749729, 121
687797 11*31*201749729, 49104
688541 7*19*31*16713689, 8512
689005 5*41*33615125, 1764
692189 17*19*214349098049, 41694208
697885 5*29*4813199809, 194996
697965 3*5*19*31*792401, 31
702149 7*37*27112368, 343
702429 3*7*13*31*8352, 31
704685 3*5*109*431867, 5
713765 5*13*79*139256, 139
716885 5*13*41*2693721, 1076
719693 13*23*29*831472, 725
720853 7*29*53*6731329, 19604
730229 313*23332817, 1849
731645 5*41*43*831805, 1764
732669 3*7*139*251139, 112
735005 5*29*37*13724505, 14641
739381 17*23*31*61153, 31
740485 5*23*47*13792, 45
744765 3*5*7*41*173189, 16
749405 5*71*2111174845, 24964
749989 17*157*2815041, 17
751485 3*5*7*17*421961, 119
753445 5*7*11*19*1031369, 721
758693 13*17*343327889, 425
759829 7*19*29*197197, 64
762013 7*23*4733196249, 40401
764949 3*17*53*2838957, 5476
765669 3*31*823337975, 27889
770109 3*23*11161284592, 27889
774701 47*53*31115625, 2799
779605 5*155921271441, 40401
780845 5*13*41*2933645, 164
785437 41*191575968026009, 191570000
786445 5*11*79*18189401, 711
789965 5*11*53*27115884, 1521
790621 13*61*997529, 468
791245 5*7*13*37*473388, 333
791997 3*41*47*1373364, 3075
800101 23*43*809985527, 108241
801413 53*151213348238378325, 93450743716
801933 3*11*19*127924300, 1
802405 5*16048117341582643878249, 14458796624428001
802869 3*31*89*97265225, 361
804221 11*113*6474056196, 188771
808917 3*47*57373481, 2256
811941 3*13*109*1912401, 2292
812181 3*97*27912500, 291
813101 569*142946541924, 1758276
813365 5*7*17*13671372, 5
818845 5*389*421405, 16
821541 3*7*19*29*71961, 464
821845 5*19*41*2112009, 2000
824261 101*8161161264601, 41139601
831405 3*5*43*1289734449, 196209
839613 3*17*101*16310609, 1616
840885 3*5*61*9197381, 4624
844781 7*17*31*2294761, 3025
845189 17*83*5991328, 729
848309 7*11*23*47922253, 11236
859677 3*7*13*47*67343, 268
861445 5*13*29*457261, 196
868749 3*7*41*1009525, 484
868965 3*5*19*30499196, 49
871653 3*29*43*2332785561, 313664
873717 3*13*43*521325, 196
876509 61*14369116488849, 76572401
877533 3*73*4007480000, 187489
880157 113*778976454401621, 62033033700
880469 29*97*3132817, 2809
883277 337*2621261942763589, 260857390564
887565 3*5*7*79*107107, 28
889413 3*7*41*10331029, 4
889477 257*3461491908469, 482329444
891429 3*7*11*17*22730899, 27
897005 5*17*61*17378125, 10309
898885 5*13*1382926569, 1089
901901 7*11*13*17*531849, 153
903901 29*71*439375769, 320031
906373 13*113*617697225, 602177
917165 5*53*346196603801985, 92580310224
930741 3*7*23*41*475329, 329
931917 3*7*199*22334969, 33631
933069 3*31*79*12779, 48
933317 7*11*17*23*31275, 252
935085 3*5*17*19*193125, 68
939605 5*187921421201, 233280
950629 353*26933177, 2209
951405 3*5*7*13*17*4180, 39
955245 3*5*43*14811161, 320
956685 3*5*23*47*5934347, 245
957885 3*5*19*3361502445, 8427
961653 3*7*11*23*181756, 575
961989 3*7*19*24112299, 112
966485 5*13*1486942661205, 42542253
970669 7*23*602917161, 5103
971741 41*137*17329237, 21316
975605 5*1951211673951058961, 1447069363920
975973 19*31*16571216, 441
978437 457*2141173289420582269, 6925497509956
979685 5*7*23*121711881, 289
982085 5*13*29*521637, 405
982181 229*428978074977561, 55801815761
984885 3*5*11*47*12780, 47
987861 3*7*470411297321, 1054729
989485 5*7*17*16632268, 605
989989 7*11*13*23*43343, 44
992165 5*61*32531896129, 1230004
992381 13*23*33196647, 9
996149 7*11*17*761761, 625