``Congruent number curves'' ny2=x3-x
with |n| congruent to 7 mod 8 and less than a million:
Positivity of rank, and list of curves of analytic 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. Fermat proved in effect that the curve E(1) has no rational points of infinite order, and deduced the exponent-4 case of his ``last theorem''. Thus 1 is not a congruent number. For general n, the curve E(n) is a ``quadratic twist'' of E(1), and its group of rational torsion points consists of the point at infinity and the 2-torsion points (0,0), (n,0), (-n,0). Thus n is congruent if and only if E(n) has positive rank. For more information about elliptic curves in general and the family of curves E(n) in particular see N.Koblitz's text Introduction to elliptic curves and modular forms, New York: Springer 1984.

It is known that the first congruent numbers are 5, 6, and 7; for instance, 6 is the area of the 3-4-5 triangle, and the common difference of {(1/2)2, (5/2)2, (7/2)2}, while 5 is the area of the triangle of sides 9/6, 40/6, 41/6 and the common difference of {(31/12)2, (41/12)2, (49/12)2}. In each of these three cases, E(n) has rank 1. In fact it is known that the sign of the functional equation of L(E(n),s) is +1 for n congruent to 1, 2, or 3 mod 8, and -1 for n congruent to 5, 6, or 7 mod 8. Thus the conjecture of Birch and Swinnerton-Dyer implies that the rank of E(n) is even if n is of the form 8k+1, 8k+2, or 8k+3, and odd if n is of the form 8k+5, 8k+6, or 8k+7. In particular, if n=8k+5, 8k+6, or 8k+7 then n is conjectured to be congruent.

Theorem. If n=8k+7 and n<106 then n is a congruent number.

The proof combines computation and theory. In my paper ``Heegner point computations'' [Lecture Notes in Computer Science 877 (proceedings of ANTS-1, 5/1994), 122-133], I announced this result with for n<200000; the extension to 106 combines the same theoretical tools with new software (Cremona's MWRANK) and faster hardware.

The main tool is the use of Heegner points of discriminant (-n) on the modular curve X0(32) to construct a rational point P(n) on E(n). The only possible stumbling block is that P(n) may be a torsion point. But this can be detected in time essentially Cn1/2 by calculating a real approximation to P(n) with low precision -- we used 19 digits in gp. We found only 339 cases, tabulated below, where P(n) is a torsion point.

Now by the celebrated formula of B.H.Gross and D.Zagier [see their paper ``Heegner points and derivatives of L-series'', Invent. Math. 84 (1986), 225-320], the canonical height of P(n) is proportional to the value at s=1 of the first derivative of the L-series attached to E(n). [Perhaps unexpectedly, testing whether P(n) is a torsion point is faster than directly computing L'(E(n),1), which takes time >>n.] Thus P(n) is a torsion point if and only if that derivative vanishes, that is, if and only if E(n) has analytic rank at least 3. So, it is precisely when the Heegner construction fails that we expect E(n) to have many rational points.

Now the usual conjectures and heuristics suggest that the regulator of E(n) can grow roughly as n1/2. Thus the minimal nonzero height of a rational point on E(n) can grow roughly as n1/2 when E(n) has rank 1, but only as n1/6 when E(n) has rank 3. Still, as n increases, even a point of height roughly n1/6 can be hard to find, let alone three independent points. Even with MWRANK, we were not able to find a rank-3 subgroup of each of our 339 curves E(n). Nevertheless, in every case MWRANK found at least one nontorsion point, and this sufficed to complete the proof that every positive integer n=8k+7<106 is congruent. This computation took several days on a 500MHz machine.

Below we list the 339 values of n<106 for which the curve has analytic rank at least 3, together with their prime factorizations. (For the conjectural extension of this table to 107, go here.) We also give the numerator and denominator of the x-coordinate of a nontrivial rational solution of ny2=x3-x. In all 339 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 7 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 7 exceptional cases are:
n = 68839 = 23 * 41 * 73
n = 450079 = 7 * 113 * 569
n = 473263 = 7 * 17 * 41 * 97
n = 488047 = 7 * 113 * 617
n = 606599 = 7 * 193 * 449
n = 900271 = 31 * 113 * 257
n = 990647 = 7 * 137 * 1033
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.

Caveat: It is in principle conceivable that for one or more of these n the point P(n) might be actually a nontorsion point that is extremely close to the origin of E(n) over R; in that case E(n) would have analytic rank 1, not 3. But in that case, V.A. Kolyvagin's theorem (see his ``Euler systems'' on pages 435-483 of The Grothendieck Festscrhift Vol.II, Birkhäuser: Boston 1990) shows that E(n) would also have arithmetic rank 1. Thus it is enough to exhibit 2 independent points on E(n) to prove that E(n) has analytic rank at least 3. Even when MWRANK finds only one nontrivial point, say Q, on E(n), the Heegner point P(n) would have to be a small nonzero multiple of Q; since P(n) is very close to the origin, the real elliptic logarithm of Q would thus have a very good rational approximation. This can be ruled out by calculating a continued fraction of this elliptic logarithm. The only difficulty is giving an a priori upper bound on the height of P(n), and thus on the denominator of a rational approximation to log(Q), that is small enough to make this continued-fraction calculation feasible. Such a bound must be straightforward if not pleasant to obtain, starting from the Gross-Zagier formula relating the height of P(n) with L'(E(n),1). Once this is done, it will be easy to complete a rigorous proof that the analytic rank E(n) is exactly 3 for each of the 339 tabulated values of n, and 1 for each of the other 100990 squarefree n=8k+7 less than a million.


n n, factored numerator and denominator of x
419913*17*19 324, 1
48955*11*89 45, 44
66717*953 841, 112
90153*5*601 4805, 3
101997*31*47 63, 31
129355*13*199 1369, 199
156555*31*101 529, 279
1615131*521 400, 121
168873*13*433 5929, 5329
233597*47*71 729, 71
280077*4001 720801, 716800
290553*5*13*149 637, 108
320397*23*199 567, 169
325997*4657 7252249, 462343
3571923*1553 3025, 81
365193*7*37*47 289, 7
411513*11*29*43 44, 43
419433*11*31*41 1024, 1
505833*13*1297 63001, 1849
513597*11*23*29 575, 63
552797*53*149 361, 63
586953*5*7*13*43 28, 15
594153*5*17*233 125, 108
6011979*761 176400, 116281
604155*43*281 605, 43
6084771*857 39601, 3249
618153*5*13*317 637, 3
655353*5*17*257 256, 1
656397*9377 2155652041, 110339159
682953*5*29*157 784, 1
6883923*41*73 1936, 73
690153*5*43*107 108, 107
709593*7*31*109 109, 108
7215123*3137 4545424, 1010025
730555*19*769 7569, 121
7315113*17*331 484, 153
764793*13*37*53 1936, 25
800155*13*1231 12493, 2645
8195941*1999 2624, 625
83159137*607 35072, 5329
946555*11*1721 2601, 841
950955*7*11*13*19 171, 11
992313*11*31*97 32761, 25
10071113*61*127 244, 117
1031597*14737 18769, 4032
10372713*79*101 729, 79
1057353*5*7*19*53 80, 53
10671111*89*109 660969, 47081
1121193*7*19*281 169, 112
1139193*13*23*127 75, 52
1192555*17*23*61 245, 61
1248153*5*53*157 157, 108
12826347*2729 6330256, 45369
1289355*107*241 223729, 69649
1299355*13*1999 17797, 1805
1325597*29*653 10609, 4732
13711113*53*199 225, 199
13767131*4441 366025, 361584
1420713*23*29*71 4900, 1
1424717*20353 36481, 4225
143903151*953 33489, 14161
1454397*79*263 166464, 21025
1482153*5*41*241 1156, 49
1494955*29*1031 2782224, 504745
1520955*19*1601 14161, 1849
15343911*13*29*37 400, 81
1575917*47*479 2601, 752
1599993*7*19*401 400, 1
164119337*487 26896, 3033
16882379*2137 573049, 557424
170671103*1657 16037942881, 9265557919
17263931*5569 255025, 17856
1862955*19*37*53 3721, 95
1917355*31*1237 178929, 130321
1944393*7*47*197 169, 28
2022797*11*37*71 225, 71
2026233*17*29*137 2209, 2175
2029353*5*83*163 83, 80
2060793*73*941 1338649, 525625
2110555*13*17*191 2401, 191
2119353*5*71*199 135, 64
2128155*31*1373 3771364, 323761
2132313*17*37*113 1813, 1587
2134555*11*3881 38809, 1
2158953*5*37*389 3721, 169
2160393*23*31*101 124, 23
2202153*5*53*277 13520, 53
2218797*29*1093 50625, 12769
2237277*31*1031 389376, 165649
22560717*23*577 4464769, 410881
22615923*9833 8415801, 1033712
2263353*5*79*191 507, 125
2308155*13*53*67 2197, 53
2342553*5*7*23*97 484, 1
2359113*13*23*263 289, 263
2387913*7*83*137 137, 112
24103941*5879 48891024, 207025
24235123*41*257 22201, 1127
24547913*23*821 729, 92
24655979*3121 1600, 1521
2491353*5*17*977 5329, 3375
2502717*35753 22201, 13552
2545353*5*71*239 961, 956
2560397*79*463 2333983, 2178576
2574393*7*13*23*41 325, 3
2593113*13*61*109 61, 48
25983111*13*23*79 961, 92
2600155*7*17*19*23 207, 17
2619197*17*31*71 567, 425
2634955*151*349 38809, 7399
2666955*11*13*373 4805, 44
2674155*79*677 361, 316
2677353*5*13*1373 6877, 12
2695353*5*7*17*151 135, 16
2715513*7*67*193 268, 75
27302337*47*157 11458225, 293327
2812713*29*53*61 1600, 169
2870793*13*17*433 637, 204
2886153*5*71*271 529, 284
2934153*5*31*631 496, 135
2934555*19*3089 18605, 9196
2940633*7*11*19*67 100, 33
3013353*5*20089 11449, 8640
30663113*103*229 18498601, 173143
307247113*2719 2832196165569, 2547127792831
3102955*229*271 729, 271
3186555*101*631 1352569, 5679
31987179*4049 15681600, 7921
32491131*47*223 16129, 2303
3275433*23*47*101 3003289, 1216700
3277113*313*349 3374569, 3192913
3290637*29*1621 91809, 2209
3294153*5*21961 2292196, 452929
3312153*5*71*311 311, 169
3340555*71*941 6889, 639
33463111*29*1049 219501, 116699
3363993*7*83*193 1600, 249
342127359*953 13132243216, 12731150825
34431111*113*277 103041, 5537
3508953*5*149*157 23104, 289
3517997*29*1733 54289, 46225
3528393*337*349 235225, 1
3538555*17*23*181 879844, 9409
3556077*37*1373 62001, 39601
3624555*71*1021 6241, 3969
36508731*11777 743324332800625, 730848666278513
36780711*29*1153 69629, 11979
3722155*17*29*151 3721, 2416
37371113*17*19*89 2304, 2057
3744153*5*109*229 169, 60
3819837*197*277 45892529, 10763200
3822153*5*83*307 1452, 83
3838317*54833 62001, 7168
3855355*83*929 4356, 289
3922317*137*409 4096, 1233
39367911*13*2753 9477, 3971
3947193*13*29*349 24389, 11907
39492713*17*1787 806404, 790321
3949353*5*113*233 57121, 735
3953193*313*421 529, 313
39679913*131*233 139129, 12321
4011355*7*73*157 22801, 121
4045517*57793 81830895721, 214007479
4050153*5*13*31*67 80, 13
40831923*41*433 102400, 12769
408559127*3217 16474800673, 4255344289
4120555*7*61*193 305, 81
413663569*727 40370855625, 39392001367
41687919*37*593 21316, 625
42359911*97*397 1343281, 582169
4244793*11*19*677 1100, 931
4245355*197*431 1214404, 450241
433879137*3167 9705599289, 4350140528
4341355*13*6679 35344, 28665
4354555*17*47*109 16641, 799
4378713*7*29*719 18769, 719
4423353*5*37*797 676, 121
44866331*41*353 6031936, 243089
4500797*113*569 462400, 12321
4502553*5*13*2309 17161, 5929
4511913*13*23*503 1127, 121
4583117*233*281 1628176, 8649
4595433*7*79*277 5887, 4225
4632873*11*101*139 220323, 13475
46572723*20249 6926400625, 2347729807
4732637*17*41*97 369, 175
4782393*23*29*239 6889, 1392
4880477*113*617 44521, 25200
4882155*7*13*29*37 2116, 289
4886073*7*53*439 469225, 329623
49091911*13*3433 243997, 3179
49247913*43*881 1053, 172
4936473*7*11*2137 63175, 5476
4980955*13*79*97 7569, 5041
5063913*23*41*179 3481, 537
5083353*5*33889 148996, 20449
5094153*5*33961 1972488841, 107765161
5096155*227*449 676, 449
51187919*29*929 115101, 70699
51338313*17*23*101 2209, 391
5147597*151*487 37249, 36288
5212555*7*53*281 88804, 289
5240955*11*13*733 85805, 44
5241677*103*727 42925456, 700569
5252155*17*37*167 500, 333
52967941*12919 16400, 3481
53050331*109*157 11242609, 2447791
5343597*23*3319 69169, 7168
5382153*5*53*677 5929, 841
5420955*181*599 923521, 56169
55051113*17*47*53 5041, 153
5537355*7*13*1217 19321, 12321
555719263*2113 7500000528, 2643913561
5726313*23*43*193 7803, 1075
6014553*5*101*397 505, 289
6035193*7*29*991 1600, 609
6065997*193*449 256, 193
6113913*11*97*191 32279, 25921
617999409*1511 665856, 483025
6223917*11*59*137 1859, 59
62267923*27073 501264, 175561
62543923*71*383 687241, 255600
63103141*15391 237145025, 15760384
6316877*31*41*71 2624, 1849
6379353*5*71*599 54289, 11760
64015913*23*2141 16104169, 15959447
6408153*5*7*17*359 359, 121
6422955*19*6761 8281, 1520
6453513*7*79*389 16807, 13924
6506955*181*719 729, 719
6538155*43*3041 11236, 3969
6560955*11*79*151 1062961, 79
6610555*29*47*97 11664, 7105
6642153*5*44281 10731125, 15123
6676155*13*10271 13456, 3185
6682473*29*7681 237169, 139200
6691517*109*877 4489, 2527
6755913*7*53*607 607, 289
692479263*2633 2831446577, 2774519552
69499111*23*41*67 369, 368
6982953*5*13*3581 17797, 108
70015137*127*149 43609469, 23025100
7053993*73*3221 158404, 76729
7081353*5*17*2777 94249, 169
71688723*71*439 8970025, 6647
7220553*5*37*1301 9409, 999
7297353*5*48649 12243001, 1816560
7306555*29*5039 5184, 145
7322793*19*29*443 228484, 1225
7372953*5*13*19*199 147, 52
7382553*5*7*79*89 84, 5
7383355*13*37*307 320, 13
739159839*881 17606416, 11256025
7403913*47*59*89 841, 752
74339947*15817 76729, 60912
7455033*11*19*29*41 19456, 1331
74635117*43*1021 8228, 961
7465113*23*31*349 581839, 244007
754151761*991 779264, 255025
7580955*13*107*109 287296, 182329
7585353*5*61*829 1805, 147
7625597*41*2657 2169729, 633983
7709597*241*457 2639632, 2152089
77743913*79*757 21169201, 16935151
7812555*37*41*103 144, 41
783439313*2503 1957815313, 62575
7894953*5*7*73*103 8464, 945
789559127*6217 900601, 896112
7897353*5*17*19*163 243, 80
7907277*37*43*71 3025, 28
79214311*23*31*101 2401, 124
8000793*7*31*1229 868, 361
8018793*13*29*709 361, 348
8118555*11*29*509 1315609, 16641
8170713*17*37*433 6400, 961
8212155*23*37*193 35344, 361
8226477*17*31*223 1092025, 217
8247355*281*587 224676, 12769
8300313*337*821 484, 337
8303193*7*19*2081 1101297, 1100401
8358633*7*53*751 16129, 3004
8365917*119513 238521861769, 233910963113
83939911*137*557 49649348, 815409
83975931*103*263 9801, 1648
840487223*3769 183739760044096, 128118356800929
8503677*29*59*71 982081, 465831
8522955*17*37*271 117649, 39601
8526633*7*19*2137 78961, 27889
8576793*19*41*367 5041, 367
86222317*67*757 320356, 1369
87387147*18593 30625, 6561
87614353*61*271 26244, 11881
8808153*5*13*4517 12493, 10092
8880633*11*17*1583 1600, 17
8899353*5*79*751 22201, 3004
8912155*13*13711 50960, 37249
8935353*5*71*839 839, 484
8975193*7*79*541 553, 529
8982153*5*233*257 245, 12
8982477*128321 6316397063383696, 2937212749235529
90027131*113*257 144, 113
9042555*11*41*401 20187049, 18740241
905519337*2687 876096, 744433
9056955*7*113*229 700569, 38759
9088953*5*13*59*79 59, 20
9110313*11*19*1453 6859, 1859
9162555*29*71*89 17161, 11999
9163917*31*41*103 17161, 12400
9182553*5*13*17*277 277, 243
9255197*109*1213 8743849, 8179927
9288555*23*41*197 5476, 2601
92959113*23*3109 70225, 10609
9299997*132857 100489, 32368
9341997*317*421 173889, 93025
9358797*133697 263169, 4225
9371673*7*11*4057 19899275, 604803
9382317*134033 221841, 46225
93831111*197*433 16296388, 1315609
9394555*11*19*29*31 551, 441
9412155*11*109*157 38809, 441
94123913*17*4259 27019204, 881721
94615123*31*1327 7744, 6417
94656741*23087 266505625, 946567
95455113*101*727 104329, 87967
9546555*13*19*773 10125, 76
9576153*5*63841 3214849, 86640
9598713*11*17*29*59 539, 464
9614955*19*29*349 3509, 19
965543383*2521 177037225, 25692023
9656717*29*67*71 26569, 25631
9668477*37*3733 177241, 9409
9770155*13*15031 114594493, 27130955
9779433*181*1801 645866101, 91310700
9787113*41*73*109 1168, 841
9816153*5*31*2111 57121, 124
9856473*13*127*199 24649, 624
9906477*137*1033 187553, 161728
9910397*31*4567 389872, 381951
9960797*142297 4669104177721, 131414267737