## ``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 4199 13*17*19 324, 1 4895 5*11*89 45, 44 6671 7*953 841, 112 9015 3*5*601 4805, 3 10199 7*31*47 63, 31 12935 5*13*199 1369, 199 15655 5*31*101 529, 279 16151 31*521 400, 121 16887 3*13*433 5929, 5329 23359 7*47*71 729, 71 28007 7*4001 720801, 716800 29055 3*5*13*149 637, 108 32039 7*23*199 567, 169 32599 7*4657 7252249, 462343 35719 23*1553 3025, 81 36519 3*7*37*47 289, 7 41151 3*11*29*43 44, 43 41943 3*11*31*41 1024, 1 50583 3*13*1297 63001, 1849 51359 7*11*23*29 575, 63 55279 7*53*149 361, 63 58695 3*5*7*13*43 28, 15 59415 3*5*17*233 125, 108 60119 79*761 176400, 116281 60415 5*43*281 605, 43 60847 71*857 39601, 3249 61815 3*5*13*317 637, 3 65535 3*5*17*257 256, 1 65639 7*9377 2155652041, 110339159 68295 3*5*29*157 784, 1 68839 23*41*73 1936, 73 69015 3*5*43*107 108, 107 70959 3*7*31*109 109, 108 72151 23*3137 4545424, 1010025 73055 5*19*769 7569, 121 73151 13*17*331 484, 153 76479 3*13*37*53 1936, 25 80015 5*13*1231 12493, 2645 81959 41*1999 2624, 625 83159 137*607 35072, 5329 94655 5*11*1721 2601, 841 95095 5*7*11*13*19 171, 11 99231 3*11*31*97 32761, 25 100711 13*61*127 244, 117 103159 7*14737 18769, 4032 103727 13*79*101 729, 79 105735 3*5*7*19*53 80, 53 106711 11*89*109 660969, 47081 112119 3*7*19*281 169, 112 113919 3*13*23*127 75, 52 119255 5*17*23*61 245, 61 124815 3*5*53*157 157, 108 128263 47*2729 6330256, 45369 128935 5*107*241 223729, 69649 129935 5*13*1999 17797, 1805 132559 7*29*653 10609, 4732 137111 13*53*199 225, 199 137671 31*4441 366025, 361584 142071 3*23*29*71 4900, 1 142471 7*20353 36481, 4225 143903 151*953 33489, 14161 145439 7*79*263 166464, 21025 148215 3*5*41*241 1156, 49 149495 5*29*1031 2782224, 504745 152095 5*19*1601 14161, 1849 153439 11*13*29*37 400, 81 157591 7*47*479 2601, 752 159999 3*7*19*401 400, 1 164119 337*487 26896, 3033 168823 79*2137 573049, 557424 170671 103*1657 16037942881, 9265557919 172639 31*5569 255025, 17856 186295 5*19*37*53 3721, 95 191735 5*31*1237 178929, 130321 194439 3*7*47*197 169, 28 202279 7*11*37*71 225, 71 202623 3*17*29*137 2209, 2175 202935 3*5*83*163 83, 80 206079 3*73*941 1338649, 525625 211055 5*13*17*191 2401, 191 211935 3*5*71*199 135, 64 212815 5*31*1373 3771364, 323761 213231 3*17*37*113 1813, 1587 213455 5*11*3881 38809, 1 215895 3*5*37*389 3721, 169 216039 3*23*31*101 124, 23 220215 3*5*53*277 13520, 53 221879 7*29*1093 50625, 12769 223727 7*31*1031 389376, 165649 225607 17*23*577 4464769, 410881 226159 23*9833 8415801, 1033712 226335 3*5*79*191 507, 125 230815 5*13*53*67 2197, 53 234255 3*5*7*23*97 484, 1 235911 3*13*23*263 289, 263 238791 3*7*83*137 137, 112 241039 41*5879 48891024, 207025 242351 23*41*257 22201, 1127 245479 13*23*821 729, 92 246559 79*3121 1600, 1521 249135 3*5*17*977 5329, 3375 250271 7*35753 22201, 13552 254535 3*5*71*239 961, 956 256039 7*79*463 2333983, 2178576 257439 3*7*13*23*41 325, 3 259311 3*13*61*109 61, 48 259831 11*13*23*79 961, 92 260015 5*7*17*19*23 207, 17 261919 7*17*31*71 567, 425 263495 5*151*349 38809, 7399 266695 5*11*13*373 4805, 44 267415 5*79*677 361, 316 267735 3*5*13*1373 6877, 12 269535 3*5*7*17*151 135, 16 271551 3*7*67*193 268, 75 273023 37*47*157 11458225, 293327 281271 3*29*53*61 1600, 169 287079 3*13*17*433 637, 204 288615 3*5*71*271 529, 284 293415 3*5*31*631 496, 135 293455 5*19*3089 18605, 9196 294063 3*7*11*19*67 100, 33 301335 3*5*20089 11449, 8640 306631 13*103*229 18498601, 173143 307247 113*2719 2832196165569, 2547127792831 310295 5*229*271 729, 271 318655 5*101*631 1352569, 5679 319871 79*4049 15681600, 7921 324911 31*47*223 16129, 2303 327543 3*23*47*101 3003289, 1216700 327711 3*313*349 3374569, 3192913 329063 7*29*1621 91809, 2209 329415 3*5*21961 2292196, 452929 331215 3*5*71*311 311, 169 334055 5*71*941 6889, 639 334631 11*29*1049 219501, 116699 336399 3*7*83*193 1600, 249 342127 359*953 13132243216, 12731150825 344311 11*113*277 103041, 5537 350895 3*5*149*157 23104, 289 351799 7*29*1733 54289, 46225 352839 3*337*349 235225, 1 353855 5*17*23*181 879844, 9409 355607 7*37*1373 62001, 39601 362455 5*71*1021 6241, 3969 365087 31*11777 743324332800625, 730848666278513 367807 11*29*1153 69629, 11979 372215 5*17*29*151 3721, 2416 373711 13*17*19*89 2304, 2057 374415 3*5*109*229 169, 60 381983 7*197*277 45892529, 10763200 382215 3*5*83*307 1452, 83 383831 7*54833 62001, 7168 385535 5*83*929 4356, 289 392231 7*137*409 4096, 1233 393679 11*13*2753 9477, 3971 394719 3*13*29*349 24389, 11907 394927 13*17*1787 806404, 790321 394935 3*5*113*233 57121, 735 395319 3*313*421 529, 313 396799 13*131*233 139129, 12321 401135 5*7*73*157 22801, 121 404551 7*57793 81830895721, 214007479 405015 3*5*13*31*67 80, 13 408319 23*41*433 102400, 12769 408559 127*3217 16474800673, 4255344289 412055 5*7*61*193 305, 81 413663 569*727 40370855625, 39392001367 416879 19*37*593 21316, 625 423599 11*97*397 1343281, 582169 424479 3*11*19*677 1100, 931 424535 5*197*431 1214404, 450241 433879 137*3167 9705599289, 4350140528 434135 5*13*6679 35344, 28665 435455 5*17*47*109 16641, 799 437871 3*7*29*719 18769, 719 442335 3*5*37*797 676, 121 448663 31*41*353 6031936, 243089 450079 7*113*569 462400, 12321 450255 3*5*13*2309 17161, 5929 451191 3*13*23*503 1127, 121 458311 7*233*281 1628176, 8649 459543 3*7*79*277 5887, 4225 463287 3*11*101*139 220323, 13475 465727 23*20249 6926400625, 2347729807 473263 7*17*41*97 369, 175 478239 3*23*29*239 6889, 1392 488047 7*113*617 44521, 25200 488215 5*7*13*29*37 2116, 289 488607 3*7*53*439 469225, 329623 490919 11*13*3433 243997, 3179 492479 13*43*881 1053, 172 493647 3*7*11*2137 63175, 5476 498095 5*13*79*97 7569, 5041 506391 3*23*41*179 3481, 537 508335 3*5*33889 148996, 20449 509415 3*5*33961 1972488841, 107765161 509615 5*227*449 676, 449 511879 19*29*929 115101, 70699 513383 13*17*23*101 2209, 391 514759 7*151*487 37249, 36288 521255 5*7*53*281 88804, 289 524095 5*11*13*733 85805, 44 524167 7*103*727 42925456, 700569 525215 5*17*37*167 500, 333 529679 41*12919 16400, 3481 530503 31*109*157 11242609, 2447791 534359 7*23*3319 69169, 7168 538215 3*5*53*677 5929, 841 542095 5*181*599 923521, 56169 550511 13*17*47*53 5041, 153 553735 5*7*13*1217 19321, 12321 555719 263*2113 7500000528, 2643913561 572631 3*23*43*193 7803, 1075 601455 3*5*101*397 505, 289 603519 3*7*29*991 1600, 609 606599 7*193*449 256, 193 611391 3*11*97*191 32279, 25921 617999 409*1511 665856, 483025 622391 7*11*59*137 1859, 59 622679 23*27073 501264, 175561 625439 23*71*383 687241, 255600 631031 41*15391 237145025, 15760384 631687 7*31*41*71 2624, 1849 637935 3*5*71*599 54289, 11760 640159 13*23*2141 16104169, 15959447 640815 3*5*7*17*359 359, 121 642295 5*19*6761 8281, 1520 645351 3*7*79*389 16807, 13924 650695 5*181*719 729, 719 653815 5*43*3041 11236, 3969 656095 5*11*79*151 1062961, 79 661055 5*29*47*97 11664, 7105 664215 3*5*44281 10731125, 15123 667615 5*13*10271 13456, 3185 668247 3*29*7681 237169, 139200 669151 7*109*877 4489, 2527 675591 3*7*53*607 607, 289 692479 263*2633 2831446577, 2774519552 694991 11*23*41*67 369, 368 698295 3*5*13*3581 17797, 108 700151 37*127*149 43609469, 23025100 705399 3*73*3221 158404, 76729 708135 3*5*17*2777 94249, 169 716887 23*71*439 8970025, 6647 722055 3*5*37*1301 9409, 999 729735 3*5*48649 12243001, 1816560 730655 5*29*5039 5184, 145 732279 3*19*29*443 228484, 1225 737295 3*5*13*19*199 147, 52 738255 3*5*7*79*89 84, 5 738335 5*13*37*307 320, 13 739159 839*881 17606416, 11256025 740391 3*47*59*89 841, 752 743399 47*15817 76729, 60912 745503 3*11*19*29*41 19456, 1331 746351 17*43*1021 8228, 961 746511 3*23*31*349 581839, 244007 754151 761*991 779264, 255025 758095 5*13*107*109 287296, 182329 758535 3*5*61*829 1805, 147 762559 7*41*2657 2169729, 633983 770959 7*241*457 2639632, 2152089 777439 13*79*757 21169201, 16935151 781255 5*37*41*103 144, 41 783439 313*2503 1957815313, 62575 789495 3*5*7*73*103 8464, 945 789559 127*6217 900601, 896112 789735 3*5*17*19*163 243, 80 790727 7*37*43*71 3025, 28 792143 11*23*31*101 2401, 124 800079 3*7*31*1229 868, 361 801879 3*13*29*709 361, 348 811855 5*11*29*509 1315609, 16641 817071 3*17*37*433 6400, 961 821215 5*23*37*193 35344, 361 822647 7*17*31*223 1092025, 217 824735 5*281*587 224676, 12769 830031 3*337*821 484, 337 830319 3*7*19*2081 1101297, 1100401 835863 3*7*53*751 16129, 3004 836591 7*119513 238521861769, 233910963113 839399 11*137*557 49649348, 815409 839759 31*103*263 9801, 1648 840487 223*3769 183739760044096, 128118356800929 850367 7*29*59*71 982081, 465831 852295 5*17*37*271 117649, 39601 852663 3*7*19*2137 78961, 27889 857679 3*19*41*367 5041, 367 862223 17*67*757 320356, 1369 873871 47*18593 30625, 6561 876143 53*61*271 26244, 11881 880815 3*5*13*4517 12493, 10092 888063 3*11*17*1583 1600, 17 889935 3*5*79*751 22201, 3004 891215 5*13*13711 50960, 37249 893535 3*5*71*839 839, 484 897519 3*7*79*541 553, 529 898215 3*5*233*257 245, 12 898247 7*128321 6316397063383696, 2937212749235529 900271 31*113*257 144, 113 904255 5*11*41*401 20187049, 18740241 905519 337*2687 876096, 744433 905695 5*7*113*229 700569, 38759 908895 3*5*13*59*79 59, 20 911031 3*11*19*1453 6859, 1859 916255 5*29*71*89 17161, 11999 916391 7*31*41*103 17161, 12400 918255 3*5*13*17*277 277, 243 925519 7*109*1213 8743849, 8179927 928855 5*23*41*197 5476, 2601 929591 13*23*3109 70225, 10609 929999 7*132857 100489, 32368 934199 7*317*421 173889, 93025 935879 7*133697 263169, 4225 937167 3*7*11*4057 19899275, 604803 938231 7*134033 221841, 46225 938311 11*197*433 16296388, 1315609 939455 5*11*19*29*31 551, 441 941215 5*11*109*157 38809, 441 941239 13*17*4259 27019204, 881721 946151 23*31*1327 7744, 6417 946567 41*23087 266505625, 946567 954551 13*101*727 104329, 87967 954655 5*13*19*773 10125, 76 957615 3*5*63841 3214849, 86640 959871 3*11*17*29*59 539, 464 961495 5*19*29*349 3509, 19 965543 383*2521 177037225, 25692023 965671 7*29*67*71 26569, 25631 966847 7*37*3733 177241, 9409 977015 5*13*15031 114594493, 27130955 977943 3*181*1801 645866101, 91310700 978711 3*41*73*109 1168, 841 981615 3*5*31*2111 57121, 124 985647 3*13*127*199 24649, 624 990647 7*137*1033 187553, 161728 991039 7*31*4567 389872, 381951 996079 7*142297 4669104177721, 131414267737