Subject: Congruent Numbers
    Date: Thu, 27 Jan 2000 09:54:00 +0000
    From: Allan MacLeod
      To: kc2h-msm@asahi-net.or.jp



Hisanori,

Apologies for not being in contact recently - I have been busy with
University teaching and administration.

I have also been trying to understand and program a very powerful
method for congruent numbers developed by Noam Elkies of Harvard.
This uses the theory of modular forms, Heegner points and arithmetic
in quadratic fields to find points on the associated elliptic curve.

Eventually, I have managed to get a program to work in most cases,
though it is still very rough and needs much improvement. The power
and
speed are staggering.

I have used the program to compute the solutions to all the unsolved
values
in your table of solutions for N in 1 to 999. I append a copy of the
output.
I think all the values are correct but you had better check.

The values satisfy    g * square =  m * n * ( m + n ) * ( m - n ) in
accordance
with your table.

Allan MacLeod



 g =  367

 m =  2  6837 5455 3009 9241 3724 9155 8895 9538 1717 2918 3296

 n =  24  6962 9099 8107 5733 7255 2161 5532 8078 1143 5729



 g =  373

 m =  2528  3977 2306 2320 9731 7918 8676 8272 1549 3925

 n =  268  2186 4374 3370 5561 7547 4368 8693 9517 6964



 g =  503

 m =  158  8156 9125 1403 5978 0032 9846 6006 1332 5986 3766 7216

 n =  154  9240 9457 3177 6002 5384 6760 7808 1128 5576 5070 8009



 g =  599

 m =  3  0388 7091 1771 0118 8635 4576 3697 2957 1392 5202 3401 0000

 n =  1  6849 8831 6181 2889 7400 2607 1117 6358 5731 0710 2633 8361



 g =  607

 m =  1  2529 3941 1590 1357 8901 9686 1842 5335 4519
              1978 0530 0156 7885 0455 8847 2327 2256

 n =  2810  6647 0235 4661 6494 5654 1073 5723 3619
            5036 9596 8347 4461 1141 4772 5359 4769



 g =  647

 m =  1  3221 6061 2567 0673 2051 8390 3999 3435 5553
         8791 0049 9050 4499 9699 1864 0279 7429 3776

 n =  5826  7537 4825 3676 2231 0345 7197 3399 3297
       7708 1642 4257 8958 2416 3140 8485 2088 9449



 g =  653

 m =  1  6039 9063 5100 2089 6467 2660 1004 6139
         5608 3758 6455 4726 9701 7463 9826 5925

 n =  1  4182 0598 9183 9277 7995 1537 6046 5575
         6788 4269 2446 1686 4333 5856 3527 6676



 g =  677

 m =  4792  7715 4558 9072 5716 8696 9551 0411 0879
            3031 2812 5110 7904 4688 0382 4268 0325

 n =  3709  2798 3854 1840 2372 4208 3736 3147 4182
            2616 0953 7793 7176 7824 3119 1803 8756



 g =  701

 m =  424  4415 7147 5461 2081 1336 9493 0755 0489 9525

 n =  415  3875 6451 0365 8613 7296 8392 9127 3876 2084



 g =  718

 m =  4094  9496 1582 7525 3352 9715 1824 5000

 n =  649  9888 4815 6032 3372 9614 5156 5601



 g =  727

 m =  262  8662 8344 5837 5654 7819 7874 9230 0555 0901
           5247 1350 9432 2282 8208 7904 3392 8203 1376

 n =  134  7438 0738 4934 8860 4816 8596 6948 1699 2039
           9310 0700 3491 1744 2207 1019 4086 7994 5849



 g =  733

 m =  422  0204 7131 5819 9638 8978 2046 4606
           2320 7774 7694 5363 0517 1806 5925

 n =  308  7570 5726 2939 5706 7602 3276 8095
           5336 2788 4575 2827 1358 2713 4404



 g =  743

 m =  4420  8191 2085 1415 8006 8205 9243 8113
            0314 2845 3368 3543 5846 8789 1856

 n =  49  2354 5999 3945 0250 6171 9331 2552
          4998 7968 5494 5340 5719 1699 6169



 g =  757

 m =  1360  2273 1076 9475 6028 7984 4202 1854 6332 1979
       0498 4924 0729 1327 1204 1327 2088 8378 1918 9925

 n =  624  4756 7155 7461 4879 4554 6825 2566 4692 1945
      2570 7667 4703 9517 8208 5802 9581 9437 6270 8004



 g =  758

 m =  294  4848 7134 9968 3481 5262 3041 4885 1250

 n =  1  8606 2309 8621 7412 7714 9298 9603 3329



 g =  797

 m =  337  7442 2602 9191 4058 5730 4114 8996 9262 1955 8640
           9011 7955 6563 9723 4285 8629 0973 9042 8739 2325

 n =  59  9664 3567 8747 4158 7549 7079 5456 7755 8730 6537
          0996 8681 5412 8163 4105 9484 1412 9621 5084 4516



 g =  823

 m =  324  2635 4552 6458 5956 6015 1861 9150 7035 3454 9412
                6763 4772 5149 4026 4277 8377 5097 0916 8016

 n =  119  5374 8708 0585 5749 9617 5911 8845 1486 9228 6556
                1334 0986 6986 9108 0531 8758 5765 3787 8009



 g =  829

 m =  1030  0552 4084 3789 5031 1365 0334 9266 9476 1339 1011 2202 6725

 n =  177  7505 7363 7318 8221 0454 6934 6144 1343 6162 6939 2747 7444



 g =  838

 m =  128  3003 9953 6533 7446 2149 4450

 n =  3  0132 9945 2291 5750 0127 5249



 g =  853

 m =  20  6364 4701 2147 1356 0377 8013 8733 2325

 n =  17  1457 3714 6384 7530 3870 5752 0181 5556



 g =  862

 m =  2016  6436 6890 9039 5690 8515 2743 0984 1328 7200

 n =  1974  3307 1696 0706 7042 2812 9795 6672 3139 5761



 g =  863

 m =  186  5955 0413 4843 2071 7134 1957 0491 7433 0489 6576

 n =  185  8782 5751 2140 2159 1774 6719 9382 2210 0554 6449



 g =  877

 m =  8  4260 2397 8582 9115 0383 0444 6902 6760 1350 4338 3142 1925

 n =  813  0712 5890 3813 6408 9453 1466 7053 8087 0027 4795 3764



 g =  887

 m =  453  6847 8993 9719 4814 7823 8311 2783 7281
           0415 2935 4097 8786 3620 8877 3109 8256

 n =  37  6228 2878 9623 1431 7323 7823 2468 7168
          1076 9990 2241 8840 8143 5674 6097 6969



 g =  911

 m =  103  3760 3501 1683 6147 9121 0723 1758
           9371 8231 3326 3769 2693 0946 5600

 n =  102  5103 8973 8429 8879 2231 4506 2287
           5404 4122 2967 2018 1761 3692 7201



 g =  958

 m =  82  2722 4286 0637 9779 8196 8208 9217 2800

 n =  75  3645 9073 4894 8753 5964 6031 3445 4081



 g =  967

 m =  1669  1589 1789 5536 2517 0458 9403 7705 6310 4478 6868 6474 5616

 n =  991  9363 7345 1122 6302 6388 5658 1596 4423 8540 6084 5569 9609



 g =  983

 m =  3  4499 6141 6670 3093 9643 2781 0353 7861
         5423 2194 0812 0248 1758 7529 7175 6816

 n =  2  7595 0111 2582 6459 2226 7797 7624 3785
         6566 6052 7271 9181 9391 3004 0012 9209



 g =  997

 m =  42  2137 0976 8307 5141 7168 6429 8903 6348 8527 3173 1642 7348
          8445 0430 7265 3296 5501 5861 1521 9772 6745 5373 0824 8325

 n =  3  7615 5528 4456 8642 4185 4415 3311 5738 6556 1361 2025 3716
         4833 7903 7244 1219 0017 1126 5289 4274 8431 9356 4492 2916



 g =  998

 m =  99  0175 0748 1041 7659 1907 8929 8392 4723 4450

 n =  45  6840 9252 1200 9253 2538 6025 7161 1273 7489