``` Subject: Congruent Numbers
Date: Tue, 09 Feb 1999 16:45:16 +0000
From: Allan MacLeod
To: kc2h-msm@asahi-net.or.jp

Dear Hisanori,

Strangely, I have spent a lot of time over the last few months
computing solutions to the congruent number problem.

At the end of this message are some solutions from your list of
unsolved values. What I give is the congruent number N, and the
two sides of a right-angled triangle with area N. From this you
can work out the solutions for the different characterisations
of what is a congruent number.

These values were all computed by looking for rational points
on the elliptic curve

y^2  =  x^3  -  N^2 x

using several descent arguments. Do you want the precise technical
details - this would take some time?

Allan MacLeod

I think I might have some more results in my computer at home - I'll
check.

N= 53
a= 1472112483//202332130
b= 21447205780//1472112483

N= 118
a= 332550100//93221931
b= 5500093929//83137525

N= 142
a= 2540793240//583374253
b= 41419571963//635198310

N= 157
a= 6803298487826435051217540//411340519227716149383203
b= 411340519227716149383203//21666555693714761309610

N= 173
a= 418416739097462232963//181421867613059954270
b= 62771966194118744177420//418416739097462232963

N= 181
a= 52138479900//2820733021
b= 2820733021//144028950

N= 197
a= 12767323300497220//1988610855961437
b= 1988610855961437//32404373859130

N= 237
a= 1542182460//212116267
b= 16757185093//257030410

N= 269
a= 3811563648242244176683700//1973646324592395897327837
b= 1973646324592395897327837//7084690795989301443650

N= 277
a= 133019165336738929060120082660340//14945906772955209419887378179997
b= 14945906772955209419887378179997//240106796636712868339566936210

N= 278
a= 661590775791//694746018145
b= 2778984072580//4759645869

N= 293
a= 5353759948354490004946563//889611509588919417084830
b= 521312344619106778411710380//5353759948354490004946563

N= 303
a= 522134246505//33621060538
b= 67242121076//1723215335

N= 317
a= 31511701858714809904340//2746649821545910809117
b= 2746649821545910809117//49702999777152697010

N= 326
a= 91423276//177496671
b= 28931957373//22855819

N= 327
a= 581084251//382113550
b= 249902261700//581084251

N= 334
a= 5435272857048//237922439923
b= 39733047467141//1358818214262

N= 349
a= 14440029059//847629690
b= 591645523620//14440029059

N= 365
a= 1031459691860//80455590957
b= 80455590957//1412958482

N= 382
a= 45779646960//2342317103
b= 447382566673//11444911740

N= 389
a= 827613354444026494185599057300//83886451016062654719097093923
b= 83886451016062654719097093923//1063770378462758989955782850

N= 397
a= 1989582752483//225617812530
b= 179140543148820//1989582752483

N= 398
a= 7933818298539//529160111350
b= 2116640445400//39868433661

N= 407
a= 61508083//231077850
b= 188097369900//61508083

N= 413
a= 9053788345//398340282
b= 47004153276//1293398335

N= 415
a= 930794716//41582073
b= 17256560295//465397358

N= 421
a= 5240283014136092580//451131541511971043
b= 451131541511971043//6223614031040490

N= 453
a= 8360011553071//944762392490
b= 5668574354940//55364314921

N= 454
a= 7846463625939//655485776279
b= 2621943105116//34565919057

N= 485
a= 505507183220//18792245997
b= 18792245997//521141426

N= 493
a= 1039002998182500//59040450498163
b= 59040450498163//1053755576250

N= 501
a= 223103496//18472213
b= 3084859571//37183916

N= 519
a= 73340867756//14309764395
b= 7426767721005//36670433878

N= 533
a= 4083995037//828570050
b= 883255673300//4083995037

N= 541
a= 2570149591160061699024541//134697139591400928750390
b= 145742305037895804907921980//2570149591160061699024541

N= 543
a= 2773655779498208400900//127913672431195199179
b= 127913672431195199179//2554010846683433150

N= 557
a= 522684119835266127363//70219707123123640670
b= 78224753735159735706380//522684119835266127363

N= 583
a= 818273757//169245850
b= 197340661100//818273757

N= 638
a= 125737417//8833125
b= 388657500//4335773

N= 695
a= 51398696163//1492008646
b= 2073892017940//51398696163

N= 703
a= 30448097447495//1035323333154
b= 2070646666308//43311660665

N= 717
a= 169466362194960//349489856794871
b= 83528075773974169//28244393699160

N= 742
a= 23873388264//707967485
b= 4955772395//112610322

N= 767
a= 112413104047908//7599942551825
b= 5829155937249775//56206552023954

N= 781
a= 19083874447337//1203329887650
b= 26473257528300//268786964047

N= 789
a= 79169904013//4433301444
b= 26599808664//301026251

N= 807
a= 260082211999324//14562450054205
b= 11751897193743435//130041105999662

N= 815
a= 47730888213020415//3820672155837442
b= 7641344311674884//58565507009841

N= 822
a= 817723977783//39614611555
b= 158458446220//1989596053

N= 831
a= 108223547882689044//6629297547215035
b= 6629297547215035//65116454803062

N= 886
a= 90369852983223//4040928651851
b= 16163714607404//203995153461

N= 989
a= 181448605791//4936200542
b= 424513246612//7889069817
```

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima