# Congruent Numbers Part II

### Using Elliptic Curves to Compute the Triangles

The method of my first post is too slow to find the side lengths for a triangle of area , which has many digits:

In the 1984 book by Koblitz,  Introduction to Elliptic Curves and Modular Forms, computation of the side lengths for is credited to the mathematician Don Zagier. Now, however, it can be computed quickly by anyone with access to the SageMath system. The Python code given below computes the values for in less than one minute on my Mac laptop.

Put the code in the next section into a file called (say) congruent.py, fire up SageMath, and then type load(‘congruent.py’) followed by print_range(157, 157, 13). This produces the answer in about 15 seconds.

This post has three parts: I derive equations that produce triangle side lengths for congruent numbers; I give Python code that implements the equations; and I present two tables of congruent numbers. The first table is for , and is a variant of the web table. The second is an incomplete table for constructed using the Python code.

### The equations

Exhaustive search is too slow to find the sides of the triangle with area ; finding them requires a new idea. That idea is to reduce the search to a problem in elliptic curves (the same idea used to prove Fermat’s Last Theorem!).

An elliptic curve is the next level of complexity after a quadratic curve. A quadratic curve is ; an elliptic curve is . For the congruent number problem, Koblitz’s book Introduction to Elliptic Curves and Modular Forms shows the connection between a congruent and the elliptic curve . I summarize it here.

The connection involves adding points. For a conic section like a circle each point can be represented by , and two points , with angles , can be added to get a third point with angle . In a formula

with a similar formula for . So the formula for is a rational function of , where is our special addition, not vector addition. In the same way, two points on an elliptic curve can be added, and there is a formula for the sum. If and then the formula for (I don’t need ) is

If is a point of

this simplifies to

In that equation , and are rational numbers. I need a formula in terms of their numerators and denominators. So write

(1)

Then

Next, use Koblitz page 47, which says that if is the sum of a point on with itself, then you get rational sides with area for sides , with

so

Or since ,

This gives an algorithm that can find the sides’ lengths for area , and is fast enough to work for . Use SageMath to find a point on . Next use equation (1) to define , , , and the equation after that to get and . And use the equations directly above to get the side lengths and .

### Python code for SageMath

Here is the code that implements the algorithm of the previous section.

# congruent in python for use in SageMath

import re

# print sides of congruent number from lo to hi, inclusive
def print_range(lo, hi, lim=12):
for i in range(lo, hi+1):
if (squarefree(i) == 1):
if  ((i%2 == 1) and odd(i)) or ((i%2 == 0) and even(i)):
print_congruent(i, lim)

# return 1 if n squarefree
def squarefree(n):
if n % 4 == 0:
return 0
rt = floor(sqrt(n))
for i in range(3, rt+1, 2):
if n % (i*i) == 0:
return 0
return 1

# print sides of triangle with area N.  'lim' controls how much searching is done
def print_congruent(N, lim=12):
e = EllipticCurve([-N*N, 0])
try:
g = e.gens(descent_second_limit = lim)
except (RuntimeError) as msg:
#  msg might be of the form "generators found=[[323144640, -6423052104, 274625]])."
match = re.search(r"\[([0-9-]*),(.*),([0-9- ]*)", str(msg))
if match is not None:
preparser(False)
a = int(match.group(1))
b = int(match.group(2))
c = int(match.group(3))
preparser(True)
print_points(a, c, b, c, N)
else:
print N, "Cannot find Solution"
else:
for i in range(0, len(g)):
print_points(g[i][0].numerator(), g[i][0].denominator(), g[i][1].numerator(),
g[i][1].denominator(), N)

# Get the side lengths alpha = a/d, beta = b/e, compute
# the corresponding values of P,Q, write them as a square
# times a factor of N,  and print them
def print_points(s,t,u,v, N):  # pts x=s/t, y=u/v on elliptic curve
s1, t1, u1, v1 = double(s, t, u, v, N)
a, d, b, e = point_to_sides(s1, t1, N)
print N, "   alpha=", a, "/", d
print "      beta=", b, "/", e

#  compute A,B,C so that A^2 + B^2 = C^2
A = a*e
B = b*d
g = gcd(A,B)
A = A/g
B = B/g
C = sqrt(A*A + B*B)

# compute P,Q so that A = 2PQ, B = P^2 - Q^2
if B % 2 == 1:
P = sqrt((C+B)/2)
Q = sqrt((C-B)/2)
print "     P=", P, " Q=", Q
else:
P = sqrt((C+A)/2)
Q = sqrt((C-A)/2)
print "     P=", P, " Q=",  Q

# factor P = i0*P0^2,  Q = j0*Q0^2
for i in range(1, N+1):
if ( P%i == 0 and floor(sqrt(P/i))**2 == P/i):
P0 = sqrt(P/i)
i0 = i
break
for j in range(1, N+1):
if (Q%j == 0 and  floor(sqrt(Q/j))**2 == Q/j):
Q0 = sqrt(Q/j)
j0 = j
break
print "    P= ", i0, "*", P0, "^2, Q=", j0, "*", Q0, "^2"

# x=s/t, y=u/v is a point on y^2 = x^3 - N^2x.  Returns (x,y) + (x,y)
def double(s,t,u,v, N):
num = -8*s*t**3*u*u + v*v*(3*s*s - t*t*N*N)**2
denom = 2*u*t*t
s1 = num
t1 = denom*denom
u1 = -2*u*u*t1*t**3 + v*v*(s*t1 - t*s1)*(3*s*s - t*t*N*N)
v1 = 2*u*v*t1*t**3
g1 = gcd(s1, t1)
g2 = gcd(u1, v1)
return (s1/g1, t1/g1, u1/g2, v1/g2) # don't need last two elements

# Given (x,y) + (x,y) = (s/t, u/v), compute the corresponding alpha = a/d, beta = b/e
def point_to_sides(s, t, N):
p1 = sqrt(s + t*N)  # sqrt(x+N)
p2 = sqrt(s - t*N)  # sqrt(x-N)

a = p1 - p2
d =  sqrt(t)
b = p1 + p2
e  = d
g1 = gcd(a, d)
g2 = gcd(b, e)
return(a/g1, d/g1, b/g2, e/g2)

# test for n congruent when n is odd
def odd(n):
cnt1 = 0
cnt2 = 0
m = floor(sqrt(n))
for x in range(-m, m+1):
for y in range(-m, m+1):
for z in range(-m, m+1):
if 2*x*x + y*y + 32*z*z == n:
cnt1 = cnt1 + 1
for x in range(-m, m+1):
for y in range(-m, m+1):
for z in range(-m, m+1):
if 2*x*x + y*y + 8*z*z == n:
cnt2 = cnt2 + 1
return(2*cnt1 == cnt2)

# test for n congruent when n is even
def even(n):
cnt1 = 0
cnt2 = 0
n = n/2
m = floor(sqrt(n))
for x in range(-m, m+1):
for y in range(-m, m+1):
for z in range(-m, m+1):
if 4*x*x + y*y + 32*z*z == n:
cnt1 = cnt1 + 1
for x in range(-m, m+1):
for y in range(-m, m+1):
for z in range(-m, m+1):
if 4*x*x + y*y + 8*z*z == n:
cnt2 = cnt2 + 1
return(2*cnt1 == cnt2)


### The table through 1000

Here is a table with the side lengths of congruent numbers. Like the web tables, it gives and rather than the side lengths , , but I give and in factored form. In addition it differs from the web table at , , , and . The changes at and give fractions with a smaller denominator. The change at fixes the glitch wherein both and are both odd.

The change at adds a second solution with the same denominator. I remarked before that there are multiple values of , (or equivalently , , ) for each , and that I would pick the one with the smallest . This doesn’t always select a unique triangle. For example has both and , both with . Other such examples are

And now, here’s the table. Also available as a text file.


N                                             P                                             Q
5                                         5*1^2                                           2^2
6                                         2*1^2                                           1^2
7                                           4^2                                           3^2
13                                        13*5^2                                           6^2
14                                         2*2^2                                           1^2
15                                           2^2                                           1^2
21                                           2^2                                         3*1^2
22                                         2*5^2                                           7^2
23                                         156^2                                         133^2
29                                       29*13^2                                          70^2
30                                         3*1^2                                         2*1^2
31                                          40^2                                           9^2
34                                           3^2                                         2*2^2
37                                      37*145^2                                          42^2
38                                        2*25^2                                          17^2
39                                        13*1^2                                         3*2^2
41                                           5^2                                           4^2
46                                         2*6^2                                           7^2
47                                        3816^2                                        1513^2
53                                     53*5945^2                                       34034^2
55                                         5*5^2                                        11*2^2
61                                      61*445^2                                        3198^2
62                                       2*140^2                                         151^2
65                                           3^2                                           2^2
69                                         3*8^2                                          13^2
70                                         7*1^2                                         2*1^2
71                                          60^2                                          11^2
77                                       11*16^2                                          53^2
78                                        26*1^2                                           1^2
79                                       13000^2                                       12921^2
85                                        85*1^2                                           6^2
86                                        2*13^2                                           7^2
87                                         134^2                                          13^2
93                                          38^2                                         3*5^2
94                                        2*84^2                                          23^2
95                                        5*17^2                                        19*2^2
101                                   101*21041^2                                       63050^2
102                                         2*5^2                                           1^2
103                                    93704884^2                                    86901837^2
109                                       109*5^2                                          42^2
110                                        10*1^2                                           1^2
111                                        37*1^2                                         3*2^2
118                                     2*18925^2                                        4393^2
119                                          12^2                                           5^2
127                                 17501923504^2                                 17467450497^2
133                                         506^2                                       19*87^2
134                                      2*1021^2                                         119^2
137                                       137*5^2                                          56^2
138                                         6*2^2                                           1^2
141                                         3*4^2                                           1^2
142                                     2*49590^2                                       12809^2
143                                         318^2                                          43^2
145                                        29*1^2                                         5*2^2
149                                      149*25^2                                         238^2
151                                         340^2                                         189^2
154                                           3^2                                         2*1^2
157                             157*53156661805^2                                407598125202^2
158                                       2*620^2                                         761^2
159                                         182^2                                         107^2
161                                           4^2                                         7*1^2
165                                           4^2                                        11*1^2
166                                      2*9941^2                                        9799^2
167                                      339444^2                                       58717^2
173                              173*3728226965^2                                 48661701478^2
174                                         3*3^2                                         2*1^2
181                                    181*4121^2                                       34950^2
182                                         7*7^2                                         2*3^2
183                                      61*541^2                                       3*390^2
190                                        10*1^2                                           3^2
191                                   636182040^2                                   620449561^2
194                                        97*1^2                                         2*6^2
197                                 197*3179345^2                                    10192154^2
199                                     3732820^2                                      523149^2
205                                         5*3^2                                           2^2
206                                      2*1394^2                                        1103^2
210                                         5*1^2                                         2*1^2
210                                         6*1^2                                           1^2
213                                      3*1600^2                                         851^2
214                                       2*313^2                                         287^2
215                                          62^2                                          19^2
219                                        73*1^2                                         3*4^2
221                                        13*1^2                                           2^2
222                                       74*13^2                                          59^2
223                                  7137389056^2                                  3792733167^2
226                                           9^2                                         2*4^2
229                                   229*61793^2                                       87030^2
230                                        23*7^2                                        2*19^2
231                                         7*1^2                                           2^2
237                                       43454^2                                      3*5915^2
238                                        2*30^2                                          41^2
239                                         120^2                                         119^2
246                                        2*17^2                                          23^2
247                                        5998^2                                        5877^2
253                                         354^2                                       11*91^2
254                                       2*272^2                                         161^2
255                                           4^2                                           1^2
257                                         153^2                                         104^2
262                                      2*7345^2                                        9553^2
263                           49143127346631084^2                           46867792486220437^2
265                                           7^2                                           2^2
269                             269*85656402325^2                                 82710580922^2
271                                   300940360^2                                   101904681^2
277                         277*232579772276105^2                            1032363194300802^2
278                                    2*701185^2                                      990817^2
285                                        19*2^2                                           7^2
286                                        11*1^2                                         2*1^2
287                                      137904^2                                      137903^2
291                                           7^2                                         3*4^2
293                            293*231520447985^2                               3842474897278^2
295                                       5*433^2                                       59*62^2
299                                           6^2                                        13*1^2
301                                        43*8^2                                          27^2
302                                2*2167791890^2                                  3004234751^2
303                                      202046^2                                      166403^2
309                                        3458^2                                       3*799^2
310                                        10*2^2                                           3^2
311                                436723577940^2                                353087218429^2
313                                          13^2                                          12^2
317                              317*2951368765^2                                 16840660634^2
319                                          50^2                                          49^2
323                                        17*5^2                                        19*4^2
326                                     2*33661^2                                         679^2
327                                    109*7969^2                                     3*47950^2
330                                         6*1^2                                         5*1^2
334                                   2*1279626^2                                     1061887^2
335                                        2798^2                                        2629^2
341                                         118^2                                       11*15^2
349                                    349*7453^2                                      113730^2
353                                          17^2                                           8^2
357                                        21*1^2                                           2^2
358                                  2*21866725^2                                    20489063^2
359                                    57470100^2                                    57469741^2
365                                   365*14893^2                                       94874^2
366                                       122*1^2                                          11^2
367                     16382168821648506431464^2                       496953629608513608777^2
371                                        53*1^2                                         7*2^2
373                     373*2603564129117970265^2                        16377382078444972158^2
374                                        2*29^2                                           1^2
381                                         398^2                                       3*221^2
382                                    2*129060^2                                       88679^2
383                                    20091144^2                                    17430983^2
386                                          11^2                                         2*6^2
389                          389*14699193245425^2                              72369303587042^2
390                                         2*2^2                                         5*1^2
391                                         572^2                                         555^2
395                                         5*4^2                                           1^2
397                                  397*109045^2                                     2069034^2
398                                    2*655850^2                                      806831^2
399                                           8^2                                        57*1^2
406                                        58*8^2                                           3^2
407                                    37*11225^2                                    11*20586^2
410                                        41*1^2                                        10*2^2
413                                       64114^2                                     59*6213^2
415                                       29698^2                                       15671^2
421                                421*32786945^2                                   189819882^2
422                                       2*565^2                                         217^2
426                                         6*4^2                                           5^2
429                                         3*2^2                                           1^2
430                                        43*9^2                                        2*41^2
431                                   439631880^2                                    56365561^2
434                                         2*4^2                                        31*1^2
437                                        8914^2                                     19*1247^2
438                                           7^2                                         6*2^2
439                                   266438380^2                                   207471339^2
442                                          11^2                                        26*2^2
445                                        89*1^2                                         5*4^2
446                                  2*37655576^2                                    19427153^2
447                                       22442^2                                       21481^2
453                                     1307114^2                                    3*722785^2
454                                    2*708089^2                                      925711^2
455                                           8^2                                           1^2
457                                         253^2                                         204^2
461                            461*331709140421^2                               6025359571150^2
462                                        14*1^2                                        11*1^2
463                                  1078250296^2                                   376638153^2
465                                           4^2                                        15*1^2
469                                      67*148^2                                        1101^2
470                                        47*1^2                                         2*1^2
471                                     157*949^2                                      3*1962^2
478                             2*7613312866500^2                               1278366247319^2
479                                       46320^2                                       22849^2
485                                    485*6409^2                                       81314^2
487                              31922317361596^2                              20227123841253^2
493                                  493*348125^2                                     3026946^2
494                                       19*11^2                                        2*27^2
501                                      3*8932^2                                        4163^2
502                                  2*13172945^2                                     5749553^2
503                    126022097765194972888404^2                    124468507893835378881997^2
505                                       101*1^2                                         5*2^2
509                                      509*13^2                                         170^2
510                                         2*4^2                                        17*1^2
511                                       18952^2                                        2145^2
514                                       257*1^2                                         2*4^2
517                                           6^2                                        11*1^2
518                                        74*4^2                                          29^2
519                                      570986^2                                       64223^2
526                                      2*3774^2                                         137^2
527                                          24^2                                           7^2
533                                    533*6025^2                                      137522^2
534                                       2*157^2                                         217^2
535                                         418^2                                         311^2
541                             541*82652759501^2                               1629675045390^2
542                             2*8337517165460^2                               6072862516759^2
543                               181*849929197^2                                3*3004968950^2
546                                         7*1^2                                         6*1^2
551                                         350^2                                         179^2
557                              557*1746479765^2                                 40206424678^2
559                                      13*317^2                                      43*174^2
561                                        17*1^2                                           4^2
565                                      565*41^2                                         558^2
566                                   2*3119461^2                                      508681^2
573                                 3*155482680^2                                   222747577^2
574                                        2*12^2                                           1^2
581                                         554^2                                       83*57^2
582                                       194*1^2                                          13^2
583                                     53*8825^2                                    11*19178^2
589                                      19*944^2                                        2265^2
590                                       10*41^2                                          67^2
591                                        1406^2                                         181^2
597                                  4509125978^2                                3*2061246655^2
598                                      26*186^2                                         727^2
599                   1743235759090264843329900^2                   1298070998128102765757269^2
602                                           5^2                                         2*3^2
606                                        3*27^2                                        2*13^2
607         11193477616854092718151692958821016^2          5301570241310268303909957034492887^2
609                                         7*2^2                                           1^2
613                          613*14799287186305^2                              46559016084714^2
614                                  2*59953357^2                                    57198937^2
615                                          22^2                                          19^2
622                                        2*90^2                                          31^2
623                                  3040742304^2                                   958046297^2
629                                       629*1^2                                          10^2
631                              84388339390660^2                              84042085995171^2
638                                     11*2019^2                                      2*4375^2
645                                         5*5^2                                           2^2
646                                         2*3^2                                           1^2
647       1149852430778431376586487272927719076^2        763331759345415695713922022788822893^2
651                                         7*2^2                                         3*1^2
653         653*4956148647851982833406804183365^2           119088454066039826980851213749474^2
654                                      218*13^2                                         121^2
655                                      5*7085^2                                     131*178^2
658                                         2*8^2                                        47*1^2
661                          661*52573871914105^2                            1337443404825678^2
663                                        13*5^2                                        51*2^2
669                                          14^2                                         3*3^2
670                                        67*3^2                                        2*11^2
671                                           6^2                                           5^2
674                                       337*1^2                                        2*12^2
677       677*266071911559565396617913225699185^2          6090385733713292676671917039688234^2
678                                      2*1885^2                                        2567^2
679                                     4829572^2                                     4736355^2
685                                  685*384709^2                                     3760398^2
687                                     229*181^2                                       3*370^2
689                                       689*1^2                                          20^2
694                                       2*269^2                                         329^2
695                                    5*107633^2                                   139*13862^2
701                      701*778126037665372045^2                        20381058964400398922^2
703                                     1176802^2                                      879777^2
709                                      709*17^2                                         210^2
710                                       10*12^2                                          37^2
717                                  3*42438120^2                                      665543^2
718                          2*4524903101629650^2                            2549487886137199^2
719                                       18720^2                                       16511^2
721                                         7*4^2                                           3^2
723                                         103^2                                        3*20^2
727      16213151558097443674133462376327492124^2      11607920028365757398549972945257395957^2
731                                          39^2                                        43*4^2
733            733*7587780721511267936796299165^2              175714842077423718884281543602^2
734                                       2*644^2                                         137^2
741                                           4^2                                         3*1^2
742                                    106*4334^2                                       25983^2
743              664892406397562627531460993684^2               70167984148003728340455310237^2
749                                      107*44^2                                         307^2
751                                         520^2                                         231^2
757 757*134047218103304241673334209140989001305^2    2498951123086366985427431466492641572998^2
758                        2*121343494129262725^2                           13640465896081927^2
759                                         3*2^2                                        11*1^2
761                                      761*29^2                                          40^2
766                                 2*588188916^2                                   153301273^2
767                                    14526198^2                                     3869323^2
773                          773*35770654935185^2                             628933159841954^2
777                                        37*2^2                                         3*3^2
781                                     2079450^2                                   11*578677^2
782                                       2*110^2                                          71^2
789                                     3*53196^2                                       83339^2
790                                       10*88^2                                          39^2
791                                       113*1^2                                         7*4^2
793                                       793*5^2                                         132^2
797 797*6509757429070914743924008309828829377565^2   77437998217120395728776728661988058847754^2
798                                         2*4^2                                           5^2
799                                           8^2                                        17*1^2
805                                           9^2                                         5*4^2
806                                        31*1^2                                         2*3^2
807                                    20388278^2                                     6378229^2
813                                          14^2                                         3*5^2
814                                        74*1^2                                           5^2
815                                    63306722^2                                    60351761^2
821                                     821*905^2                                        6118^2
822                                    2*178445^2                                      221999^2
823    1800731922098507686862624200977450853996^2    1093332003924633933040927920553338780747^2
829               829*1114687145192391935645405^2                  13332313139036257353598962^2
830                                        83*9^2                                        2*29^2
831                                 277*4899877^2                                  3*13289406^2
838                             2*8009381984165^2                               1735885783193^2
839                                      959700^2                                      893651^2
853                        853*1555403055469705^2                           41407411349159166^2
854                                        7*41^2                                        2*27^2
861                                       21*11^2                                          50^2
862                      2*31754083744528352940^2                        44433441426033013319^2
863                      1365999649102602102024^2                      1363371767025194841143^2
866                                          19^2                                         2*6^2
869                                      11*208^2                                         535^2
870                                       145*1^2                                         6*2^2
871                                        3646^2                                        3579^2
877                 877*98019289159914592820005^2                    285144044108203957571958^2
878                                   2*1564070^2                                     2155561^2
879                                          62^2                                          59^2
885                                         632^2                                       59*59^2
886                                   2*1812317^2                                     2229703^2
887          2129987769776435785528571874098484^2           613374508678206671797659663089437^2
889                                           8^2                                         7*3^2
890                                           7^2                                        10*2^2
893                                   314604694^2                                 19*13026563^2
894                                       3*681^2                                        2*73^2
895                                       5*169^2                                      179*22^2
901                                       17*13^2                                        53*4^2
902                                      2*1625^2                                        2297^2
903                                        43*2^2                                         3*1^2
905                                       181*1^2                                         5*6^2
910                                        65*1^2                                        14*2^2
911              101674006024983403715286392160^2              101247414652636878045579297649^2
915                                        61*1^2                                        15*2^2
917                                    79016186^2                                 131*3187071^2
919                                   415257700^2                                   307783011^2
926                           2*207949336647956^2                              65855561465047^2
933                              3*627284421900^2                                906169058509^2
934                                       2*137^2                                          17^2
935                                          52^2                                          47^2
941                           941*6949430837141^2                             212353330183010^2
943                                       41*29^2                                       23*20^2
949                                       949*1^2                                          30^2
951                                          14^2                                          11^2
957                                        29*1^2                                           2^2
958                         2*64137447275611920^2                           86812781740303009^2
959                                          72^2                                          65^2
965                                965*37658777^2                                   268747634^2
966                                        23*1^2                                         2*1^2
967                  40855341363101303203686796^2                  31495021407376795594262853^2
973                                          74^2                                       139*3^2
974                             2*1209837172766^2                                359802927673^2
982                          2*3361484998004965^2                            1361998041016823^2
985                                      985*13^2                                         408^2
987                                        21*4^2                                          17^2
989                                      211066^2                                    43*23387^2
991                                372979863880^2                                355105187961^2
995                                         5*8^2                                          11^2
997 997*205768636979011246293067486487264646177793235665^2 1939472939861978131500861593607331562307144802554^2
998                       2*7036245713483923165^2                         6759000852285855817^2


### An incomplete table for 1000 to 2000

One of the motivations for writing the code in this posting was to try computing the side lengths for . Of the 358 square-free congruent numbers between 1000 and 2000, the code was able to find sides for all but 50. I’d be thrilled to hear about more sophisticated algorithms that can compute these missing entries.  Here are the entries I was able to compute, also available as a separate file.


1003                                    63^2                                 59*4^2
1005                                 5*565^2                                 1202^2
1006                               2*36354^2                                51343^2
1007                             640869638^2                            587562763^2
1013
1015                                 29*13^2                                 35*2^2
1021                           1021*257149^2                              7820010^2
1022                                  2*20^2                                   17^2
1023                                  7664^2                               33*375^2
1030                              103*1327^2                               2*1271^2
1031                       553773154545780^2                      313927967546509^2
1037                            1037*23725^2                               735278^2
1038                                  3*33^2                                 2*23^2
1039                           19222641880^2                            464534841^2
1045                                  5*13^2                                209*2^2
1046                               2*54433^2                                62033^2
1047                            349*815413^2                            3*6269390^2
1054                                2*1692^2                                 1561^2
1055                             5*2218957^2                           211*235258^2
1057                                   247^2                                   96^2
1061                1061*11017137603102721^2                    52393391142142750^2
1063
1069
1070                             107*41187^2                             2*256331^2
1073                                    23^2                                   14^2
1077                                 3*880^2                                 1237^2
1079                                    54^2                                   29^2
1081                                    35^2                                   12^2
1085                                  35*4^2                                   23^2
1086                                 362*1^2                                    1^2
1087           534490229096555257177885984^2          504876251097996356696056863^2
1093
1094                                 2*409^2                                  511^2
1095                                    14^2                                   13^2
1101                                  6734^2                               3*3333^2
1102                             19*330627^2                             2*724355^2
1103                                   816^2                                  287^2
1105                                     7^2                                    6^2
1109
1110                                   2*4^2                                  5*1^2
1111                                    10^2                                    1^2
1113                                   7*2^2                                    5^2
1117
1118                                 26*53^2                                  197^2
1119                           373*5047393^2                           3*56058478^2
1122                                     5^2                                  2*2^2
1126                                 2*953^2                                  113^2
1131                                     4^2                                 13*1^2
1133                                104354^2                             11*31341^2
1135                               4317302^2                              1209899^2
1141                          163*20033248^2                             35220117^2
1142                    2*3498126297122005^2                     3101671470103703^2
1145                                1145*5^2                                   52^2
1146                                  2*13^2                                191*1^2
1149                            3*10180232^2                             17116727^2
1151                             126739920^2                            124197361^2
1154                                    17^2                                 2*12^2
1155                                  11*1^2                                    2^2
1157                     1157*101695523225^2                        1997676115478^2
1158                                 193*5^2                                 6*24^2
1159                                  61*5^2                                 19*6^2
1165                             1165*1037^2                                24366^2
1166                               106*229^2                                  565^2
1167                        80234354894378^2                       69040110896471^2
1169                                  7*12^2                                   29^2
1173                                69*273^2                                 1718^2
1174                         2*10970739277^2                           5524968023^2
1178                                   2*5^2                                 31*1^2
1181
1182                             3*2494283^2                             2*170977^2
1185                                     8^2                                 15*1^2
1186                                   257^2                                2*136^2
1189                               1189*65^2                                 1518^2
1190                                   2*3^2                                 17*1^2
1191                           397*4319701^2                           3*43476026^2
1195                                    22^2                                  5*7^2
1198                         2*79342901670^2                          10971111209^2
1199                            4846487934^2                           4826824675^2
1201                                    25^2                                   24^2
1205                                    31^2                                 5*12^2
1207                              11643884^2                             10025013^2
1213
1214                            2*18094076^2                             12405353^2
1217                               1217*65^2                                 1504^2
1221                                 37*17^2                                33*14^2
1222                                47*151^2                                2*731^2
1223
1229                 1229*3171885776594453^2                    22661957001363890^2
1230                                 123*1^2                                  2*1^2
1231                1093893124029377517040^2                508787759180739534831^2
1237
1238
1239                                   596^2                               177*37^2
1241                                    29^2                                   20^2
1245                                 5*449^2                                  982^2
1246                                  2*24^2                                   23^2
1247                             500503826^2                            493934857^2
1249                                   481^2                                  360^2
1253                            179*139708^2                              1576963^2
1254                                  11*1^2                                  2*2^2
1255                           5*130143077^2                          251*4799482^2
1261                             1261*6205^2                               211494^2
1262     2*5146561601772899976047772998810^2      5187845455908637659823361579711^2
1263                          421*72473677^2                          3*805599790^2
1270                              127*1561^2                               2*5381^2
1271                                  41*5^2                                 31*4^2
1277
1279               73978703287659341689600^2              38338545749238184291809^2
1282                                 641*1^2                                 2*10^2
1285                             1285*1105^2                                 8358^2
1286                      2*10448952878281^2                       10529188863199^2
1293                          3*1697884540^2                           2937063841^2
1294                             2*4872522^2                              1370959^2
1295                                     6^2                                    1^2
1301                 1301*3117192484228345^2                    77237364746203922^2
1302                                    11^2                                  6*4^2
1303
1309                                  77*3^2                                   26^2
1310                             10*153829^2                               432161^2
1311                                  23*1^2                                    2^2
1317
1318                       2*1428058929085^2                         592793094313^2
1319                2483679569975131134420^2               1474854640815874983299^2
1321                                 37669^2                                27060^2
1326                                  26*1^2                                    5^2
1327
1330                                  14*1^2                                  5*1^2
1333                                   206^2                                43*15^2
1334                                 58*48^2                                  133^2
1335                                  4258^2                                 4169^2
1339                                    34^2                                 13*9^2
1342                            11*6293233^2                           2*11110465^2
1343                            1513392816^2                             87126737^2
1346                                   239^2                                2*168^2
1349                          19*572172984^2                           1718426035^2
1351                                    92^2                                193*3^2
1357                             59*799332^2                              6119003^2
1358                                 2*170^2                                  239^2
1365                                   5*2^2                                  7*1^2
1366                      2*75067556653673^2                         276597312593^2
1367             2210029442288216604321564^2            1092492023913676770056123^2
1373
1374                             458*39481^2                               518017^2
1379                                 197*1^2                                  7*2^2
1381                    1381*5877838501505^2                       85358642942118^2
1382
1383                                  8114^2                                 6527^2
1387                                 73*25^2                                19*48^2
1389                              28962134^2                           3*15288133^2
1390                             10*529105^2                              1452477^2
1391                              13*55241^2                            107*14214^2
1393                                  1724^2                               199*57^2
1397                                  11*4^2                                    7^2
1398                                 2*205^2                                   23^2
1399                             140108980^2                             24481971^2
1405                                    19^2                                  5*4^2
1406                                  19*1^2                                  2*3^2
1407                               67*4694^2                               3*9411^2
1411                                    79^2                                 83*4^2
1414                                7*2329^2                               2*4237^2
1415                                  6998^2                                 4451^2
1419                                   3*3^2                                    4^2
1423       7846891217457546585817785748936^2      6692442382211391072755148372327^2
1429
1430                                   2*8^2                                 13*3^2
1434                                  6*10^2                                   19^2
1437                           3*359821900^2                            579736247^2
1438                      2*44553108632760^2                       37487295533089^2
1439             1008014492965566903395280^2             960164863865618935964209^2
1443                                     5^2                                  3*2^2
1446                                  2*11^2                                    1^2
1447
1453
1454                               2*59006^2                                38953^2
1455                                  1586^2                                  263^2
1461                                  9314^2                               3*2233^2
1462                                  34*1^2                                    3^2
1463                                  11*2^2                                 19*1^2
1469                                  13*3^2                                    2^2
1471          1928596035670269313433517160^2         1779886513628987622022447431^2
1477                      211*220919758676^2                        1248458528883^2
1478
1479                                    16^2                                   13^2
1482                                  19*1^2                                  6*1^2
1486                        2*239711193546^2                           8032739393^2
1487                4658308462162163630976^2               1320338709334411132607^2
1493                               1493*25^2                                  434^2
1495                                   202^2                                   87^2
1501                                  19*4^2                                   15^2
1502                          2*2028737420^2                            311366489^2
1509                         3*58309238416^2                           6189441829^2
1510                                  10*4^2                                    3^2
1511                                   780^2                                  731^2
1513                                  89*1^2                                    8^2
1517                                  1181^2                               37*160^2
1518                                   125^2                                22*13^2
1526                                218*16^2                                   19^2
1527                       591153019517366^2                      572357977499363^2
1533                                 28262^2                               3*5805^2
1534                                59*113^2                                 2*29^2
1535                          702318605942^2                         469355582779^2
1541                                    58^2                                 67*7^2
1542                                 2*265^2                                  287^2
1543
1549
1551                                 3*106^2                                11*23^2
1558                               2*24125^2                                20113^2
1559            46558955761464766823805420^2           40991718920623965561016069^2
1561                                   116^2                                223*3^2
1565                               1565*61^2                                 1078^2
1567                                  1624^2                                   57^2
1574
1581                                  93*3^2                                    2^2
1582                                    13^2                                 14*2^2
1583
1589                            1955370754^2                         227*95700609^2
1590                                 265*1^2                                  6*4^2
1591                                    22^2                                   21^2
1595                                   5*2^2                                    3^2
1597
1598                                    39^2                                 94*4^2
1599                                   236^2                                  115^2
1605                                  5*17^2                                   38^2
1606                                2*2269^2                                 1751^2
1607
1610                                  14*1^2                                    3^2
1613
1614                                3*1091^2                               2*1021^2
1615                                85*101^2                                19*42^2
1621
1622
1623                            541*818977^2                            3*8400690^2
1630                                163*81^2                                2*209^2
1631                                    11^2                                  7*4^2
1633                                   324^2                                23*25^2
1635                                 109*1^2                                 15*2^2
1637
1639                           93119931626^2                          17274220825^2
1645                                   381^2                                 5*76^2
1646                               2*10502^2                                12361^2
1649                                1649*1^2                                   40^2
1651                                  13*4^2                                    9^2
1653                                 29*61^2                                  278^2
1654                          2*5099522713^2                           7191812593^2
1655                         5*15944597077^2                       331*1593316982^2
1659                                    10^2                                 21*1^2
1661                          11*449674700^2                           1128819629^2
1662                                 554*1^2                                   23^2
1663
1669                               1669*17^2                                  690^2
1670                                167*41^2                                2*331^2
1671                            4709357954^2                           4709150447^2
1677                                 13*25^2                                129*6^2
1678                    2*2197332480074250^2                     2986757644280911^2
1679                                    83^2                                23*12^2
1685                      1685*10780079129^2                         323265236278^2
1686                               2*30193^2                                 9553^2
1687                        20020795748764^2                       14313931423077^2
1693
1695                                     8^2                                    7^2
1702                           23*83377711^2                          2*259732823^2
1703                          13*136174385^2                         131*42165966^2
1705                                   5*2^2                                 11*1^2
1709                         1709*56899397^2                           1914111430^2
1711                                29*281^2                               59*130^2
1717                                101*13^2                                   38^2
1718                       2*1622842557865^2                         432593825023^2
1726               2*232211988542532693804^2                206435450751981013463^2
1727                               157*265^2                               11*858^2
1731                                 577*1^2                                  3*4^2
1733
1735                               1247378^2                              1029289^2
1741
1742                           67*10122299^2                           2*15946527^2
1743                               3041032^2                            249*85593^2
1745                                 349*1^2                                  5*6^2
1749                                     8^2                                 11*1^2
1751                                  17*4^2                                   13^2
1757                            8065011826^2                        251*412114251^2
1758                        3*102985227451^2                       2*121557212519^2
1759
1762                                 881*1^2                                 2*20^2
1765                          1765*1965793^2                             29034642^2
1766                          2*1241178349^2                            748018649^2
1767                              19*18578^2                              3*43325^2
1770                                   2*4^2                                  3*3^2
1774                                 2*234^2                                  257^2
1781                         1781*64281065^2                           1800656762^2
1783    4792743876089805249498502926760516^2   4210256678211637838081213167457613^2
1785                                   3*2^2                                  5*1^2
1789
1790                             10*178841^2                               493243^2
1794                                  26*1^2                                 23*1^2
1797                          3*5264857000^2                           4228721773^2
1798                           31*10911449^2                            2*7818425^2
1799                                  1740^2                                   59^2
1806                                    47^2                                 6*19^2
1807                             13*408185^2                           139*121818^2
1814                             2*1024057^2                               931217^2
1821                                661934^2                             3*220917^2
1822                               2*17520^2                                23471^2
1823    7761300340225847160065004009945984^2   7073498453526479365692239139413137^2
1829                           59*29844700^2                            181206031^2
1830                                  3*29^2                                122*1^2
1831                            3565715500^2                           2898888171^2
1837
1838               2*640482037573884561770^2                768308549845289572991^2
1839                                613*97^2                                3*294^2
1846                                71*217^2                                2*439^2
1847
1853                       1853*1327014805^2                          56375498794^2
1855                                  1222^2                                  493^2
1858                                    27^2                                 2*10^2
1861
1869                                  21*3^2                                   10^2
1870                                  2*16^2                                 17*1^2
1871
1877
1878                                  2*13^2                                    5^2
1879                            4130777980^2                           1199536101^2
1885                                  29*1^2                                    4^2
1886                                   2*4^2                                    3^2
1887                                  6044^2                                 5119^2
1893                       206546260507082^2                     3*64269514053185^2
1894                        2*324087672709^2                         213795376151^2
1895
1901
1902                        3*900186672163^2                       2*946544303197^2
1903                                114182^2                                19043^2
1909                               4644138^2                             83*21329^2
1910                                  10*6^2                                   13^2
1918                                     9^2                                 14*2^2
1919                               101*197^2                               19*310^2
1927                               1654276^2                               634893^2
1933
1934                              2*127958^2                               180551^2
1939                                 277*1^2                                  7*6^2
1941                      3*22087563983576^2                       35655089110091^2
1942                  2*127615924034303765^2                    89779252381023607^2
1943                              29*16613^2                              67*2990^2
1949
1951
1957                            19*3285772^2                              7471773^2
1958                                  2*15^2                                 89*1^2
1959                        45651794958266^2                       34244727779041^2
1965                                   136^2                               131*11^2
1966                  2*378280891208203374^2                   484645319443334263^2
1967                                 12419^2                               7*4340^2
1973
1974                                  42*2^2                                   11^2
1981                                   506^2                               283*27^2
1982                           2*472837460^2                             48441881^2
1983                                 661*1^2                                 3*10^2
1990                           10*64794470^2                             85105071^2
1991                              181*1193^2                              11*1350^2
1995                                   5*2^2                                    1^2
1995                                   3*2^2                                  7*1^2
1997
1999


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