### Congruum g : 1 <= g <= 999

Definition 1.

k2g=mn(m2-n2)
k, m, n, g in N (integer > 0)

Definition 2.

(x/z2, y/z3) on elliptic curve Y2=X3-g2X
x, y, z in Z (integer)
X, Y in Q (rational)

X=mg/n, Y=kg2/n2

 Let x=i2+bj2 y=i2-bj2 g=ab then x2+y2=az2 x2-y2=bw2 and let X=x4+y4 Y=2xyzw then X2+gY2=Z2 X2-gY2=W2

The solution of the following numbers are found.

103, 199, 271, 303, 415, 431, 439, 519, 703, 767, 807, 815, 919
The solution of the following numbers are not in the range of 1 <= i, j <= 32,767.
101, 118, 127, 142, 157, 173, 191, 197,
223, 229, 237, 263, 269, 277, 278, 293,
302, 311, 317, 326, 327, 334, 358, 365, 367, 373, 382, 389, 397, 398,
407, 413, 421, 446, 453, 454, 461, 463, 478, 487, 493,
501, 502, 503, 541, 542, 543, 557, 566, 573, 583, 597, 599,
607, 613, 614, 623, 631, 638, 647, 653, 661, 662, 677, 685, 695,
701, 717, 718, 727, 733, 742, 743, 757, 758, 766, 773, 781, 789, 797,
822, 823, 829, 831, 838, 853, 862, 863, 877, 878, 886, 887, 893,
911, 917, 926, 933, 941, 958, 965, 967, 974, 982, 983, 989, 991, 997, 998

Congruum g : 1 <= g <= 999
gmnxyzaij
1038780605285453456755192927397456990440234440170596843879689620711924645872934086901837113439
26471
257
13439
19913933945152400273684876201277285508532760014598457498510248627082052314912063
25273
127
2063
271905651002769296001038456400971176124543142175047921600120787304681887800940043474680101904681120071857
30340822586116276899584091236924359314831965026198712816690166403101607109
415881971204245580241366018049660212681410970977650156715213
1079
13
213
43119327618991233440031770764668447218330203785221612640076019455728099668698422982356056365561122271943
439709894103370244004304435650645292131164351137953711600138344779425904107787787509180207471339176791039
5193260250121964124593729169206981329724220086122445784992267064223173797
1233
411
797
70313848629472047740075697299735586518844122518946152660212872597087977737545
6251
329
545
76721101042833520414971660478329161844998533101468649460880344195740498801503869323134289
25075
425
4289
8074156818798052844068180517644133545527700286418819335796423986303785812851063782292693743
8961
2987
3743
81540077410503852843642335055801121326630895606400646024626741477955295570746484506035176151719871
919172438957409290000947303818602261211584714018591375100001666935299916741184477656809300307783011110367887

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