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

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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

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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