X=mg/n, Y=kg2/n2
- 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)
Type means,
- m=(i2{j2)2, n=(i2|j2)2
- m=(i2{j2)2, n=(2ij)2
- m=(i2|j2)2, n=(2ij)2
- m=(2ij)2, n=(i2|j2)2
There are 176 congruum g : 500 <= g <= 999.
The solution of the following numbers are found.
679, 839The solution of the following numbers are not in the range of 1 <= i, j <= 10,000.
501, 502, 503, 519, 533, 541, 542, 543, 557, 566, 573, 583, 597, 599,
607, 613, 614, 623, 631, 638, 647, 653, 661, 662, 677, 685, 695,
701, 703, 717, 718, 727, 733, 742, 743, 757, 758, 766, 767, 773, 781, 789, 797,
807, 815, 822, 823, 829, 831, 838, 853, 862, 863, 877, 878, 886, 887, 893,
911, 917, 919, 926, 933, 941, 958, 965, 967, 974, 982, 983, 989, 991, 997, 998
| g | m | n | x | y | z | type | i | j |
|---|---|---|---|---|---|---|---|---|
| 511 | 359178304 | 4601025 | 183540113344 | 78625036156344328 | 2145 | (9) (8) | 103 195 | 92 11 |
| 514 | 289 | 225 | 148546 | 35930656 | 15 | (6) (7) | 4 5 | 1 3 |
| 527 | 576 | 49 | 303552 | 166637400 | 7 | (9) (8) | 4 7 | 3 1 |
| 674 | 625 | 49 | 421250 | 272565600 | 7 | (6) (7) | 4 7 | 3 1 |
| 679 | 23324765703184 | 22433058686025 | 15837515912461936 | 545829598311328775512684 | 4736355 | (9) (8) | 2398 3405 | 1007 1391 |
| 719 | 350438400 | 272613121 | 251965209600 | 79473346962216480 | 16511 | (9) (8) | 144 209 | 65 79 |
| 721 | 3025 | 2304 | 2181025 | 2087161615 | 48 | (8) (9) | 8 11 | 3 5 |
| 345997201 | 319694400 | 249463981921 | 47651516683186031 | 17880 | (8) (9) | 149 209 | 60 89 | |
| 1033319604893521 | 144897072782400 | 745023435128228641 | 636711357710981751269632561 | 12037320 | (8) (9) | 5765 6809 | 1044 4721 | |
| 751 | 270400 | 53361 | 203070400 | 2836902469960 | 231 | (9) (8) | 20 33 | 13 7 |
| 761 | 2499561 | 672400 | 1902165921 | 79902649977969 | 820 | (8) (9) | 41 51 | 10 31 |
| 1566655561 | 502656400 | 1192224881921 | 1232955420453836031 | 22420 | (7) (6) | 190 249 | 59 131 | |
| 1635683244103561 | 361605974083600 | 1244754948762809921 | 1354393778865162349324448031 | 19015940 | (7) (6) | 6170 7711 | 1541 4629 | |
| 839 | 921024090000 | 798612109801 | 772739211510000 | 10700613728104274492700 | 893651 | (9) (8) | 1050 1507 | 457 593 |
| 866 | 168922009 | 135606025 | 146286459794 | 33362602321497072 | 11645 | (6) (7) | 111 137 | 26 85 |
| 889 | 346921 | 90000 | 308412769 | 5230813038353 | 300 | (8) (9) | 25 31 | 6 19 |
| 360714354025 | 164326458384 | 320675060728225 | 5111969378875096024945 | 405372 | (8) (9) | 814 1063 | 249 565 | |
| 1434769777506025 | 160987678600464 | 1275510332202856225 | 1431445320755372492871133105 | 12688092 | (8) (9) | 6238 7255 | 1017 5221 | |
| 959 | 5184 | 4225 | 4971456 | 6423052104 | 65 | (9) (8) | 9 13 | 4 5 |
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