- Definition 1.
- k2g=mn(m2-n2)
k, m, n, g in N (integer > 0)
- Definition 2.
- x2+gy2=z2
x2-gy2=w2x, y, z, w in N (integer > 0)
m=x2, n=gy2
- Definition 3.
- (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
We are using the same characters x, y, z in the definition 2 and 3,
but I think there's no confusion.There are 176 congruum g : 500 <= g <= 499.
The solution of the following numbers are not in the range of 1 <= x <= 100,000.
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
Congruum g : 500 <= g <= 599 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 505 10601 396 13841 5761 112381201 79192080 112381201 845302548601 396 510 1313 56 1823 353 1723969 1599360 1723969 844940447 56 514 67073 2040 81473 48577 4498787329 2139062400 4498787329 265455745823233 2040 517 1417 60 1967 383 2007889 1861200 2007889 1067512537 60 546 85 2 97 71 7225 2184 7225 585395 2 365 6 391 337 133225 19656 133225 48094955 6 1073 40 1423 527 1151329 873600 1151329 804665233 40 561 545 8 577 511 297025 35904 297025 160691615 8 1345 56 1889 223 1809025 1759296 1809025 566577215 56 574 82945 408 83519 82367 6879873025 95550336 6879873025 570596029737985 408 582 66197 1430 74647 56497 4382042809 1190131800 4382042809 279174697211123 1430
Congruum g : 600 <= g <= 699 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 602 949 30 1201 599 900601 541800 900601 682709651 30 609 641 20 809 409 410881 243600 410881 212094721 20 785 12 839 727 616225 87696 616225 478813105 12 73105 2912 102511 13423 5344341025 5164164096 5344341025 100592856710065 2912 645 15641 220 16609 14609 244640881 31218000 244640881 3795146019721 220 646 325 6 359 287 105625 23256 105625 33485725 6 651 793 20 943 607 628849 260400 628849 453913993 20 30865 1144 42481 10033 952648225 851987136 952648225 13155029460145 1144 658 18593 720 26207 2143 345699649 341107200 345699649 1044212647393 720 669 39145 1092 48271 27103 1532331025 797758416 1532331025 51212969499385 1092 671 1921 60 2471 1129 3690241 2415600 3690241 5359127039 60
Congruum g : 700 <= g <= 799 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 721 12625 264 14479 10447 159390625 50250816 159390625 1909684176625 264 741 265 8 343 151 70225 47424 70225 13725145 8 759 265 4 287 241 70225 12144 70225 18329255 4 777 22633 660 29167 13183 512252689 338461200 512252689 8702582261113 660 791 25313 120 25537 25087 640747969 11390400 640747969 16216690398047 120 798 1649 40 1999 1201 2719201 1276800 2719201 3958917551 40 799 4385 144 5983 1631 19228225 16568064 19228225 42790027105 144
Congruum g : 800 <= g <= 899 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 805 12961 72 13121 12799 167987521 4173120 167987521 2176614335519 72 806 1285 42 1753 479 1651225 1421784 1651225 1078997795 42 813 44041 1540 62191 3391 1939609681 1928110800 1939609681 9287792440921 1540 814 6101 210 8551 1151 37222201 35897400 37222201 60047268301 210 870 21601 572 27409 13489 466603201 284650080 466603201 7986321741601 572 889 8065 48 8191 7937 65044225 2048256 65044225 524321513855 48 890 4001 84 4721 3119 16008001 6279840 16008001 58913920799 84
Congruum g : 900 <= g <= 999 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 903 29593 260 30607 28543 875745649 61042800 875745649 25852906480393 260 905 65161 228 65521 64799 4245955921 47045520 4245955921 276653750074919 228 910 7361 132 8369 6191 54184321 15855840 54184321 381391657919 132 915 7321 44 7441 7199 53597041 1771440 53597041 392169563639 44 7817 80 8183 7433 61105489 5856000 61105489 475463076263 80 59537 252 60023 59047 3544654369 58106160 3544654369 211009730408497 252 951 53057 1540 71207 23657 2815045249 2255391600 2815045249 89376850954943 1540 957 857 20 1057 593 734449 382800 734449 537168457 20 966 533 10 617 433 284089 96600 284089 142396813 10
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