- 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 185 congruum g : 1 <= g <= 499.
The solution of the following numbers are not in the range of 1 <= x <= 100,000.
53, 101, 103, 118, 127, 142, 157, 173, 181, 191,
197, 199, 223, 229, 237, 263, 269, 271, 277, 278,
293, 302, 303, 311, 317, 326, 327, 334, 349, 358,
365, 367, 373, 382, 389, 397, 398, 407, 413, 415,
421, 431, 439, 446, 453, 454, 461, 463, 478, 485,
487, 493
Congruum g : 1 <= g <= 99 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 5 41 12 49 31 1681 720 1681 62279 12 6 5 2 7 1 25 24 25 35 2 1201 140 1249 1151 1442401 117600 1442401 1726556399 140 7 337 120 463 113 113569 100800 113569 17631503 120 14 65 12 79 47 4225 2016 4225 241345 12 15 17 4 23 7 289 240 289 2737 4 21 25 4 31 17 625 336 625 13175 4 22 4901 210 4999 4801 24019801 970200 24019801 117624975299 210 30 13 2 17 7 169 120 169 1547 2 42961 6188 54721 26399 1845647521 1148740320 1845647521 62060587589519 6188 34 145 12 161 127 21025 4896 21025 2964815 12 353 60 497 47 124609 122400 124609 8245727 60 3425 504 4513 1759 11730625 8636544 11730625 27188906975 504 39 313 20 337 287 97969 15600 97969 30273047 20 41 881 120 1169 431 776161 590400 776161 443882159 120 46 7585 924 9839 4273 57532225 39273696 57532225 318888926495 924 55 17561 2340 24689 2689 308388721 301158000 308388721 1165852329481 2340 65 97 12 137 7 9409 9360 9409 93023 12 4481 504 6049 1889 20079361 16511040 20079361 51202419841 504 14657 660 15593 13657 214827649 28314000 214827649 3121260929857 660 69 65425 3952 73199 56593 4280430625 1077662976 4280430625 271026399632975 3952 70 53 6 73 17 2809 2520 2809 65773 6 78 677 30 727 623 458329 70200 458329 306627517 30 85 8521 924 12049 191 72607441 72570960 72607441 19609880039 924
Congruum g : 100 <= g <= 199 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 102 2501 70 2599 2399 6255001 499800 6255001 15593737501 70 110 101 6 119 79 10201 3960 10201 949501 6 111 1513 140 2113 337 2289169 2175600 2289169 1077378553 140 119 21361 1560 27311 12911 456292321 289598400 456292321 7532151788881 1560 138 577 20 623 527 332929 55200 332929 189441217 20 629 30 721 521 395641 124200 395641 236278189 30 10133 770 13583 4567 102677689 81820200 102677689 628586073613 770 141 2305 56 2399 2207 5313025 442176 5313025 12204036865 56 145 1241 84 1601 719 1540081 1023120 1540081 1428538679 84 41761 408 42049 41471 1743981121 24137280 1743981121 72823419753119 408 154 85 6 113 41 7225 5544 7225 393805 6 373 30 527 23 139129 138600 139129 4521133 30 14305 468 15439 13073 204633025 33729696 204633025 2887236042335 468 161 305 24 431 17 93025 92736 93025 2234735 24 54721 4080 75329 17729 2994387841 2680070400 2994387841 73080324567361 4080 165 377 24 487 217 142129 95040 142129 39840983 24 174 733 30 833 617 537289 156600 537289 376733413 30 190 181 6 199 161 32761 6840 32761 5799059 6 194 14593 780 18193 9743 212955649 118029600 212955649 2586673444607 780
Congruum g : 200 <= g <= 299 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 205 2041 84 2369 1649 4165681 1446480 4165681 7973127721 84 210 29 2 41 1 841 840 841 1189 2 37 2 47 23 1369 840 1369 39997 2 113 4 127 97 12769 3360 12769 1392047 4 16433 528 18127 14543 270043489 58544640 270043489 4332083252113 528 219 7633 440 10033 3983 58262689 42398400 58262689 305025663887 440 16945 1144 23953 721 287133025 286613184 287133025 292642064785 1144 221 185 12 257 49 34225 31824 34225 2329705 12 226 7585 504 10721 353 57532225 57407616 57532225 28705531105 504 231 65 4 89 23 4225 3696 4225 133055 4 255 257 8 287 223 66049 16320 66049 16448257 8 265 2417 84 2777 1993 5841889 1869840 5841889 13377033937 84 285 8177 420 10823 4073 66863329 50274000 66863329 360459159983 420 286 125 6 161 73 15625 10296 15625 1469125 6 291 4705 56 4801 4607 22137025 912576 22137025 104066163935 56 299 1465 84 2063 191 2146225 2109744 2146225 577258345 84
Congruum g : 300 <= g <= 399 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 310 1681 84 2239 799 2825761 2187360 2825761 3007243441 84 313 49297 1560 56497 40847 2430194209 761716800 2430194209 113764311679823 1560 330 61 2 71 49 3721 1320 3721 212219 2 73 4 103 7 5329 5280 5329 52633 4 1213 42 1433 943 1471369 582120 1471369 1639149947 42 22093 910 27593 14657 488100649 273273000 488100649 8935085267893 910 353 87617 4080 116417 42433 7676738689 5876179200 7676738689 432821195027137 4080 357 457 20 593 257 208849 142800 208849 69647257 20 366 29525 198 29767 29281 871725625 14348664 871725625 25734212234675 198 371 3593 180 4993 943 12909649 12020400 12909649 16917277607 180 386 19825 924 26881 7967 393030625 329557536 393030625 4245740377775 924 390 89 4 119 41 7921 6240 7921 434231 4 395 6401 72 6559 6239 40972801 2047680 40972801 261939168001 72 399 7345 176 8143 6449 53949025 12359424 53949025 385716850415 176
Congruum g : 400 <= g <= 499 g x y z w m (=x2) n (=gy2) xe(=x2) ye(=xzw) ze(=y) 410 3281 36 3361 3199 10764961 531360 10764961 35276783759 36 8161 396 11441 1519 66601921 64294560 66601921 141829031519 396 426 9841 440 13391 3791 96845281 82473600 96845281 499581130321 440 14645 546 18479 9353 214476025 126997416 214476025 2531155204115 546 429 145 4 167 119 21025 6864 21025 2881585 4 434 1985 24 2047 1921 3940225 249984 3940225 7805589695 24 3761 180 5311 289 14145121 14061600 14145121 5772679919 180 88705 3876 119953 36721 7868577025 6520145184 7868577025 390727261793665 3876 438 2977 140 4177 527 8862529 8584800 8862529 6553207583 140 442 25457 660 28993 21343 648058849 192535200 648058849 15752730477743 660 445 14321 312 15761 12719 205091041 43318080 205091041 2870847221039 312 455 4097 48 4223 3967 16785409 1048320 16785409 68635570177 48 462 317 10 383 233 100489 46200 100489 28288763 10 465 481 8 511 449 231361 29760 231361 110360159 8 30017 308 30743 29273 901020289 44111760 901020289 27013494147263 308 470 2213 42 2393 2017 4897369 829080 4897369 10681445053 42
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