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, n=j2
- m=i2, n=i2-j2
- m=i2+j2, n=i2
- m=j2, n=i2-j2
- m=i2-j2, n=j2
There are 36 congruum g : 1 <= g <= 99.
The solution of the following numbers are found.
47, 61, 79The solution of the following numbers are not in the range of 1 <= i, j <= 32,767.
53
Congruum g : 1 <= g <= 99 g m n x y z type i j 5 5 4 25 75 2 (3)
(5)2
31
29 1 45 300 1 (1) 3 1 1681 720 1681 62279 12 (2)
(4)41
4931
412401 961 12005 1205400 31 (1) 49 31 27910089 23030401 139550445 931243391100 4799 (1) 5283 4799 6 3 1 18 72 1 (5) 2 1 25 24 25 35 2 (2)
(4)5
71
549 1 294 5040 1 (1) 7 1 3267 2209 19602 2021976 47 (5) 74 47 7 16 9 112 980 3 (1)
(5)
(4)4
5
53
3
4113569 100800 113569 17631503 120 (2)
(4)337
463113
337214369 12769 1500583 1834924560 113 (1) 463 113 13 325 36 4225 272935 6 (5) 19 6 361 289 4693 192660 17 (1) 19 17 14 8 1 112 1176 1 (5) 3 1 9 7 18 48 1 (4) 4 3 4225 2016 4225 241345 12 (2)
(4)65
7947
656241 2209 87374 24155040 47 (1) 79 47 160348232 141681409 2244875248 49806282706488 11903 (5) 17379 11903 15 4 1 60 450 1 (1) 2 1 289 240 289 2737 4 (2)
(4)17
237
17529 49 7935 703800 7 (1) 23 7 487204 450241 7308060 7548294150 671 (1) 698 671 21 4 3 28 98 1 (2) 2 1 625 336 625 13175 4 (2)
(4)25
3117
25961 289 20181 2734200 17 (1) 31 17 3076516 93987 21535612 99892543366 177 (2) 1754 1727 22 50 49 1100 7260 7 (3) 7 1 99 1 2178 101640 1 (5) 10 1 24990001 23049601 549780022 4980380124720 4801 (1) 4999 4801 23 24336 17689 559728 287596140 133 (1)
(5)
(4)156
205
205133
133
15629 4901 4900 142129 1082367 70 (3)
(5)70
991
709801 1 284229 151531380 1 (1) 99 1 30 169 120 169 1547 2 (2)
(4)13
177
13289 49 8670 795600 7 (1) 17 7 186245 185761 5587350 951528600 431 (3) 431 22 31 1600 81 49600 11032280 9 (1)
(5)
(4)40
41
419
9
4034 9 8 153 867 2 (2) 3 1 25 9 850 23120 3 (1)
(2)5
53
449 32 833 18207 4 (4) 9 7 81 17 162 2016 1 (2) 9 8 961 128 16337 2069529 8 (4) 33 31 1089 833 2178 65472 7 (2) 33 16 21025 4896 21025 2964815 12 (2)
(4)145
161127
14525921 16129 881314 647683680 127 (1) 161 127 29929 27225 1017586 426374416 165 (3)
(2)
(1)165
173
17352
52
165124609 122400 124609 8245727 60 (2)
(4)353
49747
353131769 2888 2240073 3351883557 38 (2) 363 359 134657 128881 4578338 2838363792 359 (3) 359 76 247009 2209 8398306 24337175520 47 (1) 497 47 10843849 164025 368690866 7078543072336 405 (1)
(2)3293
3293405
326820367169 3094081 692483746 18011267467200 1759 (1) 4513 1759 37 777925 1764 28783225 154421605115 42 (5) 883 42 779689 776161 28848493 14723512860 881 (1) 883 881 38 1539 961 58482 11046600 31 (5) 50 31 39 25 1 975 30420 1 (1) 5 1 97969 15600 97969 30273047 20 (2)
(4)313
337287
313113569 82369 4429191 6417464040 287 (1) 337 287 41 25 16 1025 25215 4 (3)
(2)
(1)4
5
53
3
4441 400 18081 1023729 20 (1)
(5)
(4)21
29
2920
20
211025 64 42025 8598315 8 (5) 33 8 1089 961 44649 4437840 31 (1) 33 31 296225 135424 12145225 37644053445 368 (5) 657 368 431649 160801 17697609 69092327520 401 (1) 657 401 776161 590400 776161 443882159 120 (2)
(4)881
1169431
8811366561 185761 56029001 415498418160 431 (1) 1169 431 1610361 547600 66024801 504518397201 740 (1)
(5)
(4)1269
1469
1469740
740
126946 72 49 3312 139656 7 (5) 11 7 121 23 242 3696 1 (4) 12 11 96805921 18258529 4453072366 291826130053920 4273 (1) 9839 4273 47 14561856 2289169 684407232 17682275119320 1513 (1)
(5)
(4)3816
4105
41051513
1513
381655 169 81 9295 786500 9 (1) 13 9 61 22306729 1852321 1360710469 50020275760260 1361 (1) 4723 1361 62 39200 22801 2430400 3082042320 151 (5) 249 151 62001 16399 124002 42110880 23 (4) 280 249 65 9 4 585 12675 2 (1) 3 2 65 16 4225 266175 4 (5) 9 4 81 49 5265 304200 7 (1) 9 7 121 4 7865 697125 2 (1) 11 2 3481 3364 226265 27669525 58 (1) 59 58 9409 9360 9409 93023 12 (2)
(4)97
1377
9718769 49 1219985 1347504600 7 (1) 137 7 18785 12544 1221025 1004328975 112 (3)
(5)112
17779
11231329 6241 2036385 2847717600 79 (1) 177 79 6985449 702244 454054185 9626221017075 838 (1) 2643 838 20079361 16511040 20079361 51202419841 504 (2) 4481 1889 30856865 16711744 2005696225 75510873577425 4088 (3) 4088 3761 47568609 14145121 3091959585 164152327138800 3761 (1) 6897 3761 214827649 28314000 214827649 3121260929857 660 (2) 14657 13657 69 192 169 13248 723672 13 (5) 19 13 70 9 5 126 1176 1 (2) 3 2 2809 2520 2809 65773 6 (2)
(4)53
7317
535329 289 373030 227497200 17 (1) 73 17 71 3600 121 255600 129150420 11 (1)
(5)
(4)60
61
6111
11
6077 2816 2809 216832 7114800 53 (5) 75 53 78 458329 70200 458329 306627517 30 (2)
(4)677
727623
677528529 388129 41225262 179665022160 623 (1) 727 623 79 169000000 166952241 13351000000 239420967877000 12921 (5)
(4)18329
1832912921
1300085 85 36 7225 556325 6 (5) 11 6 121 49 10285 953700 7 (1) 11 7 72607441 72570960 72607441 19609880039 924 (4) 12049 8521 86 387 289 33282 4038216 17 (5) 26 17 87 17956 169 1562172 1952423550 13 (1) 134 13 93 1444 75 44764 9458162 5 (2) 38 37 94 14112 529 1326528 1526754768 23 (5) 121 23 14641 13583 29282 1870176 17 (4) 168 121 95 1521 1369 144495 23934300 37 (1) 39 37
- 1 <= ‚‡ <= 99
- 100 <= ‚‡ <= 199
- 200 <= ‚‡ <= 299
- 300 <= ‚‡ <= 399
- 400 <= ‚‡ <= 499
- 500 <= ‚‡ <= 599
- 600 <= ‚‡ <= 699
- 700 <= ‚‡ <= 799
- 800 <= ‚‡ <= 899
- 900 <= ‚‡ <= 999
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