### Congruum g : 1 <= g <= 999

Definition 1.

k2g=mn(m2-n2)
k, m, n, g in N (integer > 0)

Definition 2.

x2+gy2=z2
x2-gy2= -w2

x, y, z, w in N (integer > 0)

m=gy2, n=x2

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 361 congruum g : 1 <= g <= 999.

53, 181, 349, 485, 533

The solution of the following numbers are not in the range of 1 <= y <= 10,000.

101, 103, 118, 127, 142, 157, 173, 191, 197, 199,
223, 229, 237, 263, 269, 271, 277, 278, 293,
302, 303, 311, 317, 326, 327, 334, 358, 365, 367, 373, 382, 389, 397, 398,
407, 413, 415, 421, 431, 439, 446, 453, 454, 461, 463, 478, 487, 493,
501, 502, 503, 519, 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 : 1 <= g <= 999
gxyzwm (=gy2)n (=x2)xe(=g2y2)ye(=g2yzw)ze(=x)
521315425752
15622257528347992547024524398441273512251430549626751562
136519173253642252729356
29701399149014900142129108236770
374214588388177792517642878322515442160511542
418533311025644202585983158
368856574012962251354241214522537644053445368
53340345945550592673718731803251158313156992785572252458354977042091534034
613198445472313611207952510227204736851025106437766275353198
654197651642252661754
1121717779187851254412210251004328975112
40886896897376130856865167117442005696225755108735774254088
8561117853672255563256
10942567312725176429702512338418542
13756581173425313646922512922456556
1451211711451442102535742512
11281852497192149626251272384719580625186575081536251128
1492382538719193125566441387562541025782925238
18134950412165539430393073858021122150250055636830180138082175217854270134950
2571175286518177736119229382513810950449419513025764436436074934511752
2653658973662512961755625228125912536
5523781724136278530470496138025511602397525552
3491137307453179779803211938597594112934512900676570560340913108410776581925327113730
45710084851041710319107497825101606449126506025108881580248555951008
48581314640916289111536719921511285661196659696619329732252833038246830547892581314
50516813337239853452822443099225267026221475168
6730830779649715841478130414545303668642414558593225119951769291395072567308
50917013339239860212890043784689272883022113170
5331375226025195603208791934823312518912300484103126082556256990313848049415325137522
56555841112379994976531136453661722511743763263325558
629101272362910039564124569306110
689201331768940047472126631848120
70921017499401204901441001452748091709961412123210
761402980179964000116004870407611074846896549140
79313251934919825174241572122529735124965132
82161189052664325199672419525374299245520564300254095447139032794856118
905952531857127925421459063042300641225103099242932475952
94930143794990090060127108090130
9854081357711664651664641639680257277657725408

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E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima