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

g	x	y	z	w	m (=gy2)	n (=x2)	xe(=g2y2)	ye(=g2yzw)	ze(=x)
5	2	1	3	1	5	4	25	75	2
	1562	2257	5283	4799	25470245	2439844	127351225	143054962675	1562
13	6	5	19	17	325	36	4225	272935	6
29	70	13	99	1	4901	4900	142129	1082367	70
37	42	145	883	881	777925	1764	28783225	154421605115	42
41	8	5	33	31	1025	64	42025	8598315	8
	368	85	657	401	296225	135424	12145225	37644053445	368
53	34034	5945	55059	26737	1873180325	1158313156	99278557225	24583549770420915	34034
61	3198	445	4723	1361	12079525	10227204	736851025	10643776627535	3198
65	4	1	9	7	65	16	4225	266175	4
	112	17	177	79	18785	12544	1221025	1004328975	112
	4088	689	6897	3761	30856865	16711744	2005696225	75510873577425	4088
85	6	1	11	7	85	36	7225	556325	6
109	42	5	67	31	2725	1764	297025	123384185	42
137	56	5	81	17	3425	3136	469225	129224565	56
145	12	1	17	1	145	144	21025	357425	12
	1128	185	2497	1921	4962625	1272384	719580625	18657508153625	1128
149	238	25	387	191	93125	56644	13875625	41025782925	238
181	34950	4121	65539	43039	3073858021	1221502500	556368301801	380821752178542701	34950
257	11752	865	18177	7361	192293825	138109504	49419513025	7644364360749345	11752
265	36	5	89	73	6625	1296	1755625	2281259125	36
	552	37	817	241	362785	304704	96138025	511602397525	552
349	113730	7453	179779	80321	19385975941	12934512900	6765705603409	13108410776581925327	113730
457	1008	485	10417	10319	107497825	1016064	49126506025	10888158024855595	1008
485	81314	6409	162891	115367	19921511285	6611966596	9661932973225	28330382468305478925	81314
505	168	13	337	239	85345	28224	43099225	267026221475	168
	67308	3077	96497	15841	4781304145	4530366864	2414558593225	1199517692913950725	67308
509	170	13	339	239	86021	28900	43784689	272883022113	170
533	137522	6025	195603	20879	19348233125	18912300484	10312608255625	6990313848049415325	137522
565	558	41	1123	799	949765	311364	536617225	11743763263325	558
629	10	1	27	23	629	100	395641	245693061	10
689	20	1	33	17	689	400	474721	266318481	20
709	210	17	499	401	204901	44100	145274809	1709961412123	210
761	40	29	801	799	640001	1600	487040761	10748468965491	40
793	132	5	193	49	19825	17424	15721225	29735124965	132
821	6118	905	26643	25199	672419525	37429924	552056430025	409544713903279485	6118
905	952	53	1857	1279	2542145	906304	2300641225	103099242932475	952
949	30	1	43	7	949	900	900601	271080901	30
985	408	13	577	1	166465	166464	163968025	7277657725	408