### Congruum g : 500 <= 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=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
gxyzwm (=x2)n (=gy2)xe(=x2)ye(=xzw)ze(=y)
5051060139613841576111238120179192080112381201845302548601396
510131356182335317239691599360172396984494044756
51467073204081473485774498787329213906240044987873292654557458232332040
5171417601967383200788918612002007889106751253760
54685297717225218472255853952
365639133713322519656133225480949556
10734014235271151329873600115132980466523340
5615458577511297025359042970251606916158
134556188922318090251759296180902556657721556
5748294540883519823676879873025955503366879873025570596029737985408
58266197143074647564974382042809119013180043820428092791746972111231430

Congruum g : 600 <= g <= 699
gxyzwm (=x2)n (=gy2)xe(=x2)ye(=xzw)ze(=y)
60294930120159990060154180090060168270965130
6096412080940941088124360041088121209472120
785128397276162258769661622547881310512
731052912102511134235344341025516416409653443410251005928567100652912
645156412201660914609244640881312180002446408813795146019721220
646325635928710562523256105625334857256
6517932094360762884926040062884945391399320
3086511444248110033952648225851987136952648225131550294601451144
658185937202620721433456996493411072003456996491044212647393720
669391451092482712710315323310257977584161532331025512129694993851092
67119216024711129369024124156003690241535912703960

Congruum g : 700 <= g <= 799
gxyzwm (=x2)n (=gy2)xe(=x2)ye(=xzw)ze(=y)
721126252641447910447159390625502508161593906251909684176625264
7412658343151702254742470225137251458
7592654287241702251214470225183292554
7772263366029167131835122526893384612005122526898702582261113660
7912531312025537250876407479691139040064074796916216690398047120
79816494019991201271920112768002719201395891755140
79943851445983163119228225165680641922822542790027105144

Congruum g : 800 <= g <= 899
gxyzwm (=x2)n (=gy2)xe(=x2)ye(=xzw)ze(=y)
805129617213121127991679875214173120167987521217661433551972
8061285421753479165122514217841651225107899779542
81344041154062191339119396096811928110800193960968192877924409211540
81461012108551115137222201358974003722220160047268301210
8702160157227409134894666032012846500804666032017986321741601572
889806548819179376504422520482566504422552432151385548
89040018447213119160080016279840160080015891392079984

Congruum g : 900 <= g <= 999
gxyzwm (=x2)n (=gy2)xe(=x2)ye(=xzw)ze(=y)
9032959326030607285438757456496104280087574564925852906480393260
9056516122865521647994245955921470455204245955921276653750074919228
910736113283696191541843211585584054184321381391657919132
915732144744171995359704117714405359704139216956363944
781780818374336110548958560006110548947546307626380
5953725260023590473544654369581061603544654369211009730408497252
9515305715407120723657281504524922553916002815045249893768509549431540
95785720105759373444938280073444953716845720
966533106174332840899660028408914239681310

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