単かと思ったのですが、うまくいかなかった。調べてみると、整数Bが大きい場合には難しい
ことが分かった。（→ 参考：「２つをつなぐ」）
以下では、m*sqrt(2)の連分数展開について、分かったことを紹介する。
n=0、1、2、・・・、10について、2^n*sqrt(2) を連分数展開すると、以下のようになる。
sqrt(2)=[1, ・2]
2*sqrt(2)=[2, ・1, ・4]
4*sqrt(2)=[5, ・1, 1, 1, ・10]
8*sqrt(2)=[11, ・3, 5, 3, ・22]
16*sqrt(2)=[22, ・1, 1, 1, 2, 6, 11, 6, 2, 1, 1, 1, ・44]
32*sqrt(2)=[45, ・3, 1, 12, 5, 1, 1, 2, 1, 2, 4, 1, 21, 1, 4, 2, 1, 2, 1, 1, 5, 12, 1, 3, ・90]
64*sqrt(2)=[90, ・1, 1, 25, 2, 1, 3, 1, 2, 1, 9, 1, 10, 2, 2, 5, 2, 3, 2, 1, 1, 5, 1, 7, 45, 7, 1, 5, 1,
1, 2, 3, 2, 5, 2, 2, 10, 1, 9, 1, 2, 1, 3, 1, 2, 25, 1, 1, ・180]
128*sqrt(2)=[181, ・51, 1, 2, 1, 1, 6, 1, 4, 2, 5, 4, 1, 10, 1, 6, 1, 3, 1, 2, 2, 3, 1, 1, 22, 15, 1, 2, 3,
2, 1, 1, 4, 2, 1, 2, 2, 1, 4, 1, 20, 2, 8, 2, 1, 12, 3, 1, 89, 1, 3, 12, 1, 2, 8, 2, 20, 1, 4,
1, 2, 2, 1, 2, 4, 1, 1, 2, 3, 2, 1, 15, 22, 1, 1, 3, 2, 2, 1, 3, 1, 6, 1, 10, 1, 4, 5, 2, 4, 1,
6, 1, 1, 2, 1, 51, ・362]
256*sqrt(2)=[362, ・25, 1, 6, 14, 1, 1, 1, 2, 1, 2, 9, 1, 4, 1, 14, 1, 1, 2, 1, 4, 1, 1, 3, 11, 31, 2, 1,
1, 5, 9, 1, 2, 1, 5, 1, 1, 1, 41, 1, 16, 1, 2, 6, 7, 1, 44, 2, 1, 1, 1, 5, 1, 5, 4, 4, 10, 2,
2, 2, 1, 5, 2, 1, 1, 1, 3, 1, 6, 1, 2, 7, 1, 1, 10, 1, 3, 1, 1, 2, 2, 8, 1, 2, 1, 22, 1, 1, 1,
1, 2, 4, 2, 2, 3, 3, 2, 103, 181, 103, 2, 3, 3, 2, 2, 4, 2, 1, 1, 1, 1, 22, 1, 2, 1, 8, 2, 2,
1, 1, 3, 1, 10, 1, 1, 7, 2, 1, 6, 1, 3, 1, 1, 1, 2, 5, 1, 2, 2, 2, 10, 4, 4, 5, 1, 5, 1, 1, 1,
2, 44, 1, 7, 6, 2, 1, 16, 1, 41, 1, 1, 1, 5, 1, 2, 1, 9, 5, 1, 1, 2, 31, 11, 3, 1, 1, 4, 1, 2,
1, 1, 14, 1, 4, 1, 9, 2, 1, 2, 1, 1, 1, 14, 6, 1, 25, ・724]
512*sqrt(2)=[724, ・12, 1, 13, 7, 3, 6, 2, 1, 4, 2, 2, 2, 7, 3, 2, 10, 7, 5, 1, 1, 15, 5, 11, 4, 1, 6, 1,
2, 3, 84, 1, 7, 1, 5, 3, 15, 1, 21, 1, 2, 4, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 2, 2,
11, 1, 3, 8, 1, 2, 1, 5, 3, 1, 3, 5, 2, 1, 1, 3, 1, 4, 4, 2, 1, 2, 11, 3, 3, 1, 8, 1, 4, 1, 1,
1, 1, 4, 51, 1, 1, 90, 206, 1, 6, 1, 1, 2, 2, 1, 1, 1, 2, 4, 46, 2, 17, 1, 5, 8, 1, 4, 1, 3,
3, 1, 2, 14, 1, 1, 3, 5, 2, 1, 5, 1, 4, 5, 8, 2, 11, 1, 2, 3, 5, 22, 2, 3, 1, 1, 2, 1, 2, 34,
1, 20, 3, 12, 2, 19, 2, 1, 3, 1, 62, 5, 1, 1, 1, 3, 1, 1, 1, 6, 30, 1, 1, 1, 19, 1, 2, 1, 3,
29, 3, 2, 12, 1, 1, 361, 1, 1, 12, 2, 3, 29, 3, 1, 2, 1, 19, 1, 1, 1, 30, 6, 1, 1, 1, 3, 1,
1, 1, 5, 62, 1, 3, 1, 2, 19, 2, 12, 3, 20, 1, 34, 2, 1, 2, 1, 1, 3, 2, 22, 5, 3, 2, 1, 11, 2,
8, 5, 4, 1, 5, 1, 2, 5, 3, 1, 1, 14, 2, 1, 3, 3, 1, 4, 1, 8, 5, 1, 17, 2, 46, 4, 2, 1, 1, 1, 2,
2, 1, 1, 6, 1, 206, 90, 1, 1, 51, 4, 1, 1, 1, 1, 4, 1, 8, 1, 3, 3, 11, 2, 1, 2, 4, 4, 1, 3, 1,
1, 2, 5, 3, 1, 3, 5, 1, 2, 1, 8, 3, 1, 11, 2, 2, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 4,
2, 1, 21, 1, 15, 3, 5, 1, 7, 1, 84, 3, 2, 1, 6, 1, 4, 11, 5, 15, 1, 1, 5, 7, 10, 2, 3, 7, 2,
2, 2, 4, 1, 2, 6, 3, 7, 13, 1, 12, ・1448]
1024*sqrt(2)=[1448, ・6, 2, 6, 1, 1, 3, 6, 3, 5, 1, 1, 1, 2, 2, 2, 1, 3, 6, 1, 20, 3, 1, 1, 2, 3, 1, 7, 10,
5, 1, 1, 1, 1, 14, 2, 1, 1, 169, 1, 3, 2, 2, 1, 1, 1, 31, 1, 10, 2, 1, 9, 2, 2, 2, 4, 4, 3,
2, 2, 1, 4, 1, 23, 1, 1, 17, 2, 11, 1, 1, 7, 2, 1, 2, 3, 1, 1, 9, 2, 5, 2, 1, 5, 6, 1, 1, 18,
1, 1, 1, 4, 9, 25, 1, 3, 45, 413, 1, 2, 1, 3, 1, 5, 3, 1, 8, 23, 4, 8, 1, 11, 4, 2, 2, 2, 1,
1, 1, 1, 2, 1, 29, 7, 2, 1, 2, 12, 1, 1, 1, 1, 2, 16, 1, 23, 2, 1, 1, 10, 11, 4, 1, 1, 3, 1,
2, 1, 69, 1, 9, 1, 1, 1, 24, 1, 38, 1, 2, 1, 1, 125, 2, 1, 4, 1, 1, 3, 13, 15, 3, 40, 2, 7,
14, 1, 1, 1, 4, 6, 3, 1, 180, 3, 1, 5, 1, 2, 1, 1, 58, 1, 1, 6, 1, 9, 3, 61, 3, 3, 8, 3, 2,
1, 1, 30, 1, 8, 2, 1, 9, 4, 6, 6, 10, 2, 17, 5, 2, 3, 1, 1, 2, 1, 10, 1, 1, 2, 6, 1, 2, 5, 1,
2, 1, 3, 1, 1, 2, 9, 1, 2, 2, 1, 10, 1, 1, 3, 7, 5, 1, 1, 7, 1, 1, 1, 17, 2, 1, 35, 1, 92, 2,
5, 3, 1, 5, 14, 1, 102, 1, 1, 44, 1, 3, 25, 1, 1, 1, 1, 4, 10, 1, 3, 1, 7, 1, 1, 1, 5, 5, 2,
1, 1, 1, 1, 1, 1, 8, 5, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 4, 7, 1, 5, 4, 1, 5, 1, 1, 1, 4, 4,
3, 1, 4, 1, 2, 2, 5, 1, 10, 2, 7, 1, 1, 1, 11, 1, 3, 2, 42, 6, 1, 2, 3, 2, 2, 22, 2, 1, 1, 7,
3, 1, 2, 14, 5, 4, 1, 1, 1, 3, 4, 1, 4, 2, 2, 1, 12, 1, 1, 1, 3, 27, 1, 5, 1, 1, 723, 1, 1,
5, 1, 27, 3, 1, 1, 1, 12, 1, 2, 2, 4, 1, 4, 3, 1, 1, 1, 4, 5, 14, 2, 1, 3, 7, 1, 1, 2, 22, 2,
2, 3, 2, 1, 6, 42, 2, 3, 1, 11, 1, 1, 1, 7, 2, 10, 1, 5, 2, 2, 1, 4, 1, 3, 4, 4, 1, 1, 1, 5,
1, 4, 5, 1, 7, 4, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 5, 8, 1, 1, 1, 1, 1, 1, 2, 5, 5, 1, 1, 1,
7, 1, 3, 1, 10, 4, 1, 1, 1, 1, 25, 3, 1, 44, 1, 1, 102, 1, 14, 5, 1, 3, 5, 2, 92, 1, 35, 1,
2, 17, 1, 1, 1, 7, 1, 1, 5, 7, 3, 1, 1, 10, 1, 2, 2, 1, 9, 2, 1, 1, 3, 1, 2, 1, 5, 2, 1, 6, 2,
1, 1, 10, 1, 2, 1, 1, 3, 2, 5, 17, 2, 10, 6, 6, 4, 9, 1, 2, 8, 1, 30, 1, 1, 2, 3, 8, 3, 3, 61,
3, 9, 1, 6, 1, 1, 58, 1, 1, 2, 1, 5, 1, 3, 180, 1, 3, 6, 4, 1, 1, 1, 14, 7, 2, 40, 3, 15, 13,
3, 1, 1, 4, 1, 2, 125, 1, 1, 2, 1, 38, 1, 24, 1, 1, 1, 9, 1, 69, 1, 2, 1, 3, 1, 1, 4, 11, 10,
1, 1, 2, 23, 1, 16, 2, 1, 1, 1, 1, 12, 2, 1, 2, 7, 29, 1, 2, 1, 1, 1, 1, 2, 2, 2, 4, 11, 1, 8,
4, 23, 8, 1, 3, 5, 1, 3, 1, 2, 1, 413, 45, 3, 1, 25, 9, 4, 1, 1, 1, 18, 1, 1, 6, 5, 1, 2, 5,
2, 9, 1, 1, 3, 2, 1, 2, 7, 1, 1, 11, 2, 17, 1, 1, 23, 1, 4, 1, 2, 2, 3, 4, 4, 2, 2, 2, 9, 1, 2,
10, 1, 31, 1, 1, 1, 2, 2, 3, 1, 169, 1, 1, 2, 14, 1, 1, 1, 1, 5, 10, 7, 1, 3, 2, 1, 1, 3, 20,
1, 6, 3, 1, 2, 2, 2, 1, 1, 1, 5, 3, 6, 3, 1, 1, 6, 2, 6, ・2896]
nが大きくなるにしたがって、連分数の循環節の長さがほぼ2^nに比例して長くなる。
■類数2の虚2次体Q(sqrt(-6))の整数環Z[sqrt(-6)]に対応するj-不変量j(sqrt(-6))は、実２
次体Q(sqrt(2))の代数的整数であるが、Pari/GPで小数点以下1000桁の精度で計算して、連
分数展開すると、
gp > default(realprecision,1000)
gp > r=ellj(sqrt(-6))
%13 = 4831907.903351339745397366298049145980411172284643549875203910956201009
8941091307115758680234063466620338706791715722518679612753040559530166
0765344362482484728186418152205690831063804669240010331185949034512779
5363895234783033977030713467562218519056774938747811353562832847297446
4964617203456848998101054189828012766292697275645140884357433907454486
1928222330272600349297433598832863291500291360737579843078039157546575
6528306719061406830944955403207525455433554144709245756975013653453097
0205634265912676014815640762624676219410126945597525790508560083124663
3358704837114061498010087504502140831610902625969303554662932246432786
5129115192927104248561682336682194321821173351939606198085043129616317
4937029106923592993174890332539220760243979883695716995429768400248417
2258612054082758268682967804182149501946599311266247951944772101289963
8835697886264207345809968926120852318336486275039812794938679832154222
2844819565436826683105718931184782197709345572817930158985376138334945
50691137104964065270
gp > contfrac(r)
%14 = [4831907, 1, 9, 2, 1, 7, 1, 1, 1, 1, 2, 1, 3, 3, 3, 1, 1, 1, 7, 1, 2, 2, 2, 22, 7, 1, 22, 14, 3, 2,
5, 1, 1, 1, 12, 2, 8, 11, 2, 1, 6, 1, 7, 8, 3, 7, 3, 6, 2, 1, 2, 1, 4, 4, 1, 1, 1, 2, 3, 2, 1, 13, 1,
4, 1, 6, 1, 3, 2, 47, 1, 12, 8, 1, 1, 38, 4, 2, 4, 1, 3, 163, 3, 10, 1, 1, 1, 1, 4, 4, 7, 16, 1, 12,
3, 1, 13, 1, 3, 1, 323, 1, 2, 2, 1, 4, 1, 1, 2, 1, 3, 2, 5, 14, 14, 3, 30, 1, 5, 21, 2, 1, 6, 3, 3,
2, 5, 1, 4, 1, 3, 1, 2, 1, 2, 7, 3, 1, 1, 15, 1, 3, 2, 3, 2, 65, 1, 3, 4, 1, 1, 12, 2, 4, 6, 1, 1, 1,
1, 1, 13, 4, 6, 1, 1, 1, 1, 10, 2, 1, 2, 1, 11, 1, 2, 15, 1, 5, 20, 1, 2, 3, 5, 8, 2, 1, 3, 1, 4, 1,
1, 5, 2, 3, 3, 1, 12, 1, 1, 1, 1, 15, 58, 6, 2, 10, 1, 7, 11, 1, 2, 3, 46, 1, 2, 1, 8, 1, 2, 1, 2, 5,
1, 5, 8, 1, 7, 1, 5, 1, 21, 5, 5, 13, 2, 1, 3, 5, 1, 4, 5, 1, 1, 2, 45, 1, 1, 1, 5, 2, 1, 1, 3, 1, 4, 1,
2, 1, 10, 1, 1, 2, 27, 1, 1, 34, 1, 27, 1, 1, 1, 10, 1, 18, 1, 1, 1, 4, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3,
3, 1, 3, 3, 3, 3, 1, 23, 1, 2, 1, 43, 184, 1, 8, 1, 12, 1, 130, 15, 1, 3, 13, 1, 1, 1, 5, 2, 11, 1,
3, 13, 7, 2, 3, 1, 9, 1, 1, 5, 26, 3, 1, 52, 5, 10, 1, 4, 1, 2, 3, 3, 20, 6, 49, 2, 3, 1, 2, 1, 3, 10,
2, 2, 1, 1, 2, 1, 6, 1, 3, 2, 4, 1, 1, 1, 12, 2, 2, 1, 1, 4, 1, 5, 3, 2, 1, 11, 10, 1, 1, 1, 1, 1, 18,
1, 1, 2, 2, 2, 17, 7, 7, 3, 6, 1, 1, 1, 11, 2, 1, 21, 6, 2, 1, 7, 1, 2, 232, 1, 1, 8, 1, 2, 3, 1, 7, 21,
4, 1, 8, 1, 21, 1, 1, 1, 1, 4, 1, 2, 2, 11, 1, 5, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 11, 1, 1, 1, 736, 2,
6, 8, 1, 2, 1, 1, 1, 19, 2, 138, 1, 2, 1, 79, 1, 87, 4, 1, 30, 5, 1, 1, 1, 1, 1, 4, 6, 13, 5, 2, 2, 4,
2, 1, 1, 21, 1, 22, 3, 1, 2, 1, 8, 1, 2, 100, 2, 1, 22, 2, 2, 2, 1, 6, 2, 1, 2, 14, 5, 2, 2757, 3, 12,
111, 1, 1, 7, 13, 2, 1, 1, 1, 3, 72, 1, 2, 1, 1, 1, 3, 2, 4, 1, 1, 1, 6, 2, 2, 4, 1, 3, 6, 1, 2, 2, 2,
11, 3, 3, 2, 5, 8, 1, 1, 1, 1, 3, 4, 2, 3, 2, 1, 8, 1, 1, 4, 2, 5153, 10, 1, 1, 5, 1, 4, 1, 1, 1, 69,
1, 6, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 27, 5, 2, 2, 3, 22, 1, 2, 2, 3, 1, 1, 5, 1, 523, 8, 2, 3, 1, 2,
4, 1, 5, 2, 2, 4, 3, 1, 1, 1, 2, 1, 2, 4, 15, 1, 1, 2, 3, 22, 1, 1, 1, 1, 30, 3, 2, 3, 1, 3, 3, 9, 2, 2,
1, 1, 4, 1, 40, 2, 6, 2, 13, 2, 1, 50, 5, 2, 1, 1, 1, 26, 4, 25, 4, 1, 3, 1, 3, 1, 6, 2, 1, 5, 125, 2,
1, 1, 3, 4, 1, 3, 1, 5, 1, 3, 3, 1, 1, 1, 9, 2, 197, 1, 18, 1, 7, 2, 1, 1, 2, 1, 3, 14, 1, 4, 14, 2, 1,
5, 3, 2, 1, 5, 1, 6, 21, 2, 3, 19, 1, 1, 7, 5, 52, 17, 1, 1, 1, 6, 5, 1, 1, 8, 1, 5, 3, 2, 2, 12, 1, 32,
1, 1, 1, 1, 2, 1, 2, 3, 21, 2, 24, 2, 1, 13, 1, 4, 1, 3, 1, 2, 1, 1, 4, 2, 3, 1, 3, 1, 1, 4, 2, 7, 2, 1,
388, 1, 26, 1, 9, 1, 1, 1, 5, 2, 1, 3, 1, 2, 2, 17, 2, 25, 1, 4, 1, 1, 1, 2, 1, 1, 5, 1, 2, 1, 3, 2, 4,
3, 1, 1, 930, 1, 12, 1, 13, 1, 2, 10, 1, 6, 1, 1, 1, 11, 2, 7, 3, 3, 4, 1, 1, 4, 19, 1, 1, 3, 5, 2, 3,
9, 1, 1, 23, 1, 4, 1, 3, 1, 3, 9, 1, 2, 12, 2, 13, 22, 1, 23, 2, 2, 1, 2, 1, 5, 2, 5, 12, 4, 1, 1, 2,
1, 2, 6, 3, 1, 8, 2, 1, 17, 1, 1, 1, 152, 8, 2, 11, 8, 12, 1, 1, 11, 2, 3, 3, 3, 2, 1, 1, 1, 5, 1, 2,
7, 1, 6, 3, 1, 1178, 6, 16, 1, 1, 1, 9, 1, 6, 3, 352, 2, 3, 1, 1, 26, 1, 1, 1, 3, 4, 12, 1, 2, 1, 1,
2, 6, 2, 5, 3]
となり、循環連分数であるかどうかは分からない。精度を10000桁に増やしても同様である。
十分な精度(もっともっと高い精度)で計算すると、循環連分数になることが確認できるはず
である。
しかし、別の方法で、
j(sqrt(-6))=2417472+1707264*sqrt(2) = (24*sqrt(2)+60)^3*(sqrt(2)+1)^2
であることが分かる。
■1つめの方法は、j(sqrt(-6))とj(sqrt(-6)/2)がQ(sqrt(2))のconjugateであることを利用する。
gp > (ellj(sqrt(-6))+ellj(sqrt(-6)/2))
time = 1 ms.
%16 = 4834944.00000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000
gp > (ellj(sqrt(-6))-ellj(sqrt(-6)/2))/sqrt(2)
time = 1 ms.
%17 = 3414528.00000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000
これにより、
j(sqrt(-6))=(4834944+3414528*sqrt(2))/2=2417472+1707264*sqrt(2)
j(sqrt(-6)/2)=(4834944-3414528*sqrt(2))/2=2417472-1707264*sqrt(2)
であることが分かる。
■2つめの方法は、LLL-algorithmで、j(sqrt(-6))が満たす整数係数のmonic 2次方程式を
見つける。
3次の正方行列A=[1,0,0;0,1,0;[r^2*10^100],[r*10^100],10^100]に、LLL-algorithmを適用す
ると、
gp > A=[1,0,0;0,1,0;floor(r^2*10^100),floor(r*10^100),10^100]
%19 =
[1 0 0]
[0 1 0]
[23347333986469139993970523798554930063113114970603623607817898054218335581
4635768791494736445603753555206908370365
483190790335133974539736629804914598041117228464354987520391095620100989410
91307115758680234063466620338706
100000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000]
gp > B=qflll(A,1)
%20 =
[1
-696235687961132154949410071524777658469103976
220565253504247676770193672310435646753153206789]
[-4834944
4517004482581425660278121701120307021073601266
14745678672006243209458594413290163629098883759]
[14670139392
16255225314378004956120235127805908020275207260459875596527
-5149610710623664676463629665003104364966425545948630741218320]
よって、r=j(sqrt(-6))はmonic 2次方程式 t^2-4834944*t+14670139392 = 0 の2根
(2417472±1707264*sqrt(2))の1つであることが分かり、
r≒4831907.903351339であることから、r=2417472+1707264*sqrt(2)
が分かる。
ＧＡＩ さんからのコメントです。（令和３年９月２６日付け）
■j-不変量 j((1+sqrt(-427))/2)=A+B*sqrt(61) を求めようとして、連分数展開を使えば簡
単かと思ったのですが、うまくいかなかった。調べてみると、整数Bが大きい場合には難しい
ことが分かった。
結果を求めるだけならば、やはりpolclass( ) のflag無しを使うことで、
gp > polclass(-427) (flagは使わない。)
%823 = x^2 + 15611455512523783919812608000*x + 155041756222618916546936832000000
を返す。
%823=0 の2解x1,x2を求めるために、
gp > 15611455512523783919812608000/2
%824 = 7805727756261891959906304000
gp > %824^2-155041756222618916546936832000000
%825 = 60929385804877310217096184941254990522923912003584000000
gp > factor(%825)
%829 =
[ 2 30]
[ 3 12]
[ 5 6]
[ 7 2]
[ 11 6]
[ 13 2]
[ 23 4]
[ 29 2]
[ 37 2]
[ 61 1]
[ 71 2]
[191 2]
[359 2]
これから、
gp > 2^15*3^6*5^3*7*11^3*13*23^2*29*37*71*191*359
%830 = 999421027517377348595712000
よって、 x1=-%824 - %830*sqrt(61) 、x2=-%824 + %830*sqrt(61)
|x1|>|x2| から、
ellj((1+sqrt(427)*I)/2)=x1=-7805727756261891959906304000
- 999421027517377348595712000*sqrt(61)
それと、qflllのコマンドを用いなくとも、
gp > ellj(sqrt(6)*I)
%818 = 4831907.9033513397454
から一発で、
gp > algdep(%818,2) (flagの2はxの2次式までで解となる方程式を求めたい時に使う。）
%819 = x^2 - 4834944*x + 14670139392
をとれる。
または、
gp > polclass(-24)
%838 = x^2 - 4834944*x + 14670139392
を返すことから、この解の絶対値が大きい方を選んで、
ellj(sqrt(-24)/2)=ellj(sqrt(6)*I)=2417472+1707264*sqrt(2) （24≡0 mod 4)
で計算できていくと思います。
ＧＡＩ さんからのコメントです。（令和３年９月２７日付け）
qflll( )のコマンドについても気になり調べたことがあり、例えば、
f(x)=x^5-2*x^4+6*x^3-5*x^2+10*x-4 に対し、factor(f(x))とすれば、
%=[x^2-2*x+4 1]
[x^3+2*x-1 1]
と、見事に因数分解する原理は何だろう？に対し、Gram-Schmidtの直交化法の利用によ
るLLL algorithm を使って因数を絞り出していく様子を追跡していった経験は以前していま
した。
それはまた、任意の数値 r=3.1415926 に対するものを解とする方程式を作り出す方法と
して、このqflll（ ）コマンドを利用すれば、2*x^4+5*x^3-6*x-331=0 などの結果を引き出して
くれることも経験していました。
一方、algdep( )コマンドにも同様な機能が備わっており、この２つの結果が同じ時もある
が、異なることもある。だから確かめたわけではないが、algdep には陰ではLLL algorithm
が使われいるのかなと思っていました。
このNakaoさんの例では、qflllで求めてあるが、同じ結果を出すにはalgdepが楽かなと思い
掲載していました。
H．Nakao さんからのコメントです。（令和３年９月２８日付け）
j((1+sqrt(-427))/2) を求める別の方法
■j((1+sqrt(-427))/2)=A+B*sqrt(61)をQ(sqrt(61))のconjugateを使って求めることができ
ました。
虚2次体Q(sqrt(-427))の類体はQ(sqrt(-7),sqrt(61))であり、Q(sqrt(-427))のclass groupの
生成元が整イデアルa=(7+3*((1+sqrt(-427))/2,(1+sqrt(-427))/2)を含む同値類であるので、
j((1+sqrt(-427))/2)とj((7+sqrt(-427))/14)が実2次体Q(sqrt(61))の代数的整数であり、互い
にconjugateであることが分かる。
Pari/GPで計算すると、
gp > default(realprecision,300)
gp > r1=ellj((1+sqrt(-427))/2)
%2 = -15611455512523783919812598068.71869837878374130089822922127126371865823
58965041864476640264439915924739350560175590449591544974107685963321295
72643160161386979739020650915335942914322818203535949564509076870094654
98521268418510213835497768091272628680003517827129764411377754597829601
40870215395571997
gp > r2=ellj((7+sqrt(-427))/14)
%3 = -9931.281301621216258699101770778728736281341764103495813552335973556008
40752606494398244095504084550258923140366787042735683983861302026097934
90846640570856771817964640504354909231299053450147873158148978616450223
19087273713199964821728702355886222454021703985912978460442800331874532
97731851851759507
gp > (r1+r2)
%4 = -15611455512523783919812608000.00000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000
gp > (r1-r2)/sqrt(61)
%5 = -1998842055034754697191424000.000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000
gp > [15611455512523783919812608000,1998842055034754697191424000]/2
%6 = [7805727756261891959906304000, 999421027517377348595712000]
gp > r1+(7805727756261891959906304000+999421027517377348595712000*sqrt(61))
%7 = 8.043163417854770450 E-279
gp > r2+(7805727756261891959906304000-999421027517377348595712000*sqrt(61))
%8 = -3.062002842093458499 E-281
となり、
j((1+sqrt(-427))/2)=
-(7805727756261891959906304000+999421027517377348595712000*sqrt(61))
j((7+sqrt(-427))/14)=
-(7805727756261891959906304000-999421027517377348595712000*sqrt(61))
であることが確認できる。さらに、これらが３乗数であることが、Pari/GPによって確認できる。
gp > z=Mod(x,x^2-61)
%11 = Mod(x, x^2 - 61)
gp > (1249638720+159999840*z)^3
%12 = Mod(999421027517377348595712000*x + 7805727756261891959906304000, x^2 - 61)
よって、 j((1+sqrt(-427))/2)= -(1249638720+159999840*sqrt(61))^3 である。
H．Nakao さんからのコメントです。（令和３年９月２９日付け）
■j((1+sqrt(-267))/2)=A+B*sqrt(89)をQ(sqrt(89))のconjugateを使って求めることができた。
最初に、j((1+sqrt(-267))/2)とj((3+sqrt(-267))/6)は実2次体Q(sqrt(89))のconjugateである
ことが分かる。
gp > n267=bnfinit(x^2+267)
time = 2 ms.
%13 = [Mat(2), Mat([1, 1]), [;],
Mat([0.E-307 + 7.0094847991474131046951217515689871352573448977084736177713068
54054949732228399256714091322502394406279897948999068895190171932
22266764463980923414616556685537462706648017224780109502466251304
71028318542161681440336129694863500346548799489310716353050358752
5340954739255372001697754382316188970248472*I,
0.E-326 + 9.787927706753293029272847642343499336023011247604448450835542611
64310767868661762561311531193543133910424514752609906088927365533
04349867833553358893397456625878817386439631661116668015771544334
12179855965706023348673377000126313795013540796078221470809234760
55169439349776866178519869879546534952491*I, 0.E-307]),
[[23, [-2, 2]~, 1, 1, [4, -134; 2, 2]], [3, [1, -1]~, 2, 1, [1, 67; -1, 2]], [23, [4, 2]~, 1, 1,
[-2, -134; 2, -4]]], 0, [x^2 + 267, [0, 1], -267, 2, [
Mat([1, -0.5000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000 +
8.17006731918409592733529033786524173802592205003422845742229918814433
887748956416060475300409951336463071607689014115471030885996823491728
069192921589384801004700150702063917374324454445220493583236784313505
103253298918307518081739615585452560996587628596153470254013871277571
570127978530819417645063*I]),
[1, 7.670067319184095927335290337865241738025922050034228457422299188144338
8774895641606047530040995133646307160768901411547103088599682349172806
9192921589384801004700150702063917374324454445220493583236784313505103
2532989183075180817396155854525609965876285961534702540138712775715701
27978530819417645063; 1,
-8.67006731918409592733529033786524173802592205003422845742229918814433
8877489564160604753004099513364630716076890141154710308859968234917280
6919292158938480100470015070206391737432445444522049358323678431350510
3253298918307518081739615585452560996587628596153470254013871277571570
127978530819417645063],
[16, 123; 16, -139], [2, -1; -1, -133], [267, 134; 0, 1], [133, -1; -1, -2],
[267, [134, -67; 1, 133]], [3, 89]],
[0.E-327 + 16.340134638368191854670580675730483476051844100068456914844598376
28867775497912832120950600819902672926143215378028230942061771993
64698345613838584317876960200940030140412783474864890889044098716
64735686270102065065978366150361634792311709051219931752571923069
4050802774255514314025595706163883529013*I],
[2, x - 1], [1, 1; 0, 2], [1, 0, 0, -67; 0, 1, 1, -1]], [[2, [2], [[3, 2; 0, 1]]], 1, 1, [2, -1], []],
[Mat(-1),
Mat(4.234106504597259382201998068732721823089870872663185193733329711120487
5392034072650720168469667269031421672426372880026382267328609668576256
2829882873155274045390459549810017578635965535049147445315463850184987
5281460712515154233202124985304854850166130472645130933316603131631843
10996461866670731162
- 3.50474239957370655234756087578449356762867244885423680888565342702747
4866114199628357045661251197203139948974499534447595085966111333822319
9046170730827834276873135332400861239005475123312565235514159271080840
7201680648474317501732743997446553581765251793762670477369627686000848
877191158094485124236*I),
Mat(-8.46821300919451876440399613746544364617974174532637038746665942224097
5078406814530144033693933453806284334485274576005276453465721933715251
2565976574631054809078091909962003515727193107009829489063092770036997
5056292142503030846640424997060970970033226094529026186663320626326368
621992923733341462324
- 4.450147717014402766 E-308*I), [Mat([[-2/69, -1/69]~, 1])]~, Mat(-1),
Mat(0)], [0, 0, 0]]
gp > n267.clgp
%14 = [2, [2], [[3, 2; 0, 1]]]
gp > r1=ellj((1+sqrt(-267))/2)
%15 = -19683091854079461000701.9927370407698390265871939038084540794286931669
8929256292401499201516449074703762284652944971775218405783691971772854
7836871444742379795755356111092951811078822449146467444712208268016022
7723216405897450321528729840244882852255648819273845177974336878947301
765640143735582660046
gp > r2=ellj((3+sqrt(-267))/6)
%16 = -26999298.0072629592301609734128060961915459205713068330107074370759850
0798483550925296237715347055028224781594216308028227145216312855525762
0204244643888907048188921177550853532555287791731983977227678359410254
9678471270159755117147744351180726154822025663121052698234359856264417
339953846637174722528
gp > r1+r2
%17 = -19683091854079488000000.0000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000
gp > (r1-r2)/sqrt(89)
%18 = -2086403563729465344000.00000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000
gp > [19683091854079488000000,2086403563729465344000]/2
%19 = [9841545927039744000000, 1043201781864732672000]
gp > r1+(9841545927039744000000+1043201781864732672000*sqrt(89))
%20 = 9.456852790191458878 E-285
gp > r2+(9841545927039744000000-1043201781864732672000*sqrt(89))
%21 = 2.697336878242112232 E-287
gp > z=Mod(x,x^2-89)
%22 = Mod(x, x^2 - 89)
gp > (150000+12720*z)^3*(500+53*z)^2
%23 = Mod(1043201781864732672000*x + 9841545927039744000000, x^2 - 89)
これにより、
j((1+sqrt(-267))/2) = -(150000+12720*sqrt(89))^3*(500+53*sqrt(89))^2
j((3+sqrt(-267))/6) = -(150000-12720*sqrt(89))^3*(500-53*sqrt(89))^2
であることが確認できる。
■同様の(実2次体のconjugateを使う)方法で、
j((1+sqrt(-15))/2) = -(15+12*sqrt(5))^3*((1+sqrt(5))/2)^2
j((1+sqrt(-51))/2) = -(240+48*sqrt(17))^3*(4+sqrt(17))^2
j((1+sqrt(-123))/2) = -(3840+480*sqrt(41))^3*(32+5*sqrt(41))^2
を示すことができた。
■j((1+sqrt(-15))/2)とj((1+sqrt(-15))/4)は、Q(sqrt(5))のconjugateであるので、Pari/GPに
より、
gp> default(realprecision,300)
gp> r1=ellj((1+sqrt(-15))/2)
%2 = -191657.8328625472074713534448212730499333579879173965773293379043095388
73500111516162571283524266690868001427804042895756224129878981913058449
55451932563463284949929450181782443197036755726020723332158639866457790
89206846973153237358486113830998341183514892736086884995114432076023346
43336161425738873
gp> r2=ellj((1+sqrt(-15))/4)
%3 = 632.83286254720747135344482127304993335798791739657732933790430953887350
01115161625712835242666908680014278040428957562241298789819130584495545
19325634632849499294501817824431970367557260207233321586398664577908920
68469731532373584861138309983411835148927360868849951144320760233464333
6161425738872902 + 1.8031616113096322776 E-305*I
gp> (r1+r2)
%4 = -191025.0000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000 + 1.8031616113096322776 E-305*I
gp> (r1-r2)/sqrt(5)
%5 = -85995.00000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000 - 8.063983874612782724 E-306*I
gp > z=Mod(x,x^2-5)
%6 = Mod(x, x^2 - 5)
gp > (191025+85995*z)/2-(15+12*z)^3*((1+z)/2)^2
%7 = Mod(0, x^2 - 5)
となる。よって、
j((1+sqrt(-15))/2) = -(191025+85995*sqrt(5)) = -(15+12*sqrt(5))^3*((1+sqrt(5))/2)^2
j((1+sqrt(-15))/4) = -(191025-85995*sqrt(5)) = -(15-12*sqrt(5))^3*((1-sqrt(5))/2)^2
であることが確認できた。
■j((1+sqrt(-51))/2)とj((3+sqrt(-51))/6)は、Q(sqrt(17))のconjugateであるので、Pari/GP
により、
gp > r1=ellj((1+sqrt(-51))/2)
%8 = -5541100437.888321325135060704019494027794220348706788063248239144687097
65294235442246154734650488578146178178664103918944367996392125195893432
28970287352692060236637016285109701479555817119130427630110046287398462
81017079752257355926562047414354758340061253810034050371301005567461712
59468588083007411
gp > r2=ellj((3+sqrt(-51))/6)
%9 = -1130.111678674864939295980505972205779651293211936751760855312902347057
64557753845265349511421853821821335896081055632003607874804106567710297
12647307939763362983714890298520444182880869572369889953712601537189829
20247742644073437952585645241659938746189965949628698994432538287405314
11916992589040578
gp > (r1+r2)
%10 = -5541101568.00000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000
gp > (r1-r2)/sqrt(17)
%11 = -1343913984.00000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000
gp > [5541101568,1343913984]/2
%12 = [2770550784, 671956992]
gp > z=Mod(x,x^2-17)
%13 = Mod(x, x^2 - 17)
gp > (2770550784+671956992*z)-(240+48*z)^3*(4+z)^2
%14 = Mod(0, x^2 - 17)
となる。よって、
j((1+sqrt(-51))/2) = -(2770550784+671956992*sqrt(17)) = -(240+48*sqrt(17))^3*(4+sqrt(17))^2
j((3+sqrt(-51))/6) = -(2770550784-671956992*sqrt(17)) = -(240-48*sqrt(17))^3*(4-sqrt(17))^2
であることが確認できた。
■j((1+sqrt(-123))/2)とj((3+sqrt(-123))/2)は、Q(sqrt(41))のconjugateであるので、Pari/GP
により、
gp > r1=ellj((1+sqrt(-123))/2)
%15 = -1354146840466108.22923438159681338028061108857035687651173228199042664
5213030810974102141376803789478946237496744167196426861596333374548993
7043390390141107423184783189517095547808811451506714285649987029773447
8323778829307331464605466628189870127805033683468577620715186727621277
236658389622005025744
gp > r2=ellj((3+sqrt(-123))/6)
%16 = -109891.770765618403186619719388911429643123488267718009573354786969189
0258978586231962105210537625032558328035731384036666254510062956609609
8588925768152168104829044521911885484932857143500129702265521676221170
6926685353945333718101298721949663165314223792848132723787227633416103
779949742556542617397
gp > (r1+r2)
%17 = -1354146840576000.00000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000
gp > (r1-r2)/sqrt(41)
%18 = -211482206208000.000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000
gp > [1354146840576000,211482206208000]/2
%19 = [677073420288000, 105741103104000]
gp > z=Mod(x,x^2-41)
%20 = Mod(x, x^2 - 41)
gp > (677073420288000+105741103104000*z)-(3840+480*z)^3*(32+5*z)^2
%22 = Mod(0, x^2 - 41)
となる。よって、
j((1+sqrt(-123))/2) = -(677073420288000+105741103104000*sqrt(41))
= -(3840+480*sqrt(41))^3*(32+5*sqrt(41))^2
j((3+sqrt(-123))/6) = -(677073420288000-105741103104000*sqrt(41))
= -(3840-480*sqrt(41))^3*(32-5*sqrt(41))^2
であることが確認できた。
H．Nakao さんからのコメントです。（令和３年９月３０日付け）
■j(sqrt(-6))=A+B*sqrt(2)の連分数展開が循環連分数であることを確認するために必要
な小数点演算の精度は、小数点以下150000桁程度であることが分かった。
(以前に10000桁の精度でうまくいかなくてこの方法を断念したが、もう少し桁数を増やせば
*原理的には*計算できたと思われる。)
最初に、m*sqrt(2)の連分数展開の循環節の長さを求める。
Pari/GPにより、
gp > cf(n)=contfrac(n*sqrt(2))
%1 = (n)->contfrac(n*sqrt(2))
gp > findc(r)=local(s,w,l);l=length(r);w=0;s=r[1]*2;for(i=2,l,if(r[i]==s,w=i;break));w
%2 = (r)->local(s,w,l);l=length(r);w=0;s=r[1]*2;for(i=2,l,if(r[i]==s,w=i;break));w
gp > cyc(w)=if(w>0,w-1,0)
%3 = (w)->if(w>0,w-1,0)
gp > pr(n1,n2)=local(w);for(i=n1,n2,w=findc(cf(i));print("cycle of ",i,"*sqrt(2)=",cyc(w),";
[(l/m)*1000]=",floor((cyc(w)/i)*1000)))
%4 = (n1,n2)->local(w);for(i=n1,n2,w=findc(cf(i));print("cycle of ",i,"*sqrt(2)=",cyc(w),";
[(l/m)*1000]=",floor((cyc(w)/i)*1000)))
gp > default(realprecision,10000)
gp > pr(1,1024)
cycle of 1*sqrt(2)=1;[(l/m)*1000]=1000
cycle of 2*sqrt(2)=2;[(l/m)*1000]=1000
cycle of 3*sqrt(2)=2;[(l/m)*1000]=666
cycle of 4*sqrt(2)=4;[(l/m)*1000]=1000
cycle of 5*sqrt(2)=1;[(l/m)*1000]=200
cycle of 6*sqrt(2)=2;[(l/m)*1000]=333
cycle of 7*sqrt(2)=4;[(l/m)*1000]=571
cycle of 8*sqrt(2)=4;[(l/m)*1000]=500
cycle of 9*sqrt(2)=10;[(l/m)*1000]=1111
cycle of 10*sqrt(2)=2;[(l/m)*1000]=200
cycle of 11*sqrt(2)=10;[(l/m)*1000]=909
cycle of 12*sqrt(2)=2;[(l/m)*1000]=166
cycle of 13*sqrt(2)=5;[(l/m)*1000]=384
cycle of 14*sqrt(2)=4;[(l/m)*1000]=285
cycle of 15*sqrt(2)=8;[(l/m)*1000]=533
cycle of 16*sqrt(2)=12;[(l/m)*1000]=750
cycle of 17*sqrt(2)=2;[(l/m)*1000]=117
cycle of 18*sqrt(2)=6;[(l/m)*1000]=333
cycle of 19*sqrt(2)=14;[(l/m)*1000]=736
cycle of 20*sqrt(2)=8;[(l/m)*1000]=400
cycle of 21*sqrt(2)=8;[(l/m)*1000]=380
cycle of 22*sqrt(2)=6;[(l/m)*1000]=272
cycle of 23*sqrt(2)=16;[(l/m)*1000]=695
cycle of 24*sqrt(2)=4;[(l/m)*1000]=166
cycle of 25*sqrt(2)=13;[(l/m)*1000]=520
cycle of 26*sqrt(2)=10;[(l/m)*1000]=384
cycle of 27*sqrt(2)=26;[(l/m)*1000]=962
cycle of 28*sqrt(2)=8;[(l/m)*1000]=285
cycle of 29*sqrt(2)=1;[(l/m)*1000]=34
cycle of 30*sqrt(2)=8;[(l/m)*1000]=266
cycle of 31*sqrt(2)=24;[(l/m)*1000]=774
cycle of 32*sqrt(2)=24;[(l/m)*1000]=750
cycle of 33*sqrt(2)=6;[(l/m)*1000]=181
cycle of 34*sqrt(2)=2;[(l/m)*1000]=58
cycle of 35*sqrt(2)=2;[(l/m)*1000]=57
cycle of 36*sqrt(2)=6;[(l/m)*1000]=166
cycle of 37*sqrt(2)=15;[(l/m)*1000]=405
cycle of 38*sqrt(2)=18;[(l/m)*1000]=473
cycle of 39*sqrt(2)=16;[(l/m)*1000]=410
cycle of 40*sqrt(2)=20;[(l/m)*1000]=500
cycle of 41*sqrt(2)=4;[(l/m)*1000]=97
cycle of 42*sqrt(2)=8;[(l/m)*1000]=190
cycle of 43*sqrt(2)=30;[(l/m)*1000]=697
cycle of 44*sqrt(2)=6;[(l/m)*1000]=136
cycle of 45*sqrt(2)=10;[(l/m)*1000]=222
cycle of 46*sqrt(2)=16;[(l/m)*1000]=347
cycle of 47*sqrt(2)=28;[(l/m)*1000]=595
cycle of 48*sqrt(2)=8;[(l/m)*1000]=166
cycle of 49*sqrt(2)=36;[(l/m)*1000]=734
cycle of 50*sqrt(2)=22;[(l/m)*1000]=440
cycle of 51*sqrt(2)=2;[(l/m)*1000]=39
cycle of 52*sqrt(2)=20;[(l/m)*1000]=384
cycle of 53*sqrt(2)=19;[(l/m)*1000]=358
cycle of 54*sqrt(2)=30;[(l/m)*1000]=555
cycle of 55*sqrt(2)=10;[(l/m)*1000]=181
cycle of 56*sqrt(2)=12;[(l/m)*1000]=214
cycle of 57*sqrt(2)=14;[(l/m)*1000]=245
cycle of 58*sqrt(2)=2;[(l/m)*1000]=34
cycle of 59*sqrt(2)=14;[(l/m)*1000]=237
cycle of 60*sqrt(2)=8;[(l/m)*1000]=133
cycle of 61*sqrt(2)=25;[(l/m)*1000]=409
cycle of 62*sqrt(2)=20;[(l/m)*1000]=322
cycle of 63*sqrt(2)=4;[(l/m)*1000]=63
cycle of 64*sqrt(2)=48;[(l/m)*1000]=750
cycle of 65*sqrt(2)=13;[(l/m)*1000]=200
cycle of 66*sqrt(2)=6;[(l/m)*1000]=90
cycle of 67*sqrt(2)=54;[(l/m)*1000]=805
cycle of 68*sqrt(2)=2;[(l/m)*1000]=29
cycle of 69*sqrt(2)=32;[(l/m)*1000]=463
cycle of 70*sqrt(2)=2;[(l/m)*1000]=28
cycle of 71*sqrt(2)=52;[(l/m)*1000]=732
cycle of 72*sqrt(2)=16;[(l/m)*1000]=222
cycle of 73*sqrt(2)=30;[(l/m)*1000]=410
cycle of 74*sqrt(2)=26;[(l/m)*1000]=351
cycle of 75*sqrt(2)=40;[(l/m)*1000]=533
cycle of 76*sqrt(2)=10;[(l/m)*1000]=131
cycle of 77*sqrt(2)=6;[(l/m)*1000]=77
cycle of 78*sqrt(2)=16;[(l/m)*1000]=205
cycle of 79*sqrt(2)=20;[(l/m)*1000]=253
cycle of 80*sqrt(2)=36;[(l/m)*1000]=450
cycle of 81*sqrt(2)=78;[(l/m)*1000]=962
cycle of 82*sqrt(2)=4;[(l/m)*1000]=48
cycle of 83*sqrt(2)=58;[(l/m)*1000]=698
cycle of 84*sqrt(2)=8;[(l/m)*1000]=95
cycle of 85*sqrt(2)=14;[(l/m)*1000]=164
cycle of 86*sqrt(2)=30;[(l/m)*1000]=348
cycle of 87*sqrt(2)=8;[(l/m)*1000]=91
cycle of 88*sqrt(2)=20;[(l/m)*1000]=227
cycle of 89*sqrt(2)=34;[(l/m)*1000]=382
cycle of 90*sqrt(2)=8;[(l/m)*1000]=88
cycle of 91*sqrt(2)=34;[(l/m)*1000]=373
cycle of 92*sqrt(2)=20;[(l/m)*1000]=217
cycle of 93*sqrt(2)=52;[(l/m)*1000]=559
cycle of 94*sqrt(2)=32;[(l/m)*1000]=340
cycle of 95*sqrt(2)=46;[(l/m)*1000]=484
cycle of 96*sqrt(2)=20;[(l/m)*1000]=208
cycle of 97*sqrt(2)=38;[(l/m)*1000]=391
cycle of 98*sqrt(2)=32;[(l/m)*1000]=326
cycle of 99*sqrt(2)=2;[(l/m)*1000]=20
cycle of 100*sqrt(2)=40;[(l/m)*1000]=400
cycle of 101*sqrt(2)=31;[(l/m)*1000]=306
cycle of 102*sqrt(2)=2;[(l/m)*1000]=19
cycle of 103*sqrt(2)=20;[(l/m)*1000]=194
cycle of 104*sqrt(2)=36;[(l/m)*1000]=346
cycle of 105*sqrt(2)=4;[(l/m)*1000]=38
cycle of 106*sqrt(2)=34;[(l/m)*1000]=320
cycle of 107*sqrt(2)=74;[(l/m)*1000]=691
cycle of 108*sqrt(2)=26;[(l/m)*1000]=240
cycle of 109*sqrt(2)=39;[(l/m)*1000]=357
cycle of 110*sqrt(2)=8;[(l/m)*1000]=72
cycle of 111*sqrt(2)=52;[(l/m)*1000]=468
cycle of 112*sqrt(2)=32;[(l/m)*1000]=285
cycle of 113*sqrt(2)=14;[(l/m)*1000]=123
cycle of 114*sqrt(2)=10;[(l/m)*1000]=87
cycle of 115*sqrt(2)=52;[(l/m)*1000]=452
cycle of 116*sqrt(2)=8;[(l/m)*1000]=68
cycle of 117*sqrt(2)=64;[(l/m)*1000]=547
cycle of 118*sqrt(2)=10;[(l/m)*1000]=84
cycle of 119*sqrt(2)=18;[(l/m)*1000]=151
cycle of 120*sqrt(2)=16;[(l/m)*1000]=133
cycle of 121*sqrt(2)=102;[(l/m)*1000]=842
cycle of 122*sqrt(2)=58;[(l/m)*1000]=475
cycle of 123*sqrt(2)=8;[(l/m)*1000]=65
cycle of 124*sqrt(2)=48;[(l/m)*1000]=387
cycle of 125*sqrt(2)=51;[(l/m)*1000]=408
cycle of 126*sqrt(2)=4;[(l/m)*1000]=31
cycle of 127*sqrt(2)=96;[(l/m)*1000]=755
cycle of 128*sqrt(2)=96;[(l/m)*1000]=750
cycle of 129*sqrt(2)=34;[(l/m)*1000]=263
cycle of 130*sqrt(2)=26;[(l/m)*1000]=200
cycle of 131*sqrt(2)=98;[(l/m)*1000]=748
cycle of 132*sqrt(2)=6;[(l/m)*1000]=45
cycle of 133*sqrt(2)=42;[(l/m)*1000]=315
cycle of 134*sqrt(2)=50;[(l/m)*1000]=373
cycle of 135*sqrt(2)=22;[(l/m)*1000]=162
cycle of 136*sqrt(2)=2;[(l/m)*1000]=14
cycle of 137*sqrt(2)=9;[(l/m)*1000]=65
cycle of 138*sqrt(2)=28;[(l/m)*1000]=202
cycle of 139*sqrt(2)=102;[(l/m)*1000]=733
cycle of 140*sqrt(2)=4;[(l/m)*1000]=28
cycle of 141*sqrt(2)=60;[(l/m)*1000]=425
cycle of 142*sqrt(2)=48;[(l/m)*1000]=338
cycle of 143*sqrt(2)=68;[(l/m)*1000]=475
cycle of 144*sqrt(2)=28;[(l/m)*1000]=194
cycle of 145*sqrt(2)=5;[(l/m)*1000]=34
cycle of 146*sqrt(2)=22;[(l/m)*1000]=150
cycle of 147*sqrt(2)=56;[(l/m)*1000]=380
cycle of 148*sqrt(2)=44;[(l/m)*1000]=297
cycle of 149*sqrt(2)=53;[(l/m)*1000]=355
cycle of 150*sqrt(2)=44;[(l/m)*1000]=293
cycle of 151*sqrt(2)=116;[(l/m)*1000]=768
cycle of 152*sqrt(2)=20;[(l/m)*1000]=131
cycle of 153*sqrt(2)=14;[(l/m)*1000]=91
cycle of 154*sqrt(2)=8;[(l/m)*1000]=51
cycle of 155*sqrt(2)=20;[(l/m)*1000]=129
cycle of 156*sqrt(2)=28;[(l/m)*1000]=179
cycle of 157*sqrt(2)=61;[(l/m)*1000]=388
cycle of 158*sqrt(2)=12;[(l/m)*1000]=75
cycle of 159*sqrt(2)=68;[(l/m)*1000]=427
cycle of 160*sqrt(2)=68;[(l/m)*1000]=425
cycle of 161*sqrt(2)=48;[(l/m)*1000]=298
cycle of 162*sqrt(2)=86;[(l/m)*1000]=530
cycle of 163*sqrt(2)=122;[(l/m)*1000]=748
cycle of 164*sqrt(2)=8;[(l/m)*1000]=48
cycle of 165*sqrt(2)=6;[(l/m)*1000]=36
cycle of 166*sqrt(2)=62;[(l/m)*1000]=373
cycle of 167*sqrt(2)=120;[(l/m)*1000]=718
cycle of 168*sqrt(2)=16;[(l/m)*1000]=95
cycle of 169*sqrt(2)=1;[(l/m)*1000]=5
cycle of 170*sqrt(2)=14;[(l/m)*1000]=82
cycle of 171*sqrt(2)=34;[(l/m)*1000]=198
cycle of 172*sqrt(2)=30;[(l/m)*1000]=174
cycle of 173*sqrt(2)=59;[(l/m)*1000]=341
cycle of 174*sqrt(2)=8;[(l/m)*1000]=45
cycle of 175*sqrt(2)=14;[(l/m)*1000]=80
cycle of 176*sqrt(2)=32;[(l/m)*1000]=181
cycle of 177*sqrt(2)=6;[(l/m)*1000]=33
cycle of 178*sqrt(2)=34;[(l/m)*1000]=191
cycle of 179*sqrt(2)=18;[(l/m)*1000]=100
cycle of 180*sqrt(2)=8;[(l/m)*1000]=44
cycle of 181*sqrt(2)=77;[(l/m)*1000]=425
cycle of 182*sqrt(2)=34;[(l/m)*1000]=186
cycle of 183*sqrt(2)=88;[(l/m)*1000]=480
cycle of 184*sqrt(2)=56;[(l/m)*1000]=304
cycle of 185*sqrt(2)=37;[(l/m)*1000]=200
cycle of 186*sqrt(2)=36;[(l/m)*1000]=193
cycle of 187*sqrt(2)=14;[(l/m)*1000]=74
cycle of 188*sqrt(2)=64;[(l/m)*1000]=340
cycle of 189*sqrt(2)=18;[(l/m)*1000]=95
cycle of 190*sqrt(2)=46;[(l/m)*1000]=242
cycle of 191*sqrt(2)=132;[(l/m)*1000]=691
cycle of 192*sqrt(2)=52;[(l/m)*1000]=270
cycle of 193*sqrt(2)=62;[(l/m)*1000]=321
cycle of 194*sqrt(2)=38;[(l/m)*1000]=195
cycle of 195*sqrt(2)=60;[(l/m)*1000]=307
cycle of 196*sqrt(2)=60;[(l/m)*1000]=306
cycle of 197*sqrt(2)=5;[(l/m)*1000]=25
cycle of 198*sqrt(2)=2;[(l/m)*1000]=10
cycle of 199*sqrt(2)=8;[(l/m)*1000]=40
cycle of 200*sqrt(2)=88;[(l/m)*1000]=440
cycle of 201*sqrt(2)=54;[(l/m)*1000]=268
cycle of 202*sqrt(2)=70;[(l/m)*1000]=346
cycle of 203*sqrt(2)=18;[(l/m)*1000]=88
cycle of 204*sqrt(2)=2;[(l/m)*1000]=9
cycle of 205*sqrt(2)=20;[(l/m)*1000]=97
cycle of 206*sqrt(2)=16;[(l/m)*1000]=77
cycle of 207*sqrt(2)=96;[(l/m)*1000]=463
cycle of 208*sqrt(2)=84;[(l/m)*1000]=403
cycle of 209*sqrt(2)=38;[(l/m)*1000]=181
cycle of 210*sqrt(2)=4;[(l/m)*1000]=19
cycle of 211*sqrt(2)=158;[(l/m)*1000]=748
cycle of 212*sqrt(2)=72;[(l/m)*1000]=339
cycle of 213*sqrt(2)=100;[(l/m)*1000]=469
cycle of 214*sqrt(2)=86;[(l/m)*1000]=401
cycle of 215*sqrt(2)=94;[(l/m)*1000]=437
cycle of 216*sqrt(2)=52;[(l/m)*1000]=240
cycle of 217*sqrt(2)=20;[(l/m)*1000]=92
cycle of 218*sqrt(2)=86;[(l/m)*1000]=394
cycle of 219*sqrt(2)=18;[(l/m)*1000]=82
cycle of 220*sqrt(2)=4;[(l/m)*1000]=18
cycle of 221*sqrt(2)=52;[(l/m)*1000]=235
cycle of 222*sqrt(2)=56;[(l/m)*1000]=252
cycle of 223*sqrt(2)=52;[(l/m)*1000]=233
cycle of 224*sqrt(2)=72;[(l/m)*1000]=321
cycle of 225*sqrt(2)=32;[(l/m)*1000]=142
cycle of 226*sqrt(2)=22;[(l/m)*1000]=97
cycle of 227*sqrt(2)=58;[(l/m)*1000]=255
cycle of 228*sqrt(2)=14;[(l/m)*1000]=61
cycle of 229*sqrt(2)=15;[(l/m)*1000]=65
cycle of 230*sqrt(2)=48;[(l/m)*1000]=208
cycle of 231*sqrt(2)=6;[(l/m)*1000]=25
cycle of 232*sqrt(2)=20;[(l/m)*1000]=86
cycle of 233*sqrt(2)=74;[(l/m)*1000]=317
cycle of 234*sqrt(2)=64;[(l/m)*1000]=273
cycle of 235*sqrt(2)=104;[(l/m)*1000]=442
cycle of 236*sqrt(2)=10;[(l/m)*1000]=42
cycle of 237*sqrt(2)=28;[(l/m)*1000]=118
cycle of 238*sqrt(2)=16;[(l/m)*1000]=67
cycle of 239*sqrt(2)=4;[(l/m)*1000]=16
cycle of 240*sqrt(2)=24;[(l/m)*1000]=100
cycle of 241*sqrt(2)=30;[(l/m)*1000]=124
cycle of 242*sqrt(2)=102;[(l/m)*1000]=421
cycle of 243*sqrt(2)=234;[(l/m)*1000]=962
cycle of 244*sqrt(2)=92;[(l/m)*1000]=377
cycle of 245*sqrt(2)=28;[(l/m)*1000]=114
cycle of 246*sqrt(2)=12;[(l/m)*1000]=48
cycle of 247*sqrt(2)=126;[(l/m)*1000]=510
cycle of 248*sqrt(2)=92;[(l/m)*1000]=370
cycle of 249*sqrt(2)=62;[(l/m)*1000]=248
cycle of 250*sqrt(2)=106;[(l/m)*1000]=424
cycle of 251*sqrt(2)=50;[(l/m)*1000]=199
cycle of 252*sqrt(2)=8;[(l/m)*1000]=31
cycle of 253*sqrt(2)=104;[(l/m)*1000]=411
cycle of 254*sqrt(2)=84;[(l/m)*1000]=330
cycle of 255*sqrt(2)=16;[(l/m)*1000]=62
cycle of 256*sqrt(2)=196;[(l/m)*1000]=765
cycle of 257*sqrt(2)=46;[(l/m)*1000]=178
cycle of 258*sqrt(2)=34;[(l/m)*1000]=131
cycle of 259*sqrt(2)=78;[(l/m)*1000]=301
cycle of 260*sqrt(2)=60;[(l/m)*1000]=230
cycle of 261*sqrt(2)=36;[(l/m)*1000]=137
cycle of 262*sqrt(2)=82;[(l/m)*1000]=312
cycle of 263*sqrt(2)=200;[(l/m)*1000]=760
cycle of 264*sqrt(2)=12;[(l/m)*1000]=45
cycle of 265*sqrt(2)=21;[(l/m)*1000]=79
cycle of 266*sqrt(2)=52;[(l/m)*1000]=195
cycle of 267*sqrt(2)=34;[(l/m)*1000]=127
cycle of 268*sqrt(2)=42;[(l/m)*1000]=156
cycle of 269*sqrt(2)=9;[(l/m)*1000]=33
cycle of 270*sqrt(2)=18;[(l/m)*1000]=66
cycle of 271*sqrt(2)=200;[(l/m)*1000]=738
cycle of 272*sqrt(2)=8;[(l/m)*1000]=29
cycle of 273*sqrt(2)=60;[(l/m)*1000]=219
cycle of 274*sqrt(2)=14;[(l/m)*1000]=51
cycle of 275*sqrt(2)=36;[(l/m)*1000]=130
cycle of 276*sqrt(2)=32;[(l/m)*1000]=115
cycle of 277*sqrt(2)=101;[(l/m)*1000]=364
cycle of 278*sqrt(2)=106;[(l/m)*1000]=381
cycle of 279*sqrt(2)=48;[(l/m)*1000]=172
cycle of 280*sqrt(2)=8;[(l/m)*1000]=28
cycle of 281*sqrt(2)=86;[(l/m)*1000]=306
cycle of 282*sqrt(2)=64;[(l/m)*1000]=226
cycle of 283*sqrt(2)=210;[(l/m)*1000]=742
cycle of 284*sqrt(2)=88;[(l/m)*1000]=309
cycle of 285*sqrt(2)=38;[(l/m)*1000]=133
cycle of 286*sqrt(2)=68;[(l/m)*1000]=237
cycle of 287*sqrt(2)=16;[(l/m)*1000]=55
cycle of 288*sqrt(2)=56;[(l/m)*1000]=194
cycle of 289*sqrt(2)=118;[(l/m)*1000]=408
cycle of 290*sqrt(2)=10;[(l/m)*1000]=34
cycle of 291*sqrt(2)=30;[(l/m)*1000]=103
cycle of 292*sqrt(2)=18;[(l/m)*1000]=61
cycle of 293*sqrt(2)=37;[(l/m)*1000]=126
cycle of 294*sqrt(2)=64;[(l/m)*1000]=217
cycle of 295*sqrt(2)=34;[(l/m)*1000]=115
cycle of 296*sqrt(2)=108;[(l/m)*1000]=364
cycle of 297*sqrt(2)=10;[(l/m)*1000]=33
cycle of 298*sqrt(2)=98;[(l/m)*1000]=328
cycle of 299*sqrt(2)=128;[(l/m)*1000]=428
cycle of 300*sqrt(2)=40;[(l/m)*1000]=133
cycle of 301*sqrt(2)=98;[(l/m)*1000]=325
cycle of 302*sqrt(2)=108;[(l/m)*1000]=357
cycle of 303*sqrt(2)=152;[(l/m)*1000]=501
cycle of 304*sqrt(2)=56;[(l/m)*1000]=184
cycle of 305*sqrt(2)=63;[(l/m)*1000]=206
cycle of 306*sqrt(2)=12;[(l/m)*1000]=39
cycle of 307*sqrt(2)=226;[(l/m)*1000]=736
cycle of 308*sqrt(2)=8;[(l/m)*1000]=25
cycle of 309*sqrt(2)=44;[(l/m)*1000]=142
cycle of 310*sqrt(2)=16;[(l/m)*1000]=51
cycle of 311*sqrt(2)=232;[(l/m)*1000]=745
cycle of 312*sqrt(2)=40;[(l/m)*1000]=128
cycle of 313*sqrt(2)=64;[(l/m)*1000]=204
cycle of 314*sqrt(2)=110;[(l/m)*1000]=350
cycle of 315*sqrt(2)=4;[(l/m)*1000]=12
cycle of 316*sqrt(2)=32;[(l/m)*1000]=101
cycle of 317*sqrt(2)=109;[(l/m)*1000]=343
cycle of 318*sqrt(2)=72;[(l/m)*1000]=226
cycle of 319*sqrt(2)=44;[(l/m)*1000]=137
cycle of 320*sqrt(2)=140;[(l/m)*1000]=437
cycle of 321*sqrt(2)=94;[(l/m)*1000]=292
cycle of 322*sqrt(2)=52;[(l/m)*1000]=161
cycle of 323*sqrt(2)=40;[(l/m)*1000]=123
cycle of 324*sqrt(2)=74;[(l/m)*1000]=228
cycle of 325*sqrt(2)=81;[(l/m)*1000]=249
cycle of 326*sqrt(2)=118;[(l/m)*1000]=361
cycle of 327*sqrt(2)=160;[(l/m)*1000]=489
cycle of 328*sqrt(2)=20;[(l/m)*1000]=60
cycle of 329*sqrt(2)=100;[(l/m)*1000]=303
cycle of 330*sqrt(2)=6;[(l/m)*1000]=18
cycle of 331*sqrt(2)=250;[(l/m)*1000]=755
cycle of 332*sqrt(2)=62;[(l/m)*1000]=186
cycle of 333*sqrt(2)=180;[(l/m)*1000]=540
cycle of 334*sqrt(2)=124;[(l/m)*1000]=371
cycle of 335*sqrt(2)=146;[(l/m)*1000]=435
cycle of 336*sqrt(2)=24;[(l/m)*1000]=71
cycle of 337*sqrt(2)=18;[(l/m)*1000]=53
cycle of 338*sqrt(2)=2;[(l/m)*1000]=5
cycle of 339*sqrt(2)=18;[(l/m)*1000]=53
cycle of 340*sqrt(2)=16;[(l/m)*1000]=47
cycle of 341*sqrt(2)=36;[(l/m)*1000]=105
cycle of 342*sqrt(2)=34;[(l/m)*1000]=99
cycle of 343*sqrt(2)=200;[(l/m)*1000]=583
cycle of 344*sqrt(2)=56;[(l/m)*1000]=162
cycle of 345*sqrt(2)=108;[(l/m)*1000]=313
cycle of 346*sqrt(2)=114;[(l/m)*1000]=329
cycle of 347*sqrt(2)=266;[(l/m)*1000]=766
cycle of 348*sqrt(2)=8;[(l/m)*1000]=22
cycle of 349*sqrt(2)=143;[(l/m)*1000]=409
cycle of 350*sqrt(2)=14;[(l/m)*1000]=40
cycle of 351*sqrt(2)=182;[(l/m)*1000]=518
cycle of 352*sqrt(2)=60;[(l/m)*1000]=170
cycle of 353*sqrt(2)=12;[(l/m)*1000]=33
cycle of 354*sqrt(2)=14;[(l/m)*1000]=39
cycle of 355*sqrt(2)=152;[(l/m)*1000]=428
cycle of 356*sqrt(2)=26;[(l/m)*1000]=73
cycle of 357*sqrt(2)=16;[(l/m)*1000]=44
cycle of 358*sqrt(2)=14;[(l/m)*1000]=39
cycle of 359*sqrt(2)=264;[(l/m)*1000]=735
cycle of 360*sqrt(2)=12;[(l/m)*1000]=33
cycle of 361*sqrt(2)=290;[(l/m)*1000]=803
cycle of 362*sqrt(2)=150;[(l/m)*1000]=414
cycle of 363*sqrt(2)=102;[(l/m)*1000]=280
cycle of 364*sqrt(2)=64;[(l/m)*1000]=175
cycle of 365*sqrt(2)=18;[(l/m)*1000]=49
cycle of 366*sqrt(2)=100;[(l/m)*1000]=273
cycle of 367*sqrt(2)=276;[(l/m)*1000]=752
cycle of 368*sqrt(2)=136;[(l/m)*1000]=369
cycle of 369*sqrt(2)=36;[(l/m)*1000]=97
cycle of 370*sqrt(2)=74;[(l/m)*1000]=200
cycle of 371*sqrt(2)=42;[(l/m)*1000]=113
cycle of 372*sqrt(2)=36;[(l/m)*1000]=96
cycle of 373*sqrt(2)=147;[(l/m)*1000]=394
cycle of 374*sqrt(2)=16;[(l/m)*1000]=42
cycle of 375*sqrt(2)=224;[(l/m)*1000]=597
cycle of 376*sqrt(2)=136;[(l/m)*1000]=361
cycle of 377*sqrt(2)=19;[(l/m)*1000]=50
cycle of 378*sqrt(2)=22;[(l/m)*1000]=58
cycle of 379*sqrt(2)=42;[(l/m)*1000]=110
cycle of 380*sqrt(2)=42;[(l/m)*1000]=110
cycle of 381*sqrt(2)=184;[(l/m)*1000]=482
cycle of 382*sqrt(2)=136;[(l/m)*1000]=356
cycle of 383*sqrt(2)=280;[(l/m)*1000]=731
cycle of 384*sqrt(2)=84;[(l/m)*1000]=218
cycle of 385*sqrt(2)=4;[(l/m)*1000]=10
cycle of 386*sqrt(2)=62;[(l/m)*1000]=160
cycle of 387*sqrt(2)=86;[(l/m)*1000]=222
cycle of 388*sqrt(2)=34;[(l/m)*1000]=87
cycle of 389*sqrt(2)=25;[(l/m)*1000]=64
cycle of 390*sqrt(2)=56;[(l/m)*1000]=143
cycle of 391*sqrt(2)=80;[(l/m)*1000]=204
cycle of 392*sqrt(2)=116;[(l/m)*1000]=295
cycle of 393*sqrt(2)=98;[(l/m)*1000]=249
cycle of 394*sqrt(2)=6;[(l/m)*1000]=15
cycle of 395*sqrt(2)=48;[(l/m)*1000]=121
cycle of 396*sqrt(2)=2;[(l/m)*1000]=5
cycle of 397*sqrt(2)=143;[(l/m)*1000]=360
cycle of 398*sqrt(2)=8;[(l/m)*1000]=20
cycle of 399*sqrt(2)=34;[(l/m)*1000]=85
cycle of 400*sqrt(2)=180;[(l/m)*1000]=450
cycle of 401*sqrt(2)=154;[(l/m)*1000]=384
cycle of 402*sqrt(2)=50;[(l/m)*1000]=124
cycle of 403*sqrt(2)=160;[(l/m)*1000]=397
cycle of 404*sqrt(2)=140;[(l/m)*1000]=346
cycle of 405*sqrt(2)=74;[(l/m)*1000]=182
cycle of 406*sqrt(2)=14;[(l/m)*1000]=34
cycle of 407*sqrt(2)=176;[(l/m)*1000]=432
cycle of 408*sqrt(2)=2;[(l/m)*1000]=4
cycle of 409*sqrt(2)=76;[(l/m)*1000]=185
cycle of 410*sqrt(2)=20;[(l/m)*1000]=48
cycle of 411*sqrt(2)=36;[(l/m)*1000]=87
cycle of 412*sqrt(2)=40;[(l/m)*1000]=97
cycle of 413*sqrt(2)=30;[(l/m)*1000]=72
cycle of 414*sqrt(2)=116;[(l/m)*1000]=280
cycle of 415*sqrt(2)=66;[(l/m)*1000]=159
cycle of 416*sqrt(2)=156;[(l/m)*1000]=375
cycle of 417*sqrt(2)=90;[(l/m)*1000]=215
cycle of 418*sqrt(2)=42;[(l/m)*1000]=100
cycle of 419*sqrt(2)=94;[(l/m)*1000]=224
cycle of 420*sqrt(2)=4;[(l/m)*1000]=9
cycle of 421*sqrt(2)=153;[(l/m)*1000]=363
cycle of 422*sqrt(2)=154;[(l/m)*1000]=364
cycle of 423*sqrt(2)=192;[(l/m)*1000]=453
cycle of 424*sqrt(2)=152;[(l/m)*1000]=358
cycle of 425*sqrt(2)=88;[(l/m)*1000]=207
cycle of 426*sqrt(2)=96;[(l/m)*1000]=225
cycle of 427*sqrt(2)=138;[(l/m)*1000]=323
cycle of 428*sqrt(2)=66;[(l/m)*1000]=154
cycle of 429*sqrt(2)=62;[(l/m)*1000]=144
cycle of 430*sqrt(2)=86;[(l/m)*1000]=200
cycle of 431*sqrt(2)=324;[(l/m)*1000]=751
cycle of 432*sqrt(2)=92;[(l/m)*1000]=212
cycle of 433*sqrt(2)=166;[(l/m)*1000]=383
cycle of 434*sqrt(2)=16;[(l/m)*1000]=36
cycle of 435*sqrt(2)=28;[(l/m)*1000]=64
cycle of 436*sqrt(2)=156;[(l/m)*1000]=357
cycle of 437*sqrt(2)=160;[(l/m)*1000]=366
cycle of 438*sqrt(2)=18;[(l/m)*1000]=41
cycle of 439*sqrt(2)=328;[(l/m)*1000]=747
cycle of 440*sqrt(2)=8;[(l/m)*1000]=18
cycle of 441*sqrt(2)=56;[(l/m)*1000]=126
cycle of 442*sqrt(2)=36;[(l/m)*1000]=81
cycle of 443*sqrt(2)=102;[(l/m)*1000]=230
cycle of 444*sqrt(2)=52;[(l/m)*1000]=117
cycle of 445*sqrt(2)=94;[(l/m)*1000]=211
cycle of 446*sqrt(2)=48;[(l/m)*1000]=107
cycle of 447*sqrt(2)=220;[(l/m)*1000]=492
cycle of 448*sqrt(2)=140;[(l/m)*1000]=312
cycle of 449*sqrt(2)=150;[(l/m)*1000]=334
cycle of 450*sqrt(2)=36;[(l/m)*1000]=80
cycle of 451*sqrt(2)=48;[(l/m)*1000]=106
cycle of 452*sqrt(2)=14;[(l/m)*1000]=30
cycle of 453*sqrt(2)=216;[(l/m)*1000]=476
cycle of 454*sqrt(2)=54;[(l/m)*1000]=118
cycle of 455*sqrt(2)=26;[(l/m)*1000]=57
cycle of 456*sqrt(2)=28;[(l/m)*1000]=61
cycle of 457*sqrt(2)=76;[(l/m)*1000]=166
cycle of 458*sqrt(2)=30;[(l/m)*1000]=65
cycle of 459*sqrt(2)=40;[(l/m)*1000]=87
cycle of 460*sqrt(2)=96;[(l/m)*1000]=208
cycle of 461*sqrt(2)=169;[(l/m)*1000]=366
cycle of 462*sqrt(2)=6;[(l/m)*1000]=12
cycle of 463*sqrt(2)=348;[(l/m)*1000]=751
cycle of 464*sqrt(2)=52;[(l/m)*1000]=112
cycle of 465*sqrt(2)=48;[(l/m)*1000]=103
cycle of 466*sqrt(2)=78;[(l/m)*1000]=167
cycle of 467*sqrt(2)=342;[(l/m)*1000]=732
cycle of 468*sqrt(2)=56;[(l/m)*1000]=119
cycle of 469*sqrt(2)=154;[(l/m)*1000]=328
cycle of 470*sqrt(2)=112;[(l/m)*1000]=238
cycle of 471*sqrt(2)=232;[(l/m)*1000]=492
cycle of 472*sqrt(2)=24;[(l/m)*1000]=50
cycle of 473*sqrt(2)=82;[(l/m)*1000]=173
cycle of 474*sqrt(2)=24;[(l/m)*1000]=50
cycle of 475*sqrt(2)=36;[(l/m)*1000]=75
cycle of 476*sqrt(2)=8;[(l/m)*1000]=16
cycle of 477*sqrt(2)=80;[(l/m)*1000]=167
cycle of 478*sqrt(2)=4;[(l/m)*1000]=8
cycle of 479*sqrt(2)=340;[(l/m)*1000]=709
cycle of 480*sqrt(2)=56;[(l/m)*1000]=116
cycle of 481*sqrt(2)=107;[(l/m)*1000]=222
cycle of 482*sqrt(2)=26;[(l/m)*1000]=53
cycle of 483*sqrt(2)=96;[(l/m)*1000]=198
cycle of 484*sqrt(2)=86;[(l/m)*1000]=177
cycle of 485*sqrt(2)=34;[(l/m)*1000]=70
cycle of 486*sqrt(2)=254;[(l/m)*1000]=522
cycle of 487*sqrt(2)=128;[(l/m)*1000]=262
cycle of 488*sqrt(2)=180;[(l/m)*1000]=368
cycle of 489*sqrt(2)=130;[(l/m)*1000]=265
cycle of 490*sqrt(2)=24;[(l/m)*1000]=48
cycle of 491*sqrt(2)=366;[(l/m)*1000]=745
cycle of 492*sqrt(2)=12;[(l/m)*1000]=24
cycle of 493*sqrt(2)=28;[(l/m)*1000]=56
cycle of 494*sqrt(2)=104;[(l/m)*1000]=210
cycle of 495*sqrt(2)=2;[(l/m)*1000]=4
cycle of 496*sqrt(2)=168;[(l/m)*1000]=338
cycle of 497*sqrt(2)=144;[(l/m)*1000]=289
cycle of 498*sqrt(2)=62;[(l/m)*1000]=124
cycle of 499*sqrt(2)=374;[(l/m)*1000]=749
cycle of 500*sqrt(2)=232;[(l/m)*1000]=464
cycle of 501*sqrt(2)=248;[(l/m)*1000]=495
cycle of 502*sqrt(2)=66;[(l/m)*1000]=131
cycle of 503*sqrt(2)=360;[(l/m)*1000]=715
cycle of 504*sqrt(2)=12;[(l/m)*1000]=23
cycle of 505*sqrt(2)=41;[(l/m)*1000]=81
cycle of 506*sqrt(2)=104;[(l/m)*1000]=205
cycle of 507*sqrt(2)=8;[(l/m)*1000]=15
cycle of 508*sqrt(2)=176;[(l/m)*1000]=346
cycle of 509*sqrt(2)=179;[(l/m)*1000]=351
cycle of 510*sqrt(2)=8;[(l/m)*1000]=15
cycle of 511*sqrt(2)=22;[(l/m)*1000]=43
cycle of 512*sqrt(2)=368;[(l/m)*1000]=718
cycle of 513*sqrt(2)=122;[(l/m)*1000]=237
cycle of 514*sqrt(2)=54;[(l/m)*1000]=105
cycle of 515*sqrt(2)=64;[(l/m)*1000]=124
cycle of 516*sqrt(2)=30;[(l/m)*1000]=58
cycle of 517*sqrt(2)=200;[(l/m)*1000]=386
cycle of 518*sqrt(2)=82;[(l/m)*1000]=158
cycle of 519*sqrt(2)=256;[(l/m)*1000]=493
cycle of 520*sqrt(2)=128;[(l/m)*1000]=246
cycle of 521*sqrt(2)=43;[(l/m)*1000]=82
cycle of 522*sqrt(2)=36;[(l/m)*1000]=68
cycle of 523*sqrt(2)=386;[(l/m)*1000]=738
cycle of 524*sqrt(2)=94;[(l/m)*1000]=179
cycle of 525*sqrt(2)=28;[(l/m)*1000]=53
cycle of 526*sqrt(2)=188;[(l/m)*1000]=357
cycle of 527*sqrt(2)=100;[(l/m)*1000]=189
cycle of 528*sqrt(2)=28;[(l/m)*1000]=53
cycle of 529*sqrt(2)=376;[(l/m)*1000]=710
cycle of 530*sqrt(2)=34;[(l/m)*1000]=64
cycle of 531*sqrt(2)=26;[(l/m)*1000]=48
cycle of 532*sqrt(2)=44;[(l/m)*1000]=82
cycle of 533*sqrt(2)=52;[(l/m)*1000]=97
cycle of 534*sqrt(2)=26;[(l/m)*1000]=48
cycle of 535*sqrt(2)=74;[(l/m)*1000]=138
cycle of 536*sqrt(2)=76;[(l/m)*1000]=141
cycle of 537*sqrt(2)=22;[(l/m)*1000]=40
cycle of 538*sqrt(2)=18;[(l/m)*1000]=33
cycle of 539*sqrt(2)=52;[(l/m)*1000]=96
cycle of 540*sqrt(2)=18;[(l/m)*1000]=33
cycle of 541*sqrt(2)=197;[(l/m)*1000]=364
cycle of 542*sqrt(2)=204;[(l/m)*1000]=376
cycle of 543*sqrt(2)=276;[(l/m)*1000]=508
cycle of 544*sqrt(2)=12;[(l/m)*1000]=22
cycle of 545*sqrt(2)=119;[(l/m)*1000]=218
cycle of 546*sqrt(2)=60;[(l/m)*1000]=109
cycle of 547*sqrt(2)=402;[(l/m)*1000]=734
cycle of 548*sqrt(2)=32;[(l/m)*1000]=58
cycle of 549*sqrt(2)=260;[(l/m)*1000]=473
cycle of 550*sqrt(2)=32;[(l/m)*1000]=58
cycle of 551*sqrt(2)=8;[(l/m)*1000]=14
cycle of 552*sqrt(2)=72;[(l/m)*1000]=130
cycle of 553*sqrt(2)=48;[(l/m)*1000]=86
cycle of 554*sqrt(2)=190;[(l/m)*1000]=342
cycle of 555*sqrt(2)=156;[(l/m)*1000]=281
cycle of 556*sqrt(2)=98;[(l/m)*1000]=176
cycle of 557*sqrt(2)=209;[(l/m)*1000]=375
cycle of 558*sqrt(2)=44;[(l/m)*1000]=78
cycle of 559*sqrt(2)=230;[(l/m)*1000]=411
cycle of 560*sqrt(2)=20;[(l/m)*1000]=35
cycle of 561*sqrt(2)=16;[(l/m)*1000]=28
cycle of 562*sqrt(2)=98;[(l/m)*1000]=174
cycle of 563*sqrt(2)=414;[(l/m)*1000]=735
cycle of 564*sqrt(2)=80;[(l/m)*1000]=141
cycle of 565*sqrt(2)=62;[(l/m)*1000]=109
cycle of 566*sqrt(2)=206;[(l/m)*1000]=363
cycle of 567*sqrt(2)=62;[(l/m)*1000]=109
cycle of 568*sqrt(2)=188;[(l/m)*1000]=330
cycle of 569*sqrt(2)=41;[(l/m)*1000]=72
cycle of 570*sqrt(2)=30;[(l/m)*1000]=52
cycle of 571*sqrt(2)=422;[(l/m)*1000]=739
cycle of 572*sqrt(2)=48;[(l/m)*1000]=83
cycle of 573*sqrt(2)=260;[(l/m)*1000]=453
cycle of 574*sqrt(2)=12;[(l/m)*1000]=20
cycle of 575*sqrt(2)=266;[(l/m)*1000]=462
cycle of 576*sqrt(2)=136;[(l/m)*1000]=236
cycle of 577*sqrt(2)=2;[(l/m)*1000]=3
cycle of 578*sqrt(2)=102;[(l/m)*1000]=176
cycle of 579*sqrt(2)=58;[(l/m)*1000]=100
cycle of 580*sqrt(2)=32;[(l/m)*1000]=55
cycle of 581*sqrt(2)=62;[(l/m)*1000]=106
cycle of 582*sqrt(2)=26;[(l/m)*1000]=44
cycle of 583*sqrt(2)=76;[(l/m)*1000]=130
cycle of 584*sqrt(2)=44;[(l/m)*1000]=75
cycle of 585*sqrt(2)=60;[(l/m)*1000]=102
cycle of 586*sqrt(2)=74;[(l/m)*1000]=126
cycle of 587*sqrt(2)=430;[(l/m)*1000]=732
cycle of 588*sqrt(2)=56;[(l/m)*1000]=95
cycle of 589*sqrt(2)=40;[(l/m)*1000]=67
cycle of 590*sqrt(2)=42;[(l/m)*1000]=71
cycle of 591*sqrt(2)=16;[(l/m)*1000]=27
cycle of 592*sqrt(2)=220;[(l/m)*1000]=371
cycle of 593*sqrt(2)=29;[(l/m)*1000]=48
cycle of 594*sqrt(2)=10;[(l/m)*1000]=16
cycle of 595*sqrt(2)=12;[(l/m)*1000]=20
cycle of 596*sqrt(2)=208;[(l/m)*1000]=348
cycle of 597*sqrt(2)=16;[(l/m)*1000]=26
cycle of 598*sqrt(2)=116;[(l/m)*1000]=193
cycle of 599*sqrt(2)=12;[(l/m)*1000]=20
cycle of 600*sqrt(2)=88;[(l/m)*1000]=146
cycle of 601*sqrt(2)=38;[(l/m)*1000]=63
cycle of 602*sqrt(2)=110;[(l/m)*1000]=182
cycle of 603*sqrt(2)=142;[(l/m)*1000]=235
cycle of 604*sqrt(2)=212;[(l/m)*1000]=350
cycle of 605*sqrt(2)=94;[(l/m)*1000]=155
cycle of 606*sqrt(2)=144;[(l/m)*1000]=237
cycle of 607*sqrt(2)=132;[(l/m)*1000]=217
cycle of 608*sqrt(2)=104;[(l/m)*1000]=171
cycle of 609*sqrt(2)=48;[(l/m)*1000]=78
cycle of 610*sqrt(2)=122;[(l/m)*1000]=200
cycle of 611*sqrt(2)=248;[(l/m)*1000]=405
cycle of 612*sqrt(2)=8;[(l/m)*1000]=13
cycle of 613*sqrt(2)=237;[(l/m)*1000]=386
cycle of 614*sqrt(2)=238;[(l/m)*1000]=387
cycle of 615*sqrt(2)=40;[(l/m)*1000]=65
cycle of 616*sqrt(2)=8;[(l/m)*1000]=12
cycle of 617*sqrt(2)=198;[(l/m)*1000]=320
cycle of 618*sqrt(2)=40;[(l/m)*1000]=64
cycle of 619*sqrt(2)=454;[(l/m)*1000]=733
cycle of 620*sqrt(2)=40;[(l/m)*1000]=64
cycle of 621*sqrt(2)=292;[(l/m)*1000]=470
cycle of 622*sqrt(2)=220;[(l/m)*1000]=353
cycle of 623*sqrt(2)=90;[(l/m)*1000]=144
cycle of 624*sqrt(2)=80;[(l/m)*1000]=128
cycle of 625*sqrt(2)=291;[(l/m)*1000]=465
cycle of 626*sqrt(2)=60;[(l/m)*1000]=95
cycle of 627*sqrt(2)=38;[(l/m)*1000]=60
cycle of 628*sqrt(2)=232;[(l/m)*1000]=369
cycle of 629*sqrt(2)=112;[(l/m)*1000]=178
cycle of 630*sqrt(2)=4;[(l/m)*1000]=6
cycle of 631*sqrt(2)=472;[(l/m)*1000]=748
cycle of 632*sqrt(2)=72;[(l/m)*1000]=113
cycle of 633*sqrt(2)=150;[(l/m)*1000]=236
cycle of 634*sqrt(2)=234;[(l/m)*1000]=369
cycle of 635*sqrt(2)=84;[(l/m)*1000]=132
cycle of 636*sqrt(2)=84;[(l/m)*1000]=132
cycle of 637*sqrt(2)=32;[(l/m)*1000]=50
cycle of 638*sqrt(2)=44;[(l/m)*1000]=68
cycle of 639*sqrt(2)=296;[(l/m)*1000]=463
cycle of 640*sqrt(2)=292;[(l/m)*1000]=456
cycle of 641*sqrt(2)=242;[(l/m)*1000]=377
cycle of 642*sqrt(2)=74;[(l/m)*1000]=115
cycle of 643*sqrt(2)=46;[(l/m)*1000]=71
cycle of 644*sqrt(2)=92;[(l/m)*1000]=142
cycle of 645*sqrt(2)=70;[(l/m)*1000]=108
cycle of 646*sqrt(2)=28;[(l/m)*1000]=43
cycle of 647*sqrt(2)=468;[(l/m)*1000]=723
cycle of 648*sqrt(2)=164;[(l/m)*1000]=253
cycle of 649*sqrt(2)=42;[(l/m)*1000]=64
cycle of 650*sqrt(2)=142;[(l/m)*1000]=218
cycle of 651*sqrt(2)=40;[(l/m)*1000]=61
cycle of 652*sqrt(2)=110;[(l/m)*1000]=168
cycle of 653*sqrt(2)=229;[(l/m)*1000]=350
cycle of 654*sqrt(2)=156;[(l/m)*1000]=238
cycle of 655*sqrt(2)=102;[(l/m)*1000]=155
cycle of 656*sqrt(2)=52;[(l/m)*1000]=79
cycle of 657*sqrt(2)=18;[(l/m)*1000]=27
cycle of 658*sqrt(2)=108;[(l/m)*1000]=164
cycle of 659*sqrt(2)=498;[(l/m)*1000]=755
cycle of 660*sqrt(2)=6;[(l/m)*1000]=9
cycle of 661*sqrt(2)=263;[(l/m)*1000]=397
cycle of 662*sqrt(2)=234;[(l/m)*1000]=353
cycle of 663*sqrt(2)=36;[(l/m)*1000]=54
cycle of 664*sqrt(2)=124;[(l/m)*1000]=186
cycle of 665*sqrt(2)=48;[(l/m)*1000]=72
cycle of 666*sqrt(2)=176;[(l/m)*1000]=264
cycle of 667*sqrt(2)=102;[(l/m)*1000]=152
cycle of 668*sqrt(2)=244;[(l/m)*1000]=365
cycle of 669*sqrt(2)=104;[(l/m)*1000]=155
cycle of 670*sqrt(2)=138;[(l/m)*1000]=205
cycle of 671*sqrt(2)=252;[(l/m)*1000]=375
cycle of 672*sqrt(2)=56;[(l/m)*1000]=83
cycle of 673*sqrt(2)=246;[(l/m)*1000]=365
cycle of 674*sqrt(2)=14;[(l/m)*1000]=20
cycle of 675*sqrt(2)=132;[(l/m)*1000]=195
cycle of 676*sqrt(2)=8;[(l/m)*1000]=11
cycle of 677*sqrt(2)=89;[(l/m)*1000]=131
cycle of 678*sqrt(2)=14;[(l/m)*1000]=20
cycle of 679*sqrt(2)=26;[(l/m)*1000]=38
cycle of 680*sqrt(2)=12;[(l/m)*1000]=17
cycle of 681*sqrt(2)=50;[(l/m)*1000]=73
cycle of 682*sqrt(2)=36;[(l/m)*1000]=52
cycle of 683*sqrt(2)=182;[(l/m)*1000]=266
cycle of 684*sqrt(2)=34;[(l/m)*1000]=49
cycle of 685*sqrt(2)=33;[(l/m)*1000]=48
cycle of 686*sqrt(2)=208;[(l/m)*1000]=303
cycle of 687*sqrt(2)=68;[(l/m)*1000]=98
cycle of 688*sqrt(2)=132;[(l/m)*1000]=191
cycle of 689*sqrt(2)=141;[(l/m)*1000]=204
cycle of 690*sqrt(2)=92;[(l/m)*1000]=133
cycle of 691*sqrt(2)=514;[(l/m)*1000]=743
cycle of 692*sqrt(2)=256;[(l/m)*1000]=369
cycle of 693*sqrt(2)=2;[(l/m)*1000]=2
cycle of 694*sqrt(2)=258;[(l/m)*1000]=371
cycle of 695*sqrt(2)=314;[(l/m)*1000]=451
cycle of 696*sqrt(2)=24;[(l/m)*1000]=34
cycle of 697*sqrt(2)=28;[(l/m)*1000]=40
cycle of 698*sqrt(2)=282;[(l/m)*1000]=404
cycle of 699*sqrt(2)=78;[(l/m)*1000]=111
cycle of 700*sqrt(2)=28;[(l/m)*1000]=40
cycle of 701*sqrt(2)=247;[(l/m)*1000]=352
cycle of 702*sqrt(2)=176;[(l/m)*1000]=250
cycle of 703*sqrt(2)=292;[(l/m)*1000]=415
cycle of 704*sqrt(2)=128;[(l/m)*1000]=181
cycle of 705*sqrt(2)=208;[(l/m)*1000]=295
cycle of 706*sqrt(2)=8;[(l/m)*1000]=11
cycle of 707*sqrt(2)=82;[(l/m)*1000]=115
cycle of 708*sqrt(2)=10;[(l/m)*1000]=14
cycle of 709*sqrt(2)=273;[(l/m)*1000]=385
cycle of 710*sqrt(2)=160;[(l/m)*1000]=225
cycle of 711*sqrt(2)=100;[(l/m)*1000]=140
cycle of 712*sqrt(2)=64;[(l/m)*1000]=89
cycle of 713*sqrt(2)=252;[(l/m)*1000]=353
cycle of 714*sqrt(2)=12;[(l/m)*1000]=16
cycle of 715*sqrt(2)=52;[(l/m)*1000]=72
cycle of 716*sqrt(2)=18;[(l/m)*1000]=25
cycle of 717*sqrt(2)=12;[(l/m)*1000]=16
cycle of 718*sqrt(2)=244;[(l/m)*1000]=339
cycle of 719*sqrt(2)=536;[(l/m)*1000]=745
cycle of 720*sqrt(2)=20;[(l/m)*1000]=27
cycle of 721*sqrt(2)=76;[(l/m)*1000]=105
cycle of 722*sqrt(2)=270;[(l/m)*1000]=373
cycle of 723*sqrt(2)=18;[(l/m)*1000]=24
cycle of 724*sqrt(2)=296;[(l/m)*1000]=408
cycle of 725*sqrt(2)=7;[(l/m)*1000]=9
cycle of 726*sqrt(2)=102;[(l/m)*1000]=140
cycle of 727*sqrt(2)=540;[(l/m)*1000]=742
cycle of 728*sqrt(2)=128;[(l/m)*1000]=175
cycle of 729*sqrt(2)=734;[(l/m)*1000]=1006
cycle of 730*sqrt(2)=26;[(l/m)*1000]=35
cycle of 731*sqrt(2)=64;[(l/m)*1000]=87
cycle of 732*sqrt(2)=80;[(l/m)*1000]=109
cycle of 733*sqrt(2)=279;[(l/m)*1000]=380
cycle of 734*sqrt(2)=256;[(l/m)*1000]=348
cycle of 735*sqrt(2)=52;[(l/m)*1000]=70
cycle of 736*sqrt(2)=244;[(l/m)*1000]=331
cycle of 737*sqrt(2)=134;[(l/m)*1000]=181
cycle of 738*sqrt(2)=40;[(l/m)*1000]=54
cycle of 739*sqrt(2)=554;[(l/m)*1000]=749
cycle of 740*sqrt(2)=168;[(l/m)*1000]=227
cycle of 741*sqrt(2)=102;[(l/m)*1000]=137
cycle of 742*sqrt(2)=46;[(l/m)*1000]=61
cycle of 743*sqrt(2)=556;[(l/m)*1000]=748
cycle of 744*sqrt(2)=104;[(l/m)*1000]=139
cycle of 745*sqrt(2)=67;[(l/m)*1000]=89
cycle of 746*sqrt(2)=266;[(l/m)*1000]=356
cycle of 747*sqrt(2)=62;[(l/m)*1000]=82
cycle of 748*sqrt(2)=16;[(l/m)*1000]=21
cycle of 749*sqrt(2)=78;[(l/m)*1000]=104
cycle of 750*sqrt(2)=240;[(l/m)*1000]=320
cycle of 751*sqrt(2)=104;[(l/m)*1000]=138
cycle of 752*sqrt(2)=264;[(l/m)*1000]=351
cycle of 753*sqrt(2)=58;[(l/m)*1000]=77
cycle of 754*sqrt(2)=50;[(l/m)*1000]=66
cycle of 755*sqrt(2)=128;[(l/m)*1000]=169
cycle of 756*sqrt(2)=14;[(l/m)*1000]=18
cycle of 757*sqrt(2)=291;[(l/m)*1000]=384
cycle of 758*sqrt(2)=42;[(l/m)*1000]=55
cycle of 759*sqrt(2)=106;[(l/m)*1000]=139
cycle of 760*sqrt(2)=88;[(l/m)*1000]=115
cycle of 761*sqrt(2)=144;[(l/m)*1000]=189
cycle of 762*sqrt(2)=176;[(l/m)*1000]=230
cycle of 763*sqrt(2)=218;[(l/m)*1000]=285
cycle of 764*sqrt(2)=272;[(l/m)*1000]=356
cycle of 765*sqrt(2)=16;[(l/m)*1000]=20
cycle of 766*sqrt(2)=268;[(l/m)*1000]=349
cycle of 767*sqrt(2)=106;[(l/m)*1000]=138
cycle of 768*sqrt(2)=192;[(l/m)*1000]=250
cycle of 769*sqrt(2)=298;[(l/m)*1000]=387
cycle of 770*sqrt(2)=4;[(l/m)*1000]=5
cycle of 771*sqrt(2)=50;[(l/m)*1000]=64
cycle of 772*sqrt(2)=58;[(l/m)*1000]=75
cycle of 773*sqrt(2)=83;[(l/m)*1000]=107
cycle of 774*sqrt(2)=106;[(l/m)*1000]=136
cycle of 775*sqrt(2)=10;[(l/m)*1000]=12
cycle of 776*sqrt(2)=34;[(l/m)*1000]=43
cycle of 777*sqrt(2)=176;[(l/m)*1000]=226
cycle of 778*sqrt(2)=54;[(l/m)*1000]=69
cycle of 779*sqrt(2)=8;[(l/m)*1000]=10
cycle of 780*sqrt(2)=44;[(l/m)*1000]=56
cycle of 781*sqrt(2)=300;[(l/m)*1000]=384
cycle of 782*sqrt(2)=68;[(l/m)*1000]=86
cycle of 783*sqrt(2)=144;[(l/m)*1000]=183
cycle of 784*sqrt(2)=240;[(l/m)*1000]=306
cycle of 785*sqrt(2)=171;[(l/m)*1000]=217
cycle of 786*sqrt(2)=98;[(l/m)*1000]=124
cycle of 787*sqrt(2)=582;[(l/m)*1000]=739
cycle of 788*sqrt(2)=12;[(l/m)*1000]=15
cycle of 789*sqrt(2)=380;[(l/m)*1000]=481
cycle of 790*sqrt(2)=48;[(l/m)*1000]=60
cycle of 791*sqrt(2)=58;[(l/m)*1000]=73
cycle of 792*sqrt(2)=8;[(l/m)*1000]=10
cycle of 793*sqrt(2)=157;[(l/m)*1000]=197
cycle of 794*sqrt(2)=286;[(l/m)*1000]=360
cycle of 795*sqrt(2)=68;[(l/m)*1000]=85
cycle of 796*sqrt(2)=16;[(l/m)*1000]=20
cycle of 797*sqrt(2)=285;[(l/m)*1000]=357
cycle of 798*sqrt(2)=42;[(l/m)*1000]=52
cycle of 799*sqrt(2)=128;[(l/m)*1000]=160
cycle of 800*sqrt(2)=344;[(l/m)*1000]=430
cycle of 801*sqrt(2)=78;[(l/m)*1000]=97
cycle of 802*sqrt(2)=126;[(l/m)*1000]=157
cycle of 803*sqrt(2)=26;[(l/m)*1000]=32
cycle of 804*sqrt(2)=38;[(l/m)*1000]=47
cycle of 805*sqrt(2)=44;[(l/m)*1000]=54
cycle of 806*sqrt(2)=164;[(l/m)*1000]=203
cycle of 807*sqrt(2)=28;[(l/m)*1000]=34
cycle of 808*sqrt(2)=300;[(l/m)*1000]=371
cycle of 809*sqrt(2)=140;[(l/m)*1000]=173
cycle of 810*sqrt(2)=70;[(l/m)*1000]=86
cycle of 811*sqrt(2)=614;[(l/m)*1000]=757
cycle of 812*sqrt(2)=40;[(l/m)*1000]=49
cycle of 813*sqrt(2)=408;[(l/m)*1000]=501
cycle of 814*sqrt(2)=176;[(l/m)*1000]=216
cycle of 815*sqrt(2)=362;[(l/m)*1000]=444
cycle of 816*sqrt(2)=4;[(l/m)*1000]=4
cycle of 817*sqrt(2)=170;[(l/m)*1000]=208
cycle of 818*sqrt(2)=76;[(l/m)*1000]=92
cycle of 819*sqrt(2)=48;[(l/m)*1000]=58
cycle of 820*sqrt(2)=36;[(l/m)*1000]=43
cycle of 821*sqrt(2)=91;[(l/m)*1000]=110
cycle of 822*sqrt(2)=28;[(l/m)*1000]=34
cycle of 823*sqrt(2)=628;[(l/m)*1000]=763
cycle of 824*sqrt(2)=100;[(l/m)*1000]=121
cycle of 825*sqrt(2)=36;[(l/m)*1000]=43
cycle of 826*sqrt(2)=30;[(l/m)*1000]=36
cycle of 827*sqrt(2)=178;[(l/m)*1000]=215
cycle of 828*sqrt(2)=92;[(l/m)*1000]=111
cycle of 829*sqrt(2)=319;[(l/m)*1000]=384
cycle of 830*sqrt(2)=74;[(l/m)*1000]=89
cycle of 831*sqrt(2)=380;[(l/m)*1000]=457
cycle of 832*sqrt(2)=344;[(l/m)*1000]=413
cycle of 833*sqrt(2)=108;[(l/m)*1000]=129
cycle of 834*sqrt(2)=94;[(l/m)*1000]=112
cycle of 835*sqrt(2)=376;[(l/m)*1000]=450
cycle of 836*sqrt(2)=38;[(l/m)*1000]=45
cycle of 837*sqrt(2)=148;[(l/m)*1000]=176
cycle of 838*sqrt(2)=78;[(l/m)*1000]=93
cycle of 839*sqrt(2)=628;[(l/m)*1000]=748
cycle of 840*sqrt(2)=8;[(l/m)*1000]=9
cycle of 841*sqrt(2)=135;[(l/m)*1000]=160
cycle of 842*sqrt(2)=314;[(l/m)*1000]=372
cycle of 843*sqrt(2)=66;[(l/m)*1000]=78
cycle of 844*sqrt(2)=158;[(l/m)*1000]=187
cycle of 845*sqrt(2)=7;[(l/m)*1000]=8
cycle of 846*sqrt(2)=188;[(l/m)*1000]=222
cycle of 847*sqrt(2)=86;[(l/m)*1000]=101
cycle of 848*sqrt(2)=296;[(l/m)*1000]=349
cycle of 849*sqrt(2)=210;[(l/m)*1000]=247
cycle of 850*sqrt(2)=80;[(l/m)*1000]=94
cycle of 851*sqrt(2)=334;[(l/m)*1000]=392
cycle of 852*sqrt(2)=96;[(l/m)*1000]=112
cycle of 853*sqrt(2)=47;[(l/m)*1000]=55
cycle of 854*sqrt(2)=138;[(l/m)*1000]=161
cycle of 855*sqrt(2)=34;[(l/m)*1000]=39
cycle of 856*sqrt(2)=148;[(l/m)*1000]=172
cycle of 857*sqrt(2)=71;[(l/m)*1000]=82
cycle of 858*sqrt(2)=64;[(l/m)*1000]=74
cycle of 859*sqrt(2)=630;[(l/m)*1000]=733
cycle of 860*sqrt(2)=90;[(l/m)*1000]=104
cycle of 861*sqrt(2)=32;[(l/m)*1000]=37
cycle of 862*sqrt(2)=296;[(l/m)*1000]=343
cycle of 863*sqrt(2)=624;[(l/m)*1000]=723
cycle of 864*sqrt(2)=192;[(l/m)*1000]=222
cycle of 865*sqrt(2)=77;[(l/m)*1000]=89
cycle of 866*sqrt(2)=166;[(l/m)*1000]=191
cycle of 867*sqrt(2)=94;[(l/m)*1000]=108
cycle of 868*sqrt(2)=36;[(l/m)*1000]=41
cycle of 869*sqrt(2)=104;[(l/m)*1000]=119
cycle of 870*sqrt(2)=32;[(l/m)*1000]=36
cycle of 871*sqrt(2)=346;[(l/m)*1000]=397
cycle of 872*sqrt(2)=300;[(l/m)*1000]=344
cycle of 873*sqrt(2)=38;[(l/m)*1000]=43
cycle of 874*sqrt(2)=156;[(l/m)*1000]=178
cycle of 875*sqrt(2)=114;[(l/m)*1000]=130
cycle of 876*sqrt(2)=22;[(l/m)*1000]=25
cycle of 877*sqrt(2)=323;[(l/m)*1000]=368
cycle of 878*sqrt(2)=336;[(l/m)*1000]=382
cycle of 879*sqrt(2)=140;[(l/m)*1000]=159
cycle of 880*sqrt(2)=24;[(l/m)*1000]=27
cycle of 881*sqrt(2)=158;[(l/m)*1000]=179
cycle of 882*sqrt(2)=48;[(l/m)*1000]=54
cycle of 883*sqrt(2)=658;[(l/m)*1000]=745
cycle of 884*sqrt(2)=32;[(l/m)*1000]=36
cycle of 885*sqrt(2)=34;[(l/m)*1000]=38
cycle of 886*sqrt(2)=102;[(l/m)*1000]=115
cycle of 887*sqrt(2)=660;[(l/m)*1000]=744
cycle of 888*sqrt(2)=116;[(l/m)*1000]=130
cycle of 889*sqrt(2)=104;[(l/m)*1000]=116
cycle of 890*sqrt(2)=98;[(l/m)*1000]=110
cycle of 891*sqrt(2)=50;[(l/m)*1000]=56
cycle of 892*sqrt(2)=88;[(l/m)*1000]=98
cycle of 893*sqrt(2)=312;[(l/m)*1000]=349
cycle of 894*sqrt(2)=208;[(l/m)*1000]=232
cycle of 895*sqrt(2)=26;[(l/m)*1000]=29
cycle of 896*sqrt(2)=292;[(l/m)*1000]=325
cycle of 897*sqrt(2)=212;[(l/m)*1000]=236
cycle of 898*sqrt(2)=174;[(l/m)*1000]=193
cycle of 899*sqrt(2)=22;[(l/m)*1000]=24
cycle of 900*sqrt(2)=32;[(l/m)*1000]=35
cycle of 901*sqrt(2)=156;[(l/m)*1000]=173
cycle of 902*sqrt(2)=48;[(l/m)*1000]=53
cycle of 903*sqrt(2)=82;[(l/m)*1000]=90
cycle of 904*sqrt(2)=32;[(l/m)*1000]=35
cycle of 905*sqrt(2)=193;[(l/m)*1000]=213
cycle of 906*sqrt(2)=212;[(l/m)*1000]=233
cycle of 907*sqrt(2)=682;[(l/m)*1000]=751
cycle of 908*sqrt(2)=54;[(l/m)*1000]=59
cycle of 909*sqrt(2)=160;[(l/m)*1000]=176
cycle of 910*sqrt(2)=22;[(l/m)*1000]=24
cycle of 911*sqrt(2)=668;[(l/m)*1000]=733
cycle of 912*sqrt(2)=52;[(l/m)*1000]=57
cycle of 913*sqrt(2)=62;[(l/m)*1000]=67
cycle of 914*sqrt(2)=72;[(l/m)*1000]=78
cycle of 915*sqrt(2)=252;[(l/m)*1000]=275
cycle of 916*sqrt(2)=56;[(l/m)*1000]=61
cycle of 917*sqrt(2)=94;[(l/m)*1000]=102
cycle of 918*sqrt(2)=36;[(l/m)*1000]=39
cycle of 919*sqrt(2)=688;[(l/m)*1000]=748
cycle of 920*sqrt(2)=180;[(l/m)*1000]=195
cycle of 921*sqrt(2)=226;[(l/m)*1000]=245
cycle of 922*sqrt(2)=322;[(l/m)*1000]=349
cycle of 923*sqrt(2)=48;[(l/m)*1000]=52
cycle of 924*sqrt(2)=6;[(l/m)*1000]=6
cycle of 925*sqrt(2)=223;[(l/m)*1000]=241
cycle of 926*sqrt(2)=352;[(l/m)*1000]=380
cycle of 927*sqrt(2)=144;[(l/m)*1000]=155
cycle of 928*sqrt(2)=116;[(l/m)*1000]=125
cycle of 929*sqrt(2)=334;[(l/m)*1000]=359
cycle of 930*sqrt(2)=44;[(l/m)*1000]=47
cycle of 931*sqrt(2)=316;[(l/m)*1000]=339
cycle of 932*sqrt(2)=78;[(l/m)*1000]=83
cycle of 933*sqrt(2)=456;[(l/m)*1000]=488
cycle of 934*sqrt(2)=330;[(l/m)*1000]=353
cycle of 935*sqrt(2)=16;[(l/m)*1000]=17
cycle of 936*sqrt(2)=128;[(l/m)*1000]=136
cycle of 937*sqrt(2)=350;[(l/m)*1000]=373
cycle of 938*sqrt(2)=162;[(l/m)*1000]=172
cycle of 939*sqrt(2)=112;[(l/m)*1000]=119
cycle of 940*sqrt(2)=196;[(l/m)*1000]=208
cycle of 941*sqrt(2)=327;[(l/m)*1000]=347
cycle of 942*sqrt(2)=220;[(l/m)*1000]=233
cycle of 943*sqrt(2)=84;[(l/m)*1000]=89
cycle of 944*sqrt(2)=44;[(l/m)*1000]=46
cycle of 945*sqrt(2)=16;[(l/m)*1000]=16
cycle of 946*sqrt(2)=82;[(l/m)*1000]=86
cycle of 947*sqrt(2)=694;[(l/m)*1000]=732
cycle of 948*sqrt(2)=28;[(l/m)*1000]=29
cycle of 949*sqrt(2)=178;[(l/m)*1000]=187
cycle of 950*sqrt(2)=44;[(l/m)*1000]=46
cycle of 951*sqrt(2)=460;[(l/m)*1000]=483
cycle of 952*sqrt(2)=12;[(l/m)*1000]=12
cycle of 953*sqrt(2)=77;[(l/m)*1000]=80
cycle of 954*sqrt(2)=96;[(l/m)*1000]=100
cycle of 955*sqrt(2)=416;[(l/m)*1000]=435
cycle of 956*sqrt(2)=8;[(l/m)*1000]=8
cycle of 957*sqrt(2)=44;[(l/m)*1000]=45
cycle of 958*sqrt(2)=336;[(l/m)*1000]=350
cycle of 959*sqrt(2)=66;[(l/m)*1000]=68
cycle of 960*sqrt(2)=132;[(l/m)*1000]=137
cycle of 961*sqrt(2)=12;[(l/m)*1000]=12
cycle of 962*sqrt(2)=230;[(l/m)*1000]=239
cycle of 963*sqrt(2)=78;[(l/m)*1000]=80
cycle of 964*sqrt(2)=22;[(l/m)*1000]=22
cycle of 965*sqrt(2)=70;[(l/m)*1000]=72
cycle of 966*sqrt(2)=88;[(l/m)*1000]=91
cycle of 967*sqrt(2)=720;[(l/m)*1000]=744
cycle of 968*sqrt(2)=180;[(l/m)*1000]=185
cycle of 969*sqrt(2)=32;[(l/m)*1000]=33
cycle of 970*sqrt(2)=30;[(l/m)*1000]=30
cycle of 971*sqrt(2)=730;[(l/m)*1000]=751
cycle of 972*sqrt(2)=238;[(l/m)*1000]=244
cycle of 973*sqrt(2)=334;[(l/m)*1000]=343
cycle of 974*sqrt(2)=116;[(l/m)*1000]=119
cycle of 975*sqrt(2)=300;[(l/m)*1000]=307
cycle of 976*sqrt(2)=356;[(l/m)*1000]=364
cycle of 977*sqrt(2)=338;[(l/m)*1000]=345
cycle of 978*sqrt(2)=122;[(l/m)*1000]=124
cycle of 979*sqrt(2)=74;[(l/m)*1000]=75
cycle of 980*sqrt(2)=44;[(l/m)*1000]=44
cycle of 981*sqrt(2)=456;[(l/m)*1000]=464
cycle of 982*sqrt(2)=338;[(l/m)*1000]=344
cycle of 983*sqrt(2)=728;[(l/m)*1000]=740
cycle of 984*sqrt(2)=24;[(l/m)*1000]=24
cycle of 985*sqrt(2)=1;[(l/m)*1000]=1
cycle of 986*sqrt(2)=28;[(l/m)*1000]=28
cycle of 987*sqrt(2)=196;[(l/m)*1000]=198
cycle of 988*sqrt(2)=88;[(l/m)*1000]=89
cycle of 989*sqrt(2)=26;[(l/m)*1000]=26
cycle of 990*sqrt(2)=2;[(l/m)*1000]=2
cycle of 991*sqrt(2)=716;[(l/m)*1000]=722
cycle of 992*sqrt(2)=344;[(l/m)*1000]=346
cycle of 993*sqrt(2)=238;[(l/m)*1000]=239
cycle of 994*sqrt(2)=156;[(l/m)*1000]=156
cycle of 995*sqrt(2)=4;[(l/m)*1000]=4
cycle of 996*sqrt(2)=62;[(l/m)*1000]=62
cycle of 997*sqrt(2)=375;[(l/m)*1000]=376
cycle of 998*sqrt(2)=358;[(l/m)*1000]=358
cycle of 999*sqrt(2)=500;[(l/m)*1000]=500
cycle of 1000*sqrt(2)=456;[(l/m)*1000]=456
cycle of 1001*sqrt(2)=52;[(l/m)*1000]=51
cycle of 1002*sqrt(2)=244;[(l/m)*1000]=243
cycle of 1003*sqrt(2)=24;[(l/m)*1000]=23
cycle of 1004*sqrt(2)=66;[(l/m)*1000]=65
cycle of 1005*sqrt(2)=134;[(l/m)*1000]=133
cycle of 1006*sqrt(2)=364;[(l/m)*1000]=361
cycle of 1007*sqrt(2)=384;[(l/m)*1000]=381
cycle of 1008*sqrt(2)=24;[(l/m)*1000]=23
cycle of 1009*sqrt(2)=42;[(l/m)*1000]=41
cycle of 1010*sqrt(2)=86;[(l/m)*1000]=85
cycle of 1011*sqrt(2)=14;[(l/m)*1000]=13
cycle of 1012*sqrt(2)=88;[(l/m)*1000]=86
cycle of 1013*sqrt(2)=119;[(l/m)*1000]=117
cycle of 1014*sqrt(2)=8;[(l/m)*1000]=7
cycle of 1015*sqrt(2)=18;[(l/m)*1000]=17
cycle of 1016*sqrt(2)=344;[(l/m)*1000]=338
cycle of 1017*sqrt(2)=38;[(l/m)*1000]=37
cycle of 1018*sqrt(2)=374;[(l/m)*1000]=367
cycle of 1019*sqrt(2)=758;[(l/m)*1000]=743
cycle of 1020*sqrt(2)=8;[(l/m)*1000]=7
cycle of 1021*sqrt(2)=403;[(l/m)*1000]=394
cycle of 1022*sqrt(2)=22;[(l/m)*1000]=21
cycle of 1023*sqrt(2)=44;[(l/m)*1000]=43
cycle of 1024*sqrt(2)=760;[(l/m)*1000]=742
time = 13,680 ms.
のように、m*sqrt(2)の連分数展開の循環節の長さを計算できる。
m=1,...,1024の範囲では、m*sqrt(2)の連分数展開の循環節の長さは、m=9,729のときmを超
えるが、それ以外はm以下である。
次に、j(sqrt(-6))の連分数展開の循環節の長さを求める。
j(sqrt(-6))=2417472+1707264*sqrt(2)
であるので、 1707264*sqrt(2) の連分数展開の循環節の長さを求めれば十分である。
gp > findc(cf(1707264))
%7 = 0
gp > default(realprecision,10000)
gp > findc(cf(1707264))
time = 17 ms.
%10 = 0
gp > default(realprecision,100000)
gp > findc(cf(1707264))
time = 1,449 ms.
%12 = 58401
gp > length(cf(1707264))
time = 1,440 ms.
%13 = 94585
gp > default(realprecision,150000)
gp > length(cf(1707264))
time = 3,243 ms.
%15 = 141910
gp > findc(cf(1707264))
time = 3,245 ms.
%16 = 58401
により、j(sqrt(-6))の連分数展開の循環節の長さが58401-1=58400であることが確認できる。
循環連分数であることを確認するためには、連分数展開の長さが循環節の2倍以上あれ
ば良いので、小数点以下150000桁程度の精度があれば十分である。
よって、原理的には、j(sqrt(-6))の連分数展開から、その値のQ(sqrt(2))の代数的整数に
よる表示を求めることができることが分かる。
H．Nakao さんからのコメントです。（令和３年１０月１日付け）
■j(sqrt(-6))の連分数展開を使って、その値を求めることができた。
Pari/GPを使って、小数点以下150000桁の精度で、j(sqrt(-6))を計算し、連分数展開する。
gp > default(realprecision,150000)
gp > r=contfrac(ellj(sqrt(-6)));
time = 6,040 ms.
gp > default(realprecision,35)
gp > length(r)
%4 = 141909
gp > r[1]
%5 = 4831907
初項r[1]は、j(sqrt(-6))の整数部分[j(sqrt(-6)]=4831907である。
連分数の2項以降から最終項までで、最大(の整数)になる項の位置を調べる。
max{r[2],...,r[141909]}=4828870
最大になる項は、r[58401]、r[116801]であるので、循環節の長さは58400と予想できるが、
実際に、[r[2],...,r[58401]]=[r[58402],...,r[116801]]なので、循環節はr[2]..r[58401]と分かる。
gp > m=0;for(i=2,length(r),if(m<r[i],m=r[i]));m
time = 20 ms.
%6 = 4828870
gp > for(i=2,length(r),if(r[i]==4828870,print(i)))
58401
116801
time = 19 ms.
gp > for(i=2,58401,if(r[i]!=r[i+58400],print("NotEq:",i)))
time = 15 ms.
循環節をcとすると、c=[r[2],...,r[58401],c]であり、これより、cの2次方程式を導いて、その
根cを計算して、最後にj(sqrt(-6))=r[1]+(1/c)とできるが、あまり良い方法とは言えない。
j(sqrt(-6))がQ(sqrt(2))の代数的整数であること、m*sqrt(2)の連分数展開および
r[58401]/2=2414435から、循環連分数
[2414435,・r[2],...,・r[58401]]
が、ある整数mに対して、m*sqrt(2)に一致すると予想できる。
実際に、(j(sqrt(-6))-2417472)/sqrt(2)または(j(sqrt(-6))-2417472)^2を小数点以下300桁
の精度で計算すると、
gp > default(realprecision,300)
gp > 4828870/2
%12 = 2414435
gp > r[1]-2414435
%12 = 2417472
gp > (ellj(sqrt(-6))-2417472)/sqrt(2)
%13 = 1707264.00000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000
gp > (ellj(sqrt(-6))-2417472)^2
%14 = 5829500731392.00000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000
gp > z=Mod(x,x^2-2)
%15 = Mod(x, x^2 - 2)
gp > (60+24*z)^3*(1+z)^2
%16 = Mod(1707264*x + 2417472, x^2 - 2)
となるので、
j(sqrt(-6))=2417472+1707264*sqrt(2)=(60+24*sqrt(2))^3*(1+sqrt(2))^2
であることが確認できる。
【参考文献】は、以下の通り。
Pari/GPについては[1][2]、連分数については[3](p.129-153)、modular j-不変量について
は[4](p.429-439)を参照ください。
[1] PARI/GP home
[2] 「User's Guide to PARI/GP」
[3] G.H.Hardy, E.M.Wright, 「An Introduction to the Thoery of Numbers 5th edition」
Oxford University Press, 1979, ISBN0-19-853171-0.
[4] Joseph H. Silverman, 「The Arithmetic of Elliptic Curves」, GTM 106,
Springer-Verlag New York Inc., 1986, ISBN0-387-96203-4.
