本帖最后由 蔡家雄 于 2020-7-10 10:58 编辑
(4k+1)*A^2 - B^2 = ±1 和 (8k+2)*A^2 - B^2 = ±1
存在通项公式的条件:
4k+1 或 8k+2 的(奇数)素因子均具有 4d+1 的形式。
A031396 Numbers k such that Pell equation x^2 - k*y^2 = -1 is soluble.
2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97,
101, 106, 109, 113, 122, 125, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197,
202, 218, 226, 229, 233, 241, 250, 257, 265, 269, 274, 277, 281, 290, 293, 298 , ...
A031397 数据 无4k+3 的素因子,此方程 存在 正整数解,
A031397 Nonsquarefree n such that Pell equation x^2 - n y^2 = -1 is soluble.
50, 125, 250, 325, 338, 425, 845, 925, 1025, 1250, 1325, 1445, 1450, 1525, 1625, 1682, 1825, 1850,
2050, 2125, 2197, 2425, 2725, 2738, 2825, 2873, 2890, 3050, 3125, 3250, 3425, 3625, 3725, 3925,
4250, 4325, 4394, 4625, 4825, 4901, 4913 , ...
A031398 数据 无4k+3 的素因子,但方程 不存在 正整数解,
A031398 Squarefree n with no 4k+3 factors such that Pell equation x^2 - n y^2 = -1 is insoluble.
34, 146, 178, 194, 205, 221, 305, 377, 386, 410, 466, 482, 505, 514, 545, 562, 674, 689,
706, 745, 793, 802, 866, 890, 898, 905, 1154, 1186, 1202, 1205, 1234, 1282, 1345, 1346,
1394, 1405, 1469, 1513, 1517, 1537, 1538, 1717, 1762, 1802, 1858 , ...
5*A^2 - B^2 = ±1 的递推公式
A0=0, A1=1, A2=4, A(n+1)=4*An + A(n-1) ,
B0=1, B1=2, B2=7, B(n+1)=4*Bn + B(n-1) .
则 lim(n→∞) Bn/An = √5 .
求 y^2=(4n+1)*x^2 -1 的最小解之特例
若 素数4n+1=(2t+1)^2+4,
则 x=2n - 1, y=2n*(2t+1) . |