数论小猜想

Treenewbee

蔡家雄 发表于 2023-3-14 20:37
若 (10t+5)^2+4 是素数，

则 10 是素数 (10t+5)^2+4 的原根。

t=1-10000,(10t+5)^2+4 是素数的t有2204个，其中不满足10 是素数 (10t+5)^2+4 的原根的有389个：
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蔡家雄

用公式法求解 x^2 - (4k+3)*y^2=1 的最小解，

要分 8k+3 与 8k+7 两种情形，在此不再讨论。

蔡家雄

用公式法求解特殊佩尔方程

设 \(p=4k+1\) 是素数，

求 \(x^2 - p*y^2=1\) 的最小解，

设 \(x=2p*r^2 -1\) , 求 最小的 \(r=?\) ,

使 \(y=((2p*r^2)*(2p*r^2 -2)/p)^{1/2}\) 是整数。

推论：此时，

设 \(p=4k+1\) 是素数，

求 \(x^2 - p*y^2= -1\) 的最小解，

得 \(y=r\) ,  \(x=((2p*r^2)*(2p*r^2 -2)/p)^{1/2}/(2*r)\) .

蔡家雄

由 Treenewbee 程序计算，

(127, [4730624, 419775])

(511, [4188548960, 185290497]）

(2047, [...........................................................................])

wlc1

求 x^7+y^4=z^2  

Treenewbee

wlc1 发表于 2023-3-16 22:17
求 x^7+y^4=z^2

\[2^3+1=3^2\]
同乘以2^4
\[2^7+2^4=12^2\]

lusishun

求X^22+Y^14=Z^6的一组整数解

蔡家雄

设 p=8k+3 是质数，求 x^2 - p*y^2=p 的最小解，

蔡家雄

设 p=8k+7 是质数，求 x^2 - p*y^2=p 的最小解，

蔡家雄

蔡氏完全循环节问题

若 3^(2n)+2^(2n+1) 是素数，

则 10 是素数 3^(2n)+2^(2n+1) 的原根。

简记为  \(g(3^{2n}+2^{2n+1})=10\) .