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$x^2 + y^2 + z^2 = 1 \mod n$ 的解的数量

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Czhang271828 Posted at 2024-1-3 13:58:57 |Read mode
  • 求 $x_1^3+\cdots +x_{11}^3=(x_1+\cdots +x_{11})^2$ 的解的数量, 其中 $\{x_i\}_{i=1}^{11}$ 是单调不减的正整数列. OEIS.
  • 求 $x^2+y^2+z^2=1\mod 46$ 解的数量. OEIS.

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hbghlyj Posted at 2024-10-20 19:01:59
Czhang271828 发表于 2024-1-3 05:58
求 $x^2+y^2+z^2=1\mod 46$ 解的数量. OEIS.

在提供的 OEIS 链接中有一个通项公式,但没有证明$$
a(n) = n^2 \cdot \left(\frac{3}{2} \text{ if } 4 \mid n\right) \cdot \prod_{\substack{\text{prime } p \equiv 1 \bmod 4 \\ p\mid n}} \left(1 + \frac{1}{p}\right) \cdot \prod_{\substack{\text{prime } p \equiv 3 \bmod 4 \\ p\mid n}} \left(1 - \frac{1}{p}\right)
$$

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