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Questions in category: 初等数论 (Elementary Number Theory).

## 双平方和问题的解数公式

Posted by haifeng on 2023-03-19 21:59:21 last update 2023-03-19 22:29:23 | Answers (0) | 收藏

$n=2^\gamma\cdot p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_s^{\alpha_s}\cdot q_1^{\beta_1}q_2^{\beta_2}\cdots q_t^{\beta_t}$

$\delta(n)=\biggl[\prod_{i=1}^{s}(\alpha_i+1)\biggr]\biggl[\prod_{j=1}^{t}\frac{(-1)^{\beta_j}+1}{2}\biggr],$

$\varepsilon(n)=\biggl[\frac{(-1)^{\gamma}+1}{2}\biggr]\biggl[\prod_{i=1}^{s}\frac{(-1)^{\alpha_i}+1}{2}\biggr]\biggl[\prod_{j=1}^{t}\frac{(-1)^{\beta_j}+1}{2}\biggr],$

$\xi(n)=(-1)^{\gamma}\cdot\Biggl[\dfrac{(-1)^{\prod_{i=1}^{s}(\alpha_i+1)}-1}{2}\Biggr]\biggl[\prod_{j=1}^{t}\frac{(-1)^{\beta_j}+1}{2}\biggr].$

$\delta(1)=1,\quad\varepsilon(1)=1,\quad\xi(1)=-1.$

$\begin{split} r_2(n)&=\mathrm{Card}\{(x,y)\in\mathbb{Z}^2\ :\ x^2+y^2=n\}\\ &=4\delta(n). \end{split}$

$\begin{split} r_2^{+}(n)&=\mathrm{Card}\{(x,y)\in\mathbb{Z}_{+}^2\ :\ x^2+y^2=n\}\\ &=\delta(n)-\varepsilon(n). \end{split}$

$\begin{split} r_{2*}^{+}(n)&=\mathrm{Card}\{(x,y)\in\mathbb{Z}_{+}^2\ :\ x^2+y^2=n\quad\text{且}\ x\leqslant y\}\\ &=\frac{\delta(n)+\xi(n)}{2}. \end{split}$

Remark: