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arXiv · 1910.07884

Counting the number of solutions to certain infinite Diophantine equations

Nian Hong Zhou·Yalin Sun
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Abstract

Let $r, v, n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations $$ n=1^{r}\cdot |k_{1}|^{v}+2^{r}\cdot |k_{2}|^{v}+3^{r}\cdot |k_{3}|^{v}+\ldots, $$ for ${\bf k}=(k_1,k_2,k_3,\dots)\in\mathbb{Z}^{\infty}$. For each $(r,v)\in\mathbb{N}\times\{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established.

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Nian Hong Zhou, Yalin Sun. 2019-10-17. Counting the number of solutions to certain infinite Diophantine equations. https://arxiv.org/abs/1910.07884

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