Counting the number of solutions to certain infinite Diophantine equations

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 $$n=1^r\cdot |k_1{|}^v+2^r\cdot |k_2{|}^v+3^r\cdot |k_3{|}^v+\dots ,$$ for $\mathbf{k}=(k_1,k_2,k_3,\dots )\in {\mathbb{Z}}^{\infty }$. For each $(r,v)\in \mathbb{N}\times \{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established.

10 pages, accepted for publication in Taiwanese Journal of Mathematics