Counting the number of solutions to certain infinite Diophantine equations

TL;DR

This paper studies the number of solutions to a class of infinite Diophantine equations involving parameters r and v, providing generating functions and asymptotic formulas for the solution counts.

Contribution

It introduces new methods to analyze solutions of infinite Diophantine equations and derives explicit generating functions and asymptotic estimates for solution counts.

Findings

Derived generating functions for solution counts

Established asymptotic formulas for specific (r,v) cases

Analyzed the behavior of solutions as n grows large

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.