Open Access
April, 2021 Counting the Number of Solutions to Certain Infinite Diophantine Equations
Nian Hong Zhou, Yalin Sun
Author Affiliations +
Taiwanese J. Math. 25(2): 223-232 (April, 2021). DOI: 10.11650/tjm/201107


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

Funding Statement

This research was partly supported by the National Science Foundation of China (Grant No. 11971173).


The authors would like to thank the anonymous referee for his/her very helpful comments and suggestions.


Download Citation

Nian Hong Zhou. Yalin Sun. "Counting the Number of Solutions to Certain Infinite Diophantine Equations." Taiwanese J. Math. 25 (2) 223 - 232, April, 2021.


Received: 15 May 2020; Revised: 22 November 2020; Accepted: 29 November 2020; Published: April, 2021
First available in Project Euclid: 24 March 2021

Digital Object Identifier: 10.11650/tjm/201107

Primary: 11D45 , 11P82 , 11P99

Keywords: asymptotics , infinite Diophantine equations , integer partitions

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

Vol.25 • No. 2 • April, 2021
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