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

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 \[ n = 1^{r} \cdot |k_{1}|^{v} + 2^{r} \cdot |k_{2}|^{v} + 3^{r} \cdot |k_{3}|^{v} + \cdots \] 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).

Acknowledgments

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

Citation

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Nian Hong Zhou. Yalin Sun. "Counting the Number of Solutions to Certain Infinite Diophantine Equations." Taiwanese J. Math. 25 (2) 223 - 232, April, 2021. https://doi.org/10.11650/tjm/201107

Information

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

Subjects:
Primary: 11D45, 11P82, 11P99

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

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Vol.25 • No. 2 • April, 2021
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