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
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
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