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arXiv · 2404.03457

On the sum of fifth powers in arithmetic progression

Lucas Villagra Torcomian
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Abstract

In this paper we study equation $$(x-dr)^5+\cdots+x^5+\cdots+(x+dr)^5=y^p$$ under the condition $\gcd(x,r)=1$. We present a recipe for proving the non-existence of non-trivial integer solutions of the above equation, and as an application we obtain explicit results for the cases $d=2,3$ (the case $d=1$ was already solved). We also prove an asymptotic result for $d\equiv 1, 7\pmod9$. Our main tools include the modular method, employing Frey curves and their associated modular forms, as well as the symplectic argument.

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Lucas Villagra Torcomian. 2024-04-04. On the sum of fifth powers in arithmetic progression. https://arxiv.org/abs/2404.03457

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