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arXiv · 2605.18348

Sum of consecutive powers as a perfect power

Angelos Koutsianas·Nikos Tzanakis
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Abstract

In this paper we study the equation $$ x^k + (x+1)^k = y^n,\quad n\geq 3, $$ when $k\equiv 2\pmod{4}$. We prove that the only solutions are for $x=0, -1$ when $6\leq k\leq 100$ or for a $k$ with odd prime factors congruent to $3\pmod{4}$. We use linear forms in logarithms, the modular method and the resolution of Thue equations.

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Angelos Koutsianas, Nikos Tzanakis. 2026-05-18. Sum of consecutive powers as a perfect power. https://arxiv.org/abs/2605.18348

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