Sum of consecutive powers as a perfect power

Abstract

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

17 pages; Comments are welcome