Sum of consecutive powers as a perfect power

Angelos Koutsianas, Nikos Tzanakis

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…