Superelliptic equations arising from sums of consecutive powers

Michael A. Bennett, Vandita Patel, Samir Siksek

Using only elementary arguments, Cassels solved the Diophantine equation $(x-1{)}^3+x^3+(x+1{)}^3=z^2$ in integers $x$, $z$. The generalization $(x-1{)}^k+x^k+(x+1{)}^k=z^n$ (with $x$, $z$…