On perfect powers that are sums of cubes of a seven term arithmetic progression

Abstract

We prove that the equation $(x-3r{)}^3+(x-2r{)}^3+(x-r{)}^3+x^3+(x+r{)}^3+(x+2r{)}^3+(x+3r{)}^3=y^p$ only has solutions which satisfy $xy=0$ for $1\le r\le 10^6$ and $p\ge 5$ prime. This article complements the work on the equations $(x-r{)}^3+x^3+(x+r{)}^3=y^p$ and $(x-2r{)}^3+(x-r{)}^3+x^3+(x+r{)}^3+(x+2r{)}^3=y^p$ . The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.

arXiv admin note: text overlap with arXiv:1901.05382