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

Alejandro Argáez-García; Vandita Patel

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arXiv · 1911.01842

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

Alejandro Argáez-García·Vandita Patel

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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 $1\leq r\leq 10^6$ and $p\geq 5$ 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.

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Alejandro Argáez-García, Vandita Patel. 2019-11-03. On perfect powers that are sums of cubes of a seven term arithmetic progression. https://arxiv.org/abs/1911.01842

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