Perfect Powers that are Sums of Squares of an Arithmetic Progression

Debanjana Kundu; Vandita Patel

Perfect Powers that are Sums of Squares of an Arithmetic Progression

Number Theory 2019-12-23 v2

Authors: Debanjana Kundu , Vandita Patel

Abstract

In this paper, we determine all primitive solutions to the equation $(x+r)^2 +(x+2r)^2 +\cdots +(x+dr)^2 = y^n$ for $2\leq d\leq 10$ and for $1\leq r\leq 10^4$ . We make use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.

Keywords

perfect powers arithmetic progression sums of squares primitive solutions primitive divisors theorem

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@article{arxiv.1809.09167,
  title  = {Perfect Powers that are Sums of Squares of an Arithmetic Progression},
  author = {Debanjana Kundu and Vandita Patel},
  journal= {arXiv preprint arXiv:1809.09167},
  year   = {2019}
}

Comments

18 pages, 3 tables In this version, we substantially improve Theorem 1.3 from the previous version in order to solve the case $d=6$ and $1 \leq r \leq 10^4$. We are now able to record Theorems 1.1, 1.2 and 1.3 from the previous version in one concise theorem

Perfect Powers that are Sums of Squares of an Arithmetic Progression

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