Two dimensional arithmetic progressions avoiding squares
Number Theory
2026-05-13 v1
Abstract
We show that any proper symmetric two dimensional arithmetic progression contained in the interval which avoids non-zero perfect squares has at most elements. This improves on a result of Croot, Lyall and Rice. We also discuss lower bounds for this problem and their connections to bounds for the least quadratic non-residue modulo a prime.
Keywords
Cite
@article{arxiv.2605.11104,
title = {Two dimensional arithmetic progressions avoiding squares},
author = {Rainer Dietmann and Christian Elsholtz},
journal= {arXiv preprint arXiv:2605.11104},
year = {2026}
}
Comments
10 pages, to appear in Proc. Amer. Math. Soc