A system of certain linear Diophantine equations on analogs of squares
Volume 207 / 2023
Acta Arithmetica 207 (2023), 251-277
MSC: Primary 11D04; Secondary 11J71, 11B39.
DOI: 10.4064/aa220622-19-1
Published online: 28 March 2023
Abstract
We investigate the existence of tuples $(k, \ell , m)$ of integers such that all of $k$, $\ell $, $m$, $k+\ell $, $\ell +m$, $m+k$, $k+\ell +m$ belong to the set $S(\alpha )$ of all integers of the form $\lfloor \alpha n^2 \rfloor $ for $n\geq \alpha ^{-1/2}$. We show that $T(\alpha )$, the set of all such tuples, is infinite for all $\alpha \in (0,1)\cap \mathbb {Q}$ and for almost all $\alpha \in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $\alpha \gt 0$ such that $T(\alpha )$ is finite, then there is no perfect Euler brick. We also examine the set of all integers of the form $\lceil \alpha n^2 \rceil $ for $n\in \mathbb {N}$.
