A system of certain linear Diophantine equations on analogs of squares

TL;DR

This paper explores the existence of integer solutions to certain linear Diophantine equations within sets generated by scaled square functions, revealing infinite solutions for rational and almost all real parameters, and linking finiteness to the Euler brick problem.

Contribution

It introduces a novel analysis of Diophantine equations on sets of scaled squares and connects solution finiteness to the Euler brick conjecture.

Findings

Infinite solutions for rational nd almost all real re established.

Finiteness of solutions implies the non-existence of a perfect Euler brick.

Examines properties of integers of the form eil lpha n^2eil.

Abstract

This study investigates 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 $S(\alpha)$, where $S(\alpha)$ is the set of all integers of the form $\lfloor \alpha n^2 \rfloor$ for $n\geq \alpha^{-1/2}$ and $\lfloor x\rfloor$ denotes the integer part of $x$. 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 $\alpha>0$ 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 $n\in \mathbb{N}$.