论文标题
正方形类似物上某些线性二氧甘氨酸方程的系统
A system of certain linear Diophantine equations on analogs of squares
论文作者
论文摘要
这项研究调查了整数的元组$（k，\ ell，m）的存在，使得所有$ k $，$ \ ell $，$ m $，$ k+\ ell $，$ \ ell+m $，$ \ m $，$ k+k $，$ k+\+\ ell+m $ \ rfloor $ for $ n \geqα^{ - 1/2} $和$ \ lfloor x \ rfloor $表示$ x $的整数部分。我们表明，$ t（α）$是所有此类元素的集合，对于（0,1）\ cap \ mathbb {q} $而言，所有$α\对于所有$α\ is in（0,1）\ cap \ mathbb {q} $，而几乎所有$α\ in（0,1）$在Lebesgue Mesuare的意义上。此外，我们表明，如果存在$α> 0 $，则$ t（α）$是有限的，那么就没有完美的Euler砖。我们还检查了$ \lceilαn^2 \ rceil $ for $ n \ in \ mathbb {n} $的所有整数集。
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 $S(α)$, where $S(α)$ is the set of all integers of the form $\lfloor αn^2 \rfloor$ for $n\geq α^{-1/2}$ and $\lfloor x\rfloor$ denotes the integer part of $x$. We show that $T(α)$, the set of all such tuples, is infinite for all $α\in (0,1)\cap \mathbb{Q}$ and for almost all $α\in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $α>0$ such that $T(α)$ is finite, then there is no perfect Euler brick. We also examine the set of all integers of the form $\lceil αn^2 \rceil$ for $n\in \mathbb{N}$.
