论文标题
一定限制的四个正方形的总和
Sums of four squares with a certain restriction
论文作者
论文摘要
在2016年，在研究限制的整体平方之和时，太阳提出了以下猜想：每个正整数$ n $都可以写成$ x^2+y^2+y^2+z^2+w^2 $ $（x，x，y，y，z，w \ in \ mathbb {n}同时，他还推测，对于每个正整数$ n $，都存在整数$ x，y，z，w $，以便$ n = x^2+y^2+z^2+z^2+w^2 $和$ x+3y \ in \ in \ {4^k：k：k \ in \ mathbb {n} \} $。在本文中，我们通过某些三元二次形式的算术理论证实了这些猜想。
In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also conjectured that for each positive integer $n$ there exist integers $x,y,z,w$ such that $n=x^2+y^2+z^2+w^2$ and $x+3y\in\{4^k:k\in\mathbb{N}\}$. In this paper, we confirm these conjectures via some arithmetic theory of ternary quadratic forms.
