Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory175 (2017), 169–190] conjectured that 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,\ldots \})$ with $x+3y$ a square and also as $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$ with $x+3y\in \{4^k:k\in \mathbb {N}\}$ . In this paper, we confirm these conjectures via the arithmetic theory of ternary quadratic forms.

中文翻译：

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有一定限制的四个平方和

Z.-W。Sun ['提炼拉格朗日的四平方定理'，J.数论175(2017), 169–190] 推测每个正整数n可以写成$ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$和$x+3y$一个正方形，也作为$n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$和$x+3y\in \{4^k:k\in \mathbb {N}\}$. 在本文中，我们通过三元二次型的算术理论证实了这些猜想。