论文标题
具有一定限制的四个理性广场的总和
Sums of four rational squares with certain restrictions
论文作者
论文摘要
在本文中，我们主要研究具有某些限制的四个理性正方形的总和。令$ \ mathbb q _ {\ ge0} $为非负合理数字的集合。我们为理性数字建立以下四平方定理：对于任何$ a，b，c，d \ in \ mathbb q _ {\ ge0} $，每个$ r \ in \ mathbb q _ {\ ge0} $都可以写成$ x^2+y^2+y^2+y^2+z^2+z^2+w^2 $ q _ {\ ge0} $，使得$ ax+by+cz+dw $是一个理性的平方（或一个有理立方体）。本文还包含许多猜想。例如，对于任何带有$ \ gcd（a，b）= 1 $的正整数$ a $ a $ a $ a $ a $ a $ a $ a和$ b $，我们推测，每个$ r \ in \ mathbb q _ {\ ge0} $都可以写成$ aw^4+aw^4+bx^4+y^4+y^2+z^2+z^2 $ w，x，x，x，x，x，x，x，y，z in \ inth \ nath \ nath \ nath \ nmath \ bb q $。
In this paper we mainly study sums of four rational squares with certain restrictions. Let $\mathbb Q_{\ge0}$ be the set of nonnegative rational numbers. We establish the following four-square theorem for rational numbers: For any $a,b,c,d\in\mathbb Q_{\ge0}$, each $r\in\mathbb Q_{\ge0}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Q_{\ge0}$ such that $ax+by+cz+dw$ is a rational square (or a rational cube). This paper also contains many conjectures; for example, for any positive integers $a$ and $b$ with $\gcd(a,b)=1$, we conjecture that each $r\in\mathbb Q_{\ge0}$ can be written as $aw^4+bx^4+y^2+z^2$ with $w,x,y,z\in\mathbb Q$.
