New results similar to Lagrange's four-square theorem

Zhi-Wei Sun

Published 2024-11-21Version 1

In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Here we state two general results: (1) If $a$ and $b$ are odd integers with $a>0$, $b>-a$ and $\gcd(a,b)=1$, then all sufficient large integers can be written as $$\frac{w(aw+b)}2+\frac{x(ax+b)}2+\frac{y(ay+b)}2+\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers. (2) If $a$ and $b$ are integers with $a>0$, $b>-a$, $2\mid a$ and $\gcd(a,b)=1$, then all sufficient large integers can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers.