New results similar to Lagrange's four-square theorem

TL;DR

This paper extends classical four-square theorems by establishing new representations of integers using quadratic forms with parameters, including conditions for large integers and specific divisibility constraints.

Contribution

It introduces novel integer representations similar to Lagrange's four-square theorem, covering cases with parameters and divisibility conditions, and proves their validity for sufficiently large integers.

Findings

Any integer greater than 1 can be expressed in a specific quadratic form involving 5w+1.

Large integers can be represented using forms with parameters a and b under certain gcd and divisibility conditions.

New formulas generalize classical four-square results to broader classes of quadratic forms.

Abstract

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 as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Let $a$ and $b$ be integers with $a>0$, $b>-a$ and $\gcd(a,b)=1$. When $2\nmid ab$, we show that any sufficiently large integer 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. When $2\mid a$ and $2\nmid b$, we prove that any sufficiently large integer can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers.