# Refinements of Lagrange's four-square theorem

## TL;DR

This paper characterizes the possible sums of the four integers in Lagrange's four-square representations of nonnegative integers, providing new insights into the structure of these representations.

## Contribution

It offers a detailed characterization of the sums of the four squares in all representations of integers, extending understanding of Lagrange's theorem.

## Key findings

- Identifies all possible sums of the four squares for each integer n
- Provides explicit criteria for the sums in the representations
- Enhances understanding of the structure of four-square representations

## Abstract

A well-known theorem of Lagrange asserts that every nonnegative integer $n$ can be written in the form $a^2+b^2+c^2+d^2$, where $a,b,c,d \in \mathbb{Z}$. We characterize the values assumed by $a+b+c+d$ as we range over all such representations of $n$.