# A Kronecker-type identity and the representations of a number as a sum of three squares

## TL;DR

This paper derives a new proof of Gauss's formula for representing numbers as sums of three squares using a Kronecker-type identity, and also proves that every positive integer can be expressed as a sum of three triangular numbers.

## Contribution

It introduces a novel approach connecting Kronecker-type identities with classical number theory results on sums of squares and triangular numbers.

## Key findings

- New proof of Gauss's formula for three squares
- Demonstrates all positive integers as sums of three triangular numbers
- Links Kronecker identities with classical number theory results

## Abstract

By considering a limiting case of a Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares. From the Kronecker-type identity, we also deduce Gauss's theorem that every positive integer is representable as a sum of three triangular numbers.