On sums of three squares

TL;DR

This paper proves that positive integers not of the form 4^k(8m+7) can be expressed as the sum of three or fewer squares, extending classical results using recent work on mixed sums of squares and triangular numbers.

Contribution

It introduces a new proof leveraging results on mixed sums of squares and triangular numbers to characterize integers representable as sums of three or fewer squares.

Findings

Positive integers not of the form 4^k(8m+7) are representable as sums of three or fewer squares.

Utilizes recent results of Kane and Sun on mixed sums of squares and triangular numbers.

Provides a new proof of a classical number theory result.

Abstract

We prove that a positive integer not of the form, 4^{k}(8m+7) can be expressible as a sum of three or fewer squares by using some results of Kane and Sun on mixed sums of squares and triangular numbers.