Yet another Proof of an old Hat

TL;DR

This paper presents a new proof of Fermat's theorem on sums of two squares for primes of the form 1+4N, based on a combinatorial property of odd primes expressed through sums of ordered products.

Contribution

It introduces a novel combinatorial approach to prove Fermat's theorem, providing an alternative perspective on sums of two squares for specific primes.

Findings

Every odd prime p can be expressed as a sum of two products in exactly (p+1)/2 ways.

The number of such representations relates directly to Fermat's theorem on sums of two squares.

The proof offers a new combinatorial insight into classical number theory results.

Abstract

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab + cd with min(a, b) > max(c, d) of two ordered products. This gives a new proof Fermat's Theorem expressing primes of the form 1 + 4N as sums of two squares 1 .