A Quixotic Proof of Fermat's Two Squares Theorem for Prime Numbers

TL;DR

This paper presents a novel proof demonstrating that every odd prime can be expressed in multiple ways as sums of two products with specific ordering constraints, leading to a straightforward proof of Fermat's theorem on sums of two squares for primes congruent to 1 mod 4.

Contribution

It introduces a new combinatorial approach to represent odd primes as sums of two products, providing a fresh proof of Fermat's Two Squares Theorem.

Findings

Every odd prime p has exactly (p + 1)/2 representations as sums of two ordered products.

The proof offers a new perspective on expressing primes as sums of two squares for primes of the form 4N+1.

The approach simplifies the classical proof of Fermat's theorem for primes congruent to 1 mod 4.

Abstract

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two squares.