Geometry of Fermat’s Sum of Squares

In this paper, we present a novel geometric proof of Fermat’s sum of two squares theorem, which states that a prime number p $$ p $$ can be expressed as the sum of two squares if and only if p ≡ 1 ( mod 4 ) $$p \equiv 1 \ (\text{mod} \ 4)$$ . Our proof relies on classical techniques from hyperbolic geometry, specifically leveraging calculations familiar to most graduate students, and an analysis of the fixed point sets of automorphisms of the three-punctured sphere. We propose a fresh geometric perspective on this centuries-old result, which parallels the key ideas found in Heath-Brown’s celebrated proof. As such this approach offers a new connection between number theory and hyperbolic geometry, enriching the understanding of Fermat’s theorem and opening potential avenues for further exploration.