Geometry of Fermat’s Sum of Squares

Greg McShane () and Vlad Sergiescu ()
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Greg McShane: Institut Fourier 100 rue des Maths
Vlad Sergiescu: Institut Fourier 100 rue des Maths

Chapter Chapter 6 in Essays on Topology, 2025, pp 71-88 from Springer

Abstract: Abstract 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.

Keywords: Hyperbolic geometry; Two squares theorem; 11D41; 11D72 (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-81414-3_6

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DOI: 10.1007/978-3-031-81414-3_6

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