in turn was inspired by a proof of Liouville. The technique of the proof is a combinatorial analogue of the topological principle that the Euler characteristics of a topological space with an involution and of its fixed-point set have the same parity and is reminiscent of the use of sign-reversing involutions in the proofs of combinatorial bijections. This proof is equivalent to a geometric or "visual" proof using "windmill" figures, given by Alexander Spivak in 2006 and described in this MathOverflow post and this Mathologer YouTube video . Proof with partition theory In 2016, A. David Christopher gave a partition-theoretic proof by considering partitions of the odd prime having exactly two sizes , each occurring exactly times, and by showing that at least one such partition exists if is congruent to 1 modulo 4. See also Legendre's three-square theorem Lagrange's four-square theorem Landau–Ramanujan constant Thue's lemma Friedlander–Iwaniec theorem References *Richard Dedekind, The theory of algebraic integers. L. E. Dickson. History of the Theory of Numbers Vol. 2. Chelsea Publishing Co., New York 1920 Harold M. Edwards, Fermat's Last Theorem. A genetic introduction to algebraic number theory. Graduate Texts in Mathematics no. 50, Springer-Verlag, NY, 1977. C. F. Gauss, Disquisitiones Arithmeticae (English Edition). Transl. by Arthur A. Clarke. Springer-Verlag, 1986. D. R. Heath-Brown, Fermat's two squares theorem. Invariant, 11 (1984) pp. 3–5. John Stillwell, Introduction to Theory of Algebraic Integers by Richard Dedekind. Cambridge Mathematical Library, Cambridge University Press, 1996. Don Zagier, A one-sentence proof that every prime p ≡ 1 mod 4 is a sum of two squares. Amer. Math. Monthly 97 (1990), no. 2, 144, Notes External links Two more proofs at PlanetMath.org Fermat's two squares theorem, D. R. Heath-Brown, 1984. Additive number theory Squares in number theory Theorems in number theory