Proofs of Fermat's Theorem On Sums of Two Squares - Zagier's "one-sentence Proof"

Zagier's "one-sentence Proof"

If p = 4k + 1 is prime, then the set S = {(x, y, z) ∈ N3: x2 + 4yz = p} is finite and has two involutions: an obvious one (x, y, z) → (x, z, y), whose fixed points correspond to representations of p as a sum of two squares, and a more complicated one,

$(x,y,z)\mapsto \begin{cases} (x+2z, z, y-x-z),\quad \textrm{if}\,\,\, x < y-z \\ (2y-x, y, x-y+z),\quad \textrm{if}\,\,\, y-z < x < 2y\\ (x-2y, x-y+z, y),\quad \textrm{if}\,\,\, x > 2y \end{cases}$

which has exactly one fixed point, (1, 1, k); however, the number of fixed points of an involution of a finite set S has the same parity as the cardinality of S, so this number is odd for the first involution as well, proving that p is a sum of two squares.

This proof, due to Zagier, is a simplification of an earlier proof by Heath-Brown, which 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.

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