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.

Read more about this topic: Proofs Of Fermat's Theorem On Sums Of Two Squares

Famous quotes containing the word proof:

“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)