Fermat's theorem on sums of two squares - Wikipedia
https://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squaresFermat usually did not write down proofs of his claims, and he did not provide a proof of this statement. The first proof was found by Euler after much effort and is based on infinite descent. He announced it in two letters to Goldbach, on May 6, 1747 and on April 12, 1749; he published the detailed proof in two articles (between 1752 and 1755). Lagrange gave a proof in 1775 that was based on his study of quadratic forms. This proof was simplified by Gauss in his Disquisitiones Ar…
Sums of Squares
www2.math.ou.edu › ~kmartin › intro-ntTheorem 4.1.6. (Fermat’s two square theorem) Let n 2 N.Thenn is a sum of two squares, i.e., n = x2 + y2 for some x,y 2 Z,ifandonlyifeachprimewhichis3 mod 4 appears to an even power in the prime-power factorization of n. Proof. Let us write the prime-power factorization of n as n = Y pei i Y qfj j where each pi ⌘ 3 mod 4 and each qj is 2 or ...
Sums of Squares - OU Math
www2.math.ou.edu/~kmartin/intro-nt/ch4.pdfsquare, whence a sum of two squares. Also, by Theorem 4.1.5, we know each qj is a sum of two squares. Then by the composition law, n is a sum of two squares. ()) To prove the converse direction, we essentially want a kind of converse to the composition law—that if rs is a sum of two squares then r and s must each be sums of two squares.