number theory - Sums of two Gaussian Integer squares ...
math.stackexchange.com › questions › 3395338Oct 15, 2019 · Bookmark this question. Show activity on this post. The well known Sum of Two Squares Theorem states that an integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no prime congruent to 3 mod 4 raised to an odd power. I wondered about extending this to Gaussian Integers: which Gaussian Integers are expressible as the sum of two squares of two other Gaussian Integers?
Sum of two squares theorem - Wikipedia
https://en.wikipedia.org/wiki/Sum_of_two_squares_theoremThe numbers that can be represented as the sums of two squares form the integer sequences0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, ...They form the set of all norms of Gaussian integers; their square roots form the set of all lengths of line segments between pairs of points in the two-dimensional integer lattice. The number of representable numbers in the range from 0 to any number is proportional to , with …
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 Arit…
Gaussian integers + sums of squares
sites.math.washington.edu › ~mathcircle › circleSums of squares To prove the two square theorem, we’ll use the Gaussian integers and a couple of other ingredients: Theorem (Wilson’s theorem) If p is prime, then (p 1)! 1 mod p. For example, 4! = 24 4 1 mod 5: We only need Wilson’s theorem to prove: Lemma (Lagrange) If p is prime and p 1 mod 4, then there exists an integer m such that pjm2 + 1.