Proofs Of Fermat's Theorem On Sums Of Two Squares
Fermat's theorem on sums of two squares asserts that an odd prime number p can be expressed as
with integer x and y if and only if p is congruent to 1 (mod 4). The statement was announced by Fermat in 1640, but he supplied no proof.
The "only if" clause is easy: a perfect square is congruent to 0 or 1 modulo 4, hence a sum of two squares is congruent to 0, 1, or 2. An odd prime number is congruent to either 1 or 3 modulo 4, and the second possibility has just been ruled out. The first proof that such a representation exists was given by Leonhard Euler in 1747 and was quite complicated. Since then, many different proofs have been found. Among them, the proof using Minkowski's theorem about convex sets and Don Zagier's stunningly short proof based on involutions especially stand out.
Read more about Proofs Of Fermat's Theorem On Sums Of Two Squares: Euler's Proof By Infinite Descent, Lagrange's Proof Through Quadratic Forms, Dedekind's Two Proofs Using Gaussian Integers, Zagier's "one-sentence Proof", References
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