KnowraFermat's theorem on sums of two squaresFermat's theorem on sums of two squaresAn odd prime is a sum of two integer squares if and only if it is congruent to 1 modulo 4.BriefConnectFermat's little theorem: For a prime p and an integer a not divisible by p, a^(p−1) is congruent to 1 modulo p. Its quadratic-residue consequences help establish that −1 is a square modulo primes congruent to 1 modulo 4.Congruence (number theory): An equivalence relation saying two integers have the same remainder upon division by a modulus. The condition p ≡ 1 mod 4 selects exactly the odd primes covered by the representation.5: The prime number five. It is congruent to 1 modulo 4 and has the representation 5=1²+2².Pierre de Fermat: A seventeenth-century French mathematician who made foundational contributions to number theory. The theorem is named for his assertion about primes represented by two squares.Sum of two squares theorem: An integer is a sum of two squares exactly when each prime congruent to 3 modulo 4 occurs to an even exponent in its factorization. It generalizes the prime criterion to arbitrary positive integers.Quadratic reciprocity: A theorem describing when one prime is a square modulo another prime. It gives a broader framework for the residue condition underlying the theorem.Quadratic residue: A residue class modulo an integer that is congruent to a square modulo that integer. A representation p=a²+b² forces −1 to be a square modulo p when b is nonzero.13: The prime number thirteen. Its representation 13=2²+3² illustrates the theorem beyond the smallest case.Leonhard Euler: An eighteenth-century mathematician whose work shaped number theory, analysis, and mechanics. Euler supplied an early proof of Fermat’s claim using descent and arithmetic arguments.Fermat's theorem on sums of polygonal numbers: A theorem asserting that every positive integer is a sum of at most three triangular, four square, and k k-gonal numbers. It is a distinct Fermat-associated representation theorem, concerning polygonal numbers rather than prime squares.Show all 26Linked from 6 pagesFermat numberRelated: Every odd prime divisor of a Fermat number is congruent to 1 modulo a power of two, hence is a sum of two squares.Lagrange's four-square theoremCompared with: Its residue restriction contrasts with the four-square theorem's representation of every prime.Euler's four-square identityRelated: It concerns the arithmetic family whose multiplication law is the two-square analogue.Brahmagupta–Fibonacci identityRelated: The identity helps combine prime representations into representations of composite numbers.Show all 6