Sum of two squares theorem

A nonnegative integer is a sum of two integer squares exactly when every prime congruent to 3 modulo 4 has even exponent in its prime factorization.

Gaussian integers: The complex numbers whose real and imaginary parts are integers, forming the ring ℤ[i]. Unique factorization in ℤ[i] explains why primes congruent to 1 modulo 4 split into two squares.
Congruence: An equivalence relation expressing that two integers leave the same remainder upon division by a modulus. The condition distinguishes prime factors by their remainders modulo 4.
Representation function for sums of two squares: A function counting ordered integer pairs whose squares sum to a specified integer. The theorem determines when this count is nonzero.
Albert Girard: A French mathematician who stated in 1625 that primes expressible as two squares are 2 or congruent to 1 modulo 4. His statement anticipated the prime case underlying the theorem.
Fermat's theorem on sums of two squares: A theorem stating that a prime is a sum of two squares exactly when it is 2 or congruent to 1 modulo 4. The prime case supplies the central building block of the full criterion.
Fundamental theorem of arithmetic: The theorem that every integer greater than one has a unique prime factorization, up to ordering. The criterion examines the exponent of each prime in this unique factorization.
Gaussian integer norm: The function N(a+bi)=a²+b² on Gaussian integers. Representations as two squares are exactly integer values of this norm.
Pierre de Fermat: A seventeenth-century French mathematician who developed major results in number theory. Fermat gave a proof of the prime characterization later known as his two-squares theorem.
Euler's criterion for sums of two squares: A criterion stating that a prime congruent to 3 modulo 4 cannot divide a sum of two squares unless it divides both terms. Repeated divisibility forces such primes to occur to even exponents.
Prime number: An integer greater than one whose only positive divisors are one and itself. The theorem’s test is stated separately for each prime divisor.

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Brahmagupta–Fibonacci identityRelated: The identity supplies the multiplication step behind constructing such representations.