 Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes formula_187 and formula_188.Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not.This is a consequence of Fermat's theorem on sums of two squares,
which states that an odd prime formula_20 is expressible as the sum of two squares, formula_190, and therefore factorizable as formula_191, exactly when formula_20 is 1 mod 4.