Fermat's
theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem,"
states that a prime number can be represented in an essentially unique manner (up to
the order of addends) in the form for integer and iff or (which is a degenerate case with ). The theorem was stated by Fermat, but the first published
proof was by Euler.

The first few primes which are 1 or 2 (mod 4) are 2, 5, 13, 17, 29, 37, 41, 53,
61, ... (OEIS A002313) (with the only prime
congruent to 2 mod 4 being 2). The numbers such that equal these primes are (1, 1), (1, 2), (2, 3), (1, 4),
(2, 5), (1, 6), ... (OEIS A002331 and A002330).