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 signs and 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).
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 signs and 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.
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).
to the problem of representing 
for 
any integer are achieved by means
of successive applications of the genus theorem
and composition theorem.
