The fundamental theorem of arithmetic states that every integer can be resolved into a unique product of primes. Thus

777 = 3 x 7 x 37

and there is no other decomposition. This theorem is the foundation for higher arithmetic. Obviously, if 2 were not a prime number - a special one, since it is the only even prime number - on even integer can be resolved into prime numbers!

Ignoring the special prime 2, all primes belong to either the class :

5,13,17,29,37,41,…., of the form 4n+1

or the class :

3,7,11,19,23,31,…., of the form 4n+3.

Fermat's two-square theorem states that all the prime numbers belonging to the class 4n+1 alone can be expressed as the sum of two integral squares:

i.e., 5 = 12 + 22 ;	13 = 22 + 32 ;	17 = 12 + 42;
29 = 22 + 52;	37 = 12 + 62 ;	41 = 42 + 52;