A prime number p can be expressed as a sum...

Every prime of the form 4n+1 is a sum of two squares.

This is known as the "two squares theorem"

http://en.wikipedia.org/wiki/Proofs_of_Fermat's_theorem_on_sums_of_two_squares

There's also the "four squares theorem", ie, every prime is a sum of four squares

http://en.wikipedia.org/wiki/Lagrange's_four-square_theorem