The gaps between sums of two squares

Abstract

Problems concerning the setof numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identitywhich shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.