Erdős (see [4]) asked if there are infinitely many integers k which are not a difference or a sum of two powers, i.e., if there are infinitely many positive integers k with k≠|um ± vn| for u, v, m, n ε ℤ. This is certainly a very difficult problem. For example, it is known that the Catalan equation, i.e., the equation um − vn = 1 with uv≠0 and min (m, n)≥2 has only finitely many solutions (u, v, m, n), but there is no other positive integer k≥1 for which it is known that the equation

has only finitely many solution (u, v, m, n) with min (m, n)≥2.

We continue our study of the different representations of an integer n as a sum of two squares initiated in our final paper written with Paul Erdős [1]. We let r(n) denote the number of representations n = a2 + b2 counted in the usual way, that is, with regard to both the order and sign of a and b. We have

where x is the non-principal Dirichlet character (mod 4); moreover, r(n)≥0 if and only if n has no prime factor p ≡ 3 (mod 4) with odd exponent. We define the function b(n) on the sequence of representable numbers as the least possible value of |b|—for example, we have b(13) = 2,b(25) = 0, b(65) = 1—and we write

here and throughout the paper the star denotes that the sum is restricted to the representable integers. The problem considered here is to find an asymptotic formula for this sum, or, less ambitiously, to determine the order of magnitude of the function B(x).