Integral points on circles

Sixty years ago the first named author gave an example [Sch58] of a circle passing through an arbitrary number of integral points. Now we shall prove: The number N of integral points on the circle (x − a) 2 + (y − b) 2 = r 2 with radius r = 1 n √ m, where m, n ∈ Z, m, n > 0, gcd(m, n2 ) squarefree and a, b ∈ Q does not exceed r(m)/4, where r(m) is the number of representations of m as the sum of two squares, unless n|2 and n · (a, b) ∈ Z 2 ; then N ≤ r(m).