Lattice points on a circle

 Published on Friday, 20th February 2009, 09:00 pm; Solved by 2199;
Difficulty rating: 70%

Problem 233

Let f(N) be the number of points with integer coordinates that are on a circle passing through (0,0), (N,0),(0,N), and (N,N).

It can be shown that f(10000) = 36.

What is the sum of all positive integers N ≤ 1011 such that f(N) = 420 ?