## Sum of Squares

### Problem 273

Published on Saturday, 9th January 2010, 01:00 pm; Solved by 848; Difficulty rating: 70%Consider equations of the form: `a`^{2} + `b`^{2} = `N`, 0 ≤ `a` ≤ `b`, `a`, `b` and `N` integer.

For `N`=65 there are two solutions:

`a`=1, `b`=8 and `a`=4, `b`=7.

We call S(`N`) the sum of the values of `a` of all solutions of `a`^{2} + `b`^{2} = `N`, 0 ≤ `a` ≤ `b`, `a`, `b` and `N` integer.

Thus S(65) = 1 + 4 = 5.

Find ∑S(`N`), for all squarefree `N` only divisible by primes of the form 4`k`+1 with 4`k`+1 < 150.