C L's number
Let \(M\) be the smallest positive odd integer that is not a multiple of the square of any prime, and that can be expressed as a sum of squares of two integers in at least \(4\) distinct ways, ignoring signs and order. Find the last 3 digits of \(M\).
This problem is shared by C L.
Details and assumptions
If \(M=a^2+b^2\), then ignoring signs and order implies \(M = (-a)^2+b^2=b^2+a^2\) are not counted as distinct.