C L's number

Number Theory Level 5

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.


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