Roots on a Circle

Number Theory Level 5

Denote by \(N\) the number of pairs of positive integers \((n,m),\) such that \(m<n\leq 100\) and the polynomial \(x^n+x^m+1\) has a root on the unit circle. Find the last three digits of \(N.\)

Details and assumptions

A root on the unit circle is a complex root at the distance exactly \(1\) from the origin.

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