# Roots on a Circle

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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