# Reducible Polynomials

Number Theory Level 5

Find the number of pairs of integers $$(n,m)$$ with $$0\leq m < n \leq 25$$ such that the polynomial $f_{n,m}(x)=x^n+...+x^{m+1}+2x^m+...+2$ can be expressed as a product of two non-constant polynomials with integer coefficients.

Details and assumptions

The notation above means that the coefficient of $$x^i$$ in $$f_{n,m}(x)$$ equals $$1$$ if $$m<i\leq n$$ and $$2$$ if $$0\leq i\leq m.$$

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