Reducible Polynomials

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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