# Are derivatives of geometric progressions all irreducible?

Consider the polynomials $P_n(x)=1+2x+3x^2+\dots+nx^{n-1}$. Problem A5 in 2014 Putnam competition was to prove that these polynomials are pairwise relatively prime. In the solution sheet there is the following remark:

It seems likely that the individual polynomials $P_k(x)$ are all irreducible, but this appears difficult to prove.

My question is exactly about this: is it known if all these polynomials are irreducible? Or is it an open problem?

In the article Classes of polynomials having only one non-cyclotomic irreducible factor the authors (A. Borisov, M. Filaseta, T. Y. Lam, and O. Trifonov) had proved for any $$\epsilon > 0$$ for all but $$O(t^{(1/3)+\epsilon})$$ positive integers $$n\leq t$$, the derivative of the polynomial $$f(x)= 1+ x + x^2 + \cdots + x^n$$ is irreducible, and in general for all $$n\in \mathbb N$$ they conjectured $$f'(x)$$ is irreducible.