Are derivatives of geometric progressions all irreducible?

Question

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?

Thanks in advance.

Answer 1

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.