Question

Let:

$f(x)=x^n+2x^{n-1}+3x^{n-2}+4x^{n-3}+\ldots + (n-1)x^2+nx+(n+1)$.

Is $f(x)$ irreducible?

In light of the answers to this question, I now know that this is true when $n+1$ is prime. What about when $n+1$ is composite? I have checked a lot of cases and it seems to be true.

(Unnecessary background information: this is linked to this recent question of mine. At the point where I say:

"$g_n(x,−n)=x^2h(x)$,"

the $h(x)$ in question has the property that substituting $x-1$ for $x$ puts it in the form of $f(x)$ above. If I can show that $h(x)$ is irreducible, then I will have shown that the galois group of the original polynomial is doubly transitive. Not that I am trying to draw attention back to my original question!)

Answer 1

It is a conjecture that for any $k < n$ one has $$\frac{d^k}{dx^k}(1+x+\cdots+x^n)$$ is irreducible in $\mathbb Z[x]$. Your question is the case $k=1$. It is known that the set of integers $n\in [0,t]$ for which the polynomial $nx^{n-1}+(n-1)x^{n-2}+\cdots+1$ is reducible has size at most $O(t^{1/3+\epsilon})$, see the paper

A. Borisov, M. Filaseta, T. Y. Lam, O. Trifonov, "Classes of polynomials having only one non-cyclotomic irreducible factor", Acta Arith. 90 (1999) 121-153