$\gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1$ where $p_n = n$th prime.

Question

How can I prove in general that, for all $n\geq 2$:

$$ \gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1 $$

Seems to always be true:

from sympy import *
from sympy.ntheory.generate import prime


def f(n):
    return n**5 - n**3 + n**2 - n +  1

for n in range(3, 100000000):
    p = prime(n-1)
    d = gcd(f(n), p)
    if d != 1:
        print (n)

Assume true.

$$ f(X) = X^5 - X^3 + X^2 - X + 1 $$

is irreducible and modulo each prime, i.e. $f(n+1) \neq 0 \pmod {p_n}$ so that $\prod_{j=2}^n\overline{f(j+1)} \in $ the units of $\Bbb{Z}/2 \times \Bbb{Z}/3 \times \Bbb{Z}/5 \times \dots $

So far it seems like the polynomials $g(X) = f(X \pm 1)$ are such that $g(n) \neq 0 \pmod {p_n}$