There are lots of questions here and on the web about polynomials that produce prime values. I'm asking the opposite: is there a nontrivial polynomial $p(n)$ that never produces primes?
Specifically, the polynomial must:
- be irreducible over the integers
- have coefficients that as a set have a greatest common divisor of 1
- evaluate to a composite number for all integer arguments.
I suspect that such a polynomial does not exist. I've tried using the set of prime factors of $p(0)$ (the constant term) and $p(1)$ (the sum of coefficients) to generate an argument $m$ such that $p(m)$ is relatively prime to each of $p(0)$ and $p(1)$, but I can't really get anywhere with that approach. I don't have any other ideas of what to try.
The problem is that a polynomial in $\mathbb{Z}[x]$ can be primitive, yet some $d>1$ divides all of the numbers $P(n)$. No surprise, the polynomial $\frac{P(x)}{d}$ must have all values at natural numbers integral. Such polynomials are characterized as linear combinations of $\frac{x(x-1)\cdots (x-k+1)}{k!}$ with integral coefficients. Now multiplying by some integers, we can ensure that the polynomial is in $\mathbb{Z}[x]$, primitive, yet the numbers $P(n)$ have a common divisor.