primes in integer valued polynomials.

Question

For what finite degree integer-valued polynomials P does P contain a finite number of primes?

If I have a polynomial of degree n say $P(n)=a_nx^n+a_{n-1}x^{n-1}...a_1x+a_0$ is there a bound over the number of primes in the progretion for $n \geq 2$?

Answer 1

So, you want polynomials $P(x)$ which take prime values infinitely often.

It is easy to see that for this to happen, $P(x)$ must satisfy the following conditions:

1. $a_n>0$ (For if $a_n<0$, $P(n)$ is eventually negative)

2. $P(x)$ irreducible in $\mathbb{Z}[x]$. (For if $P(x)$ is reducible in $\mathbb{Z}[x]$, say $P(x)=f(x)g(x)$, then for $P(x)$ to be prime, we require either $f(x)=\pm 1$ or $g(x)=\pm 1$, which have finitely many solutions.)

3. Write $P(x)$ as a integer combination of binomial coefficients $\sum_{i=0}^{n}{c_i\binom{x}{i}}$, then $\gcd(c_1, c_2, \ldots , c_n)=1$

This becomes Bunyakovsky's conjecture, which remains open. The degree $1$ case corresponds to Dirichlet's theorem on arithmetic progression, but not much has been proved for polynomials with higher degree.

Well... almost the same. Bunyakovsky's conjecture is concerning polynomials with integer coefficients, while you are looking at integer-valued polynomials. Shouldn't make much of a difference.