Solution June 03, 2007


i) Find all infinite arithmetic progressions of positive integers (d_n)_{n\in\mathbb{N}} such that dp is prime for all sufficiently large primes p.

ii) Find all polynomials f(X) \in \mathbb{Z}[X] such that \left|f(p)\right| is prime for all sufficiently large primes p.


i) is a special case of ii) so we will only solve ii) below.

Let f(x)\, satisfy the condition in ii).

f(x) = x\, and f(x) = -x\, are obviously solutions.

Assume that f(x)\neq \pm x. Then there is a prime p\, with q:=|f(p)|\neq p\,. Then \gcd(p,q)=1\, so the sequence a_n:=n\cdot q+p contains infinitely many primes by Dirichlet's theorem ('s_theorem_on_arithmetic_progressions).

Let a_n\, be prime. Because f\, is a polynomial with integer coefficients, f(a_n)-f(p)\, is divisible by a_n-p\,, which in turn is divisible by q\, by the definition of a_n\,. We also have f(p)\equiv 0 \pmod q so f(a_n)\equiv 0\pmod q. But f(a_n)\, must be prime so that |f(a_n)| = q\,. So f(a_n)\, assumes one of the values -q\, or +q\, infinitely often which means that it is a constant polynomial.

f(x) = \pm q for prime numbers q\, are indeed solutions of ii). There are no other solutions.