# Solution June 03, 2007

### Problem

i) Find all infinite arithmetic progressions of positive integers such that *d*_{p} is prime for all sufficiently large primes *p*.

ii) Find all polynomials such that is prime for all sufficiently large primes *p*.

### Solution

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

Let satisfy the condition in ii).

and are obviously solutions.

Assume that . Then there is a prime with .
Then so the sequence contains infinitely many primes by Dirichlet's theorem (*http://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions*).

Let be prime. Because is a polynomial with integer coefficients, is divisible by , which in turn is divisible by by the definition of . We also have so . But must be prime so that . So assumes one of the values or infinitely often which means that it is a constant polynomial.

for prime numbers are indeed solutions of ii). There are no other solutions.