Problem

Source: Romanian TST 4 2007, Problem 4

Tags: algebra, polynomial, modular arithmetic, induction, number theory proposed, number theory



i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime. ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime. Dan Schwarz