Problem

Source: 2020 Austrian National Competition for Advanced Students, Part 2 problem 3

Tags: number theory, prime numbers, Sequence, Austria



Let a be a fixed positive integer and (en) the sequence, which is defined by e0=1 and en=a+n1k=0ekfor n1. Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence. (Theresia Eisenkölbl)