Problem

Source: czechoslovak national mo round

Tags: number theory, prime numbers



Let $(a_n)_{n = 0}^{\infty} $ be a sequence of positive integers such that for every $n \geq 0$ it is true that $$a_{n+2} = a_0 a_1 + a_1 a_2 + ... + a_n a_{n+1} - 1 $$a) Prove that there exist a prime number which divides infinitely many $a_n$ b) Prove that there exist infinitely many such prime numbers