MathDB

8

Part of 2020 New Zealand MO

Problems(1)

a_{n+2} = xa_{n+1} - a_n , a_1 = x + 1, a_0 = 1

Source: New Zealand MO 2020 Round 1 p8

9/20/2021
For a positive integer xx, define a sequence a0,a1,a2,...a_0, a_1, a_2, . . . according to the following rules:
a0=1a_0 = 1, a1=x+1a_1 = x + 1 and an+2=xan+1−ana_{n+2} = xa_{n+1} - a_n for all n≥0n \ge 0.
Prove that there exist infinitely many positive integers x such that this sequence does not contain a prime number.
algebraSequencerecurrence relationnumber theory