MathDB

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

Source: New Zealand MO 2020 Round 1 p8

September 20, 2021

algebraSequencerecurrence relationnumber theory

Problem Statement

For a positive integer

x

, define a sequence

a_0, a_1, a_2, . . .

according to the following rules:

a_0 = 1

a_1 = x + 1

and

a_{n+2} = xa_{n+1} - a_n

for all

n \ge 0

.
Prove that there exist infinitely many positive integers x such that this sequence does not contain a prime number.