Problem

Source: 2015 Final Korean Mathematical Olympiad Day 2 Problem 5

Tags: number theory, Recurrence



For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$. They are defined inductively, by the following recurrences. $A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$ $B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$ Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.