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Prove that all the terms are integer

Source: Brazilian M.O. 2004

October 18, 2004

inductionmodular arithmeticalgebrapolynomialnumber theory unsolvednumber theory

Problem Statement

Consider the sequence

(a_n)_{n\in \mathbb{N}}

with

a_0=a_1=a_2=a_3=1

and

a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}

. Prove that all the terms of this sequence are integer numbers.