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Prove that $a_k-22$ where $a_{k+2} = a_{k+1}a_k + 1$

Source: Kyrgyzstan 2012, Problem 5

May 2, 2013
modular arithmeticnumber theory unsolvednumber theory

Problem Statement

The sequence of natural numbers is defined as follows: for any k≥1 k\geq 1 ,ak+2=ak+1⋅ak+1 a_{k+2}= a_{k+1}\cdot a_k+1 . Prove that for k≥9 k\geq 9 the number ak−22 a_k-22 is composite.
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