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Sequence and prime divisors

Source: Middle European Mathematical Olympiad 2012 - Individuals I-4

September 14, 2012
number theoryprime numbersnumber theory proposed

Problem Statement

The sequence {an}n≥0 \{ a_n \} _ { n \ge 0 } is defined by a0=2,a1=4 a_0 = 2 , a_1 = 4 and an+1=anan−12+an+an−1 a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} for all positive integers n n . Determine all prime numbers p p for which there exists a positive integer m m such that p p divides the number am−1 a_m - 1 .
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