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VMO 2018 P6

Source: Vietnam MO 2nd day 2nd problem

January 12, 2018
Sequencenumber theory

Problem Statement

The sequence (xn)(x_n) is defined as follows: x0=2, x1=1, xn+2=xn+1+xnx_0=2,\, x_1=1,\, x_{n+2}=x_{n+1}+x_n for every non-negative integer nn. a. For each n≥1n\geq 1, prove that xnx_n is a prime number only if nn is a prime number or nn has no odd prime divisors b. Find all non-negative pairs of integers (m,n)(m,n) such that xm∣xnx_m|x_n.
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