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linear recurrence sequence with prime coefficients

Source: 2007 Bulgarian Autumn Math Competition, Problem 12.4

March 17, 2022
prime numbersnumber theorylinear recurrence

Problem Statement

Let pp and qq be prime numbers and {an}n=1∞\{a_{n}\}_{n=1}^{\infty} be a sequence of integers defined by: a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n} \forall n\geq 0 Find pp and qq if there exists an integer kk such that a3k=−3a_{3k}=-3.
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