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sequence a_{n+3} = a_{n+2}a_{n+1} + a_n

Source: Problem 5, Polish NO 1988

October 16, 2005
inductionpigeonhole principlemodular arithmeticnumber theory unsolvednumber theory

Problem Statement

The sequence a1,a2,a3,...a_1, a_2, a_3, ... is defined by a1=a2=a3=1a_1 = a_2 = a_3 = 1, an+3=an+2an+1+ana_{n+3} = a_{n+2}a_{n+1} + a_n. Show that for any positive integer rr we can find ss such that asa_s is a multiple of rr.
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