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irish lucas series

Source: irish 2011 paper 1

May 6, 2012

inductionmodular arithmeticnumber theorygreatest common divisoralgebra unsolvedalgebra

Problem Statement

The integers

a_0, a_1, a_2, a_3,\ldots

are defined as follows:

a_0 = 1

a_1 = 3

, and

a_{n+1} = a_n + a_{n-1}

for all

n \ge 1

. Find all integers

n \ge 1

for which

na_{n+1} + a_n

and

na_n + a_{n-1}

share a common factor greater than

1