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b_k = a_k + 9 and a_{k+1} = 8b_k + 8, a_k does not have perfect squares

Source: Austrian Polish 1988 APMC

April 30, 2020

Perfect Squarenumber theoryrecurrence relationSequence

Problem Statement

Two sequences

(a_k)_{k\ge 0}

and

(b_k)_{k\ge 0}

of integers are given by

b_k = a_k + 9

and

a_{k+1} = 8b_k + 8

for

k\ge 0

. Suppose that the number

1988

occurs in one of these sequences. Show that the sequence

(a_k)

does not contain any nonzero perfect square.