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a_n = a^2_{n-1} + 15a_{n-1} has no perfect squares

Source: 2021 Saudi Arabia JBMO TST 2.1

September 4, 2021

number theoryrecurrence relation

Problem Statement

Let

(a_n)_{n\ge 1}

be a sequence given by

a_1 = 45

and

a_n = a^2_{n-1} + 15a_{n-1}

for

n > 1

. Prove that the sequence contains no perfect squares.