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Perfect square in sequence

Source: 2020 CMO P5

November 27, 2019

number theoryPerfect Square

Problem Statement

Given any positive integer

c

, denote

p(c)

as the largest prime factor of

c

. A sequence

\{a_n\}

of positive integers satisfies

a_1>1

and

a_{n+1}=a_n+p(a_n)

for all

n\ge 1

. Prove that there must exist at least one perfect square in sequence

\{a_n\}