Are there positive integers such that there exist at least positive integers such that both and are perfect squares?

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#
number theory

# ELMO 2012 SL N9

# Continued Fractions and Linear Fractional Transformations

Are there positive integers such that there exist at least positive integers such that both and are perfect squares?

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This is an article written by Evan O’Dorney for Intel STS 2011. It deals with rational approximation to some square roots. The article is at here