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a_{n+1} = a_n + 2/a_n , sequence of rationals without square of rational

Source: 48th Austrian Mathematical Olympiad National Competition (Final Round, part 2) 24th May 2017 p3

May 25, 2019

number theoryrationalrecurrence relationnumber theory with sequences

Problem Statement

Let

(a_n)_{n\ge 0}

be the sequence of rational numbers with

a_0 = 2016

and

a_{n+1} = a_n + \frac{2}{a_n}

for all

n \ge 0

. Show that the sequence does not contain a square of a rational number.
Proposed by Theresia Eisenkölbl