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First term is a square

Source: 2018 Greece National Olympiad Problem 1

March 3, 2018
SequencePerfect Square

Problem Statement

Let (xn),n∈N(x_n), n\in\mathbb{N} be a sequence such that xn+1=3xn3+xn,∀n∈Nx_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N} and x1=abx_1=\frac{a}{b} where a,ba,b are positive integers such that 3∤b3\not|b. If xmx_m is a square of a rational number for some positive integer mm, prove that x1x_1 is also a square of a rational number.
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