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another number theory problem

Source: GMO 2016

September 28, 2017
GMO-Gulf Mathmatical Olympiadnumber theory

Problem Statement

Let xx be a real number that satisfies x1+x−1=3x^1 + x^{-1} = 3 Prove that xn+x−nx^n + x^{-n} is an positive integer , then prove that the positive integer x31437+x3−1437x^{3^{1437}}+x^{3^{-1437}} is divisible by at least 1439×214371439 \times 2^{1437} positive integers
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