Let $x$ be a real number that satisfies $x^1 + x^{-1} = 3$ Prove that $x^n + x^{-n}$ is an positive integer , then prove that the positive integer $x^{3^{1437}}+x^{3^{-1437}}$ is divisible by at least $1439 \times 2^{1437}$ positive integers
Source: GMO 2016
Tags: GMO-Gulf Mathmatical Olympiad, number theory
Let $x$ be a real number that satisfies $x^1 + x^{-1} = 3$ Prove that $x^n + x^{-n}$ is an positive integer , then prove that the positive integer $x^{3^{1437}}+x^{3^{-1437}}$ is divisible by at least $1439 \times 2^{1437}$ positive integers