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Irrationals and powers

Source: 2016 Nigerian Senior MO Round 2

November 16, 2022
inductionIrrational numbersalgebra

Problem Statement

Prove that (2+3)2n+(2−3)2n(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n} is an even integer and that (2+3)2n−(2−3)2n=w3(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3} for some positive integer ww, for all integers n≥1n \geq 1.
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