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\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}} = rational (2021 Auckland MO S3)

Source:

November 15, 2021
algebraNested Radicalsradical

Problem Statement

For how many integers nn between 1 1 and 20212021 does the infinite nested expression n+n+n+...\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}} give a rational number?
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