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(\sqrt{2}- 1)^n = \sqrt{m} - \sqrt{m-1}

Source: Polish MO Finals 1953 p4

August 29, 2024
algebranumber theory

Problem Statement

Prove that if n n is a natural number, then equality holds (21)n=mm1(\sqrt{2}- 1)^n = \sqrt{m} - \sqrt{m-1} where mm is a natural number.