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1,sqrt[n]{n},sqrt[m]{m} linearly dependent over Q

Source: VJIMC 2008 2.3

June 16, 2021
number theory

Problem Statement

Find all pairs of natural numbers (n,m)(n,m) with 1<n<m1<n<m such that the numbers 11, nn\sqrt[n]n and mm\sqrt[m]m are linearly dependent over the field of rational numbers Q\mathbb Q.
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