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2021 EGMO P1: {m, 2m+1, 3m} is fantabulous

Source: 2021 EGMO P1

April 13, 2021
EGMO 2021combinatoricsEGMOSets

Problem Statement

The number 2021 is fantabulous. For any positive integer mm, if any element of the set {m,2m+1,3m}\{m, 2m+1, 3m\} is fantabulous, then all the elements are fantabulous. Does it follow that the number 202120212021^{2021} is fantabulous?
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