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Each term in the sequence has a new prime divisor

Source: 2019 Pan-African Shortlist - N5

January 18, 2021
number theory

Problem Statement

Let n>1n > 1 be a positive integer. Prove that every term of the sequence n−1,nn−1,nn2−1,nn3−1,… n - 1, n^n - 1, n^{n^2} - 1, n^{n^3} - 1, \dots has a prime divisor that does not divide any of the previous terms.
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