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MathNet 2024-part-01 — 168 problems (EN–VI) Question 64

Math Olympiad Vault · MathNet 2024-part-01 — 168 problems (EN–VI)

Question 64

Is it true that for any positive integer $n \gt 1$, there exists an infinite arithmetic progression $M_n$ of positive integers, such that for any $m \in M_n$, the number $n^m - 1$ is not a perfect power (a positive integer is a perfect power if it is of the form $a^b$ for positive integers $a, b \gt 1$)?