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(n, m)-good sets

Source: MEMO 2023 I4

August 24, 2023
number theoryMEMO2023

Problem Statement

Let n,mn, m be positive integers. A set SS of positive integers is called (n,m)(n, m)-good, if:
(1) m∈Sm \in S; (2) for all a∈Sa\in S, all divisors of aa are also in SS; (3) for all distinct a,b∈Sa, b \in S, an+bn∈Sa^n+b^n \in S.
For which (n,m)(n, m), the only (n,m)(n, m)-good set is N\mathbb{N}?
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