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Prove that the set for which the number of prime divisors is bounded is finite

Source: The francophone mathematical olympiads P4

June 27, 2020
number theoryFrancophone

Problem Statement

Let (ai)i∈N(a_i)_{i\in \mathbb{N}} a sequence of positive integers, such that for any finite, non-empty subset SS of N\mathbb{N}, the integerΠk∈Sak−1\Pi_{k\in S} a_k -1is prime. Prove that the number of aia_i's with i∈Ni\in \mathbb{N} such that aia_i has less than mm distincts prime factors is finite.
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