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Slovenia 2019 TST1 P4

Source: 2019 Slovenia 1st TST Problem 4

February 18, 2019
TSTnumber theory

Problem Statement

Let PP be the set of all prime numbers. Let AA be some subset of PP that has at least two elements. Let's say that for every positive integer nn the following statement holds: If we take nn different elements p1,p2...pn∈Ap_1,p_2...p_n \in A, every prime number that divides p1p2⋯pn−1p_1 p_2 \cdots p_n-1 is also an element of AA. Prove, that AA contains all prime numbers.
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