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exist distinct n primes p_i such that M+k is divisible by p_k for any k=,1...,n

Source: 2015 Belarus TST 8.1

November 7, 2020
number theoryprimesdivisibleconsecutive

Problem Statement

Given m,n∈Nm,n \in N such that M>nn−1M>n^{n-1} and the numbers m+1,m+2,...,m+nm+1, m+2, ..., m+n are composite. Prove that exist distinct primes p1,p2,...,pnp_1,p_2,...,p_n such that M+kM+k is divisible by pkp_k for any k=1,2,...,nk=1,2,...,n.
Tuymaada Olympiad 2004, C.A.Grimm. USA
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