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exist distinct primes p_1, p_2, ..., p_n such that m + k is divisible by p_k

Source: 2021 Saudi Arabia Training Lists p40 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 5, 2022

number theorydivisible

Problem Statement

Given

m, n

such that

m > n^{n-1}

and the number

m+1

m+2

...

m+n

are composite. Prove that there exist distinct primes

p_1, p_2, ..., p_n

such that

m + k

is divisible by

p_k

for each

k = 1, 2, ...