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primes in a sequence

Source: STEMS 2021 Math Cat C Q3

January 25, 2021
Analytic Number Theory

Problem Statement

Let p∈N∖{0,1}p \in \mathbb{N} \setminus \{0, 1\} be a fixed positive integer. Prove that for every K>0K > 0, there exist infinitely many nn and NN such that there are atleast KNlog⁡(N)\dfrac{KN}{\log(N)} primes among the following NN numbers given by n+1,n+2p,n+3p,⋯ ,n+Np.n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.
Proposed by Bimit Mandal
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