On a problem of Terence Tao

Abstract

In this paper, we solve a problem of Terence Tao. We prove that for any $K\ge 2$ and sufficiently large $N$, the number of primes $p$ between $N$ and $(1+\frac{1}{K})N$ such that $\mid kp+j{a}^i+l\mid$ is composite for all $1\le a,|j|,k\le K$, $1\le i\le K\log N$ and $l$ in any set $L=L_N\subseteq \{-KN,\cdots ,KN\}$ of cardinality $K$ with $j{a}^i+l\ne 0$ is at least $C_K\frac{N}{\log N}$, where $C_K>0$ depending only on $K$.