On a problem of Terence Tao

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 $1\leq a, |j|, k\leq K$,$1\leq i \leq K\log N$and$l$inanyset$L =L_{N}\subseteq{-KN,\cdots, KN}$ofcardinality$K$with$ja^{i}+l\neq0$ is at least $C_K\frac{N}{\log N}$, where $C_K>0$ depending only on $K$.