On a problem of Terence Tao

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In this paper, we solve a problem of Terence Tao. We prove that for any K≥ 2 and sufficiently large N, the number of primes p between N and (1+1/K)N such that kp+jaⁱ+l is composite for all 1≤ a, |j|, k≤ K, 1≤ i ≤ K N and l in any set L =Lₙ{-KN,, KN} of cardinality K with jaⁱ+l≠0 is at least CₖN/ N, where Cₖ>0 depending only on K.