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Let (\mathcal {X},\mathscr {B}_\mathcal {X},\mu,T) be a measure-preserving probability system with T is invertible. Suppose that A\in \mathscr {B}_\mathcal {X} with \mu (A)>0 and \epsilon >0. For any m\geq 1, there exist infinitely many primes p_0,p_1,\ldots,p_m with p_0<\cdots 0 is a constant only depending on m, A and \epsilon. [ABSTRACT FROM AUTHOR]