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For any measure preserving system ( X ,    ,  μ , T ) and  A  ∈with  μ ( A ) &gt; 0, we show that there exist infinitely many primes  p  such that(the same holds with  p  − 1 replaced by  p  + 1). Furthermore, we show the existence of the limit in  L  2 ( μ ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form  p  − 1 (or  p  + 1) for some prime  p .

Multiple recurrence and convergence for sequences related to the prime numbers