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Suppose thatis a set of primes, such that for every, every prime factor of  p −1 is also in. We apply a new sieve method to show that eithercontains all of the primes or the counting function ofis  O ( x  1− c  ) for some  c &gt;0, where  c  depends only on the smallest prime not in. Our proof makes use of results connected with Artin's primitive root conjecture.

Sieving Very Thin Sets of Primes, and Pratt Trees with Missing Primes