Two extremes of this problem are Fermat primes and Sophie Germain primes. If either class had infinitely many members, that would contribute to an answer of your question. There is literature about the distribution of prime factors (cf. Riesel and Knuth), but I do not know the literature regarding the restriction to numbers of the form prime - 1 .

If I had to start anywhere for questions like this, I would choose one or all three of the following:

Richard Guy's Unsolved problems in Number Theory,

Hans Riesel's book on computer methods for primality proving and factorization,

Prime Numbers: a Computational perspective, with by Crandall and Pomeranceas one of the authors.

Forgive my memory if the titles or authors are missing or incorrectly spelled.

1

Two extremes of this problem are Fermat primes and Sophie Germain primes. If either class had infinitely many members, that would contribute to an answer of your question. There is literature about the distribution of prime factors (cf. Riesel and Knuth), but I do not know the literature regarding the restriction to numbers of the form prime - 1 .

If I had to start anywhere for questions like this, I would choose one or all three of the following:

Richard Guy's Unsolved problems in Number Theory,

Hans Riesel's book on computer methods for primality proving and factorization,

Prime Numbers: a Computational perspective, with Pomerance as one of the authors

Forgive my memory if the titles or authors are missing or incorrectly spelled.