TITLE: Factors of p-1 when p is prime.
QUESTION [17 upvotes]: Hi,
I have no idea where to look for, so I'm hoping you can give me some pointers.
I'm interested by numbers of form $p-1$ when $p$ is a prime number. Do they have a name, so that I can google them?
More precisely, I'm interested in their factors. Ok, obviously 2 is a factor, but what about the others? Are there a lot of small factors? Do we know the rate of the growth of its larger factor, is it linear, logarithmic?
Thanks.

REPLY [6 votes]: You should look at the paper by Erdos and Odlyzko "On the density of odd integers of the form $(p − 1)/2^n$ and related questions" in Journal of number theory, vol. 11 (1979) pp 257-263.  Among other things they prove the the set of odd divisors of $p-1$ (where $p$ runs over primes) has a positive density.
As far as the largest prime factor of $p-1$ there is a result of Fouvry "Theoreme de Brun-Titchmarsh: application au theoreme de Fermat", Invent. Math. v. 79 (1985) pp 383-407, in which he proves that the set of primes $p$ for which the largest prime factor of $p-1$ is $\ge p^{.6687}$ has positive relative density in the set of all primes.