Primes p for which p - 1 has a large prime factor - MathOverflow

Primes p for which p - 1 has a large prime factor

2010-02-09T19:00:00Z

What are the best known density results and conjectures for primes p where p - 1 has a large prime factor q, where by "large" I mean something greater than $\sqrt{p}$.

The most extreme case is that of a safe prime (Wikipedia entry), which is a prime p such that $(p - 1)/2$ is also a prime (the smaller prime is called a Sophie Germain prime). I believe it is conjectured (and not yet proved) that infinitely many safe primes exist, and that the density is roughly $c/\log^2 n$ for some constant $c$ (as it should be from a probabilistic model).

For the more general setting, where we are interested in the density of primes p for which p - 1 has a large prime factor, the only general approach I am aware of is the prime number theorem for arithmetic progressions, and some of its strengthenings such as the Bombieri-Vinogradov theorem (conditional to the GRH), the (still open) Elliott-Halberstam conjecture, Chowla's conjecture on the first Dirichlet prime, and some partial results related to this conjecture. All of these deal with the existence of primes $p \equiv a \pmod q$ for arbitrary q and arbitrary a that is coprime to q.

My question: can we expect qualitatively better results for the situation where q is prime and $a = 1$? Also, I am not interested in specifying q beforehand, so the existence of a p such that there exists any large prime q dividing $p - 1$ would be great. References to existing conjectures, conditional results, and unconditional results would be greatly appreciated.

Answer by Felipe Voloch for Primes p for which p - 1 has a large prime factor

2010-02-09T19:39:08Z

I believe that sieve methods (in particular methods similar to Chen's result on "almost" Goldbach and twin primes) should give infinitely primes $p$ such $(p-1)/2$ is a product of two primes. That's more than what you asked.

Wikipedia quotes a result of Bombieri-Friedlander-Iwaniec stating that Linnik's constant is $2$ for almost all moduli. If the same is true for infinitely many prime moduli $q$, then you are in business. A prime $p \equiv 1 \mod q, p \ll q^2$ is what you want.

Maybe an analytic number theorist will come along and give precise references.

Wikipedia - Sieve Methods

Wikipedia - Linnik's Theorem

Answer by Victor Miller for Primes p for which p - 1 has a large prime factor

2010-02-09T22:48:45Z

Fouvry showed that the relative density is positive of primes $p$ for which the largest prime factor of $p+a$ is $\ge p^{\alpha}$ for $\alpha \approx .6687$.

Etienne Fouvry, Th ́eoreme de Brun-Titchmarsh; application au th ́eoreme de Fermat, Invent. Math 79 (1985), 383–407. MR0778134 (86g:11052)

Answer by MRA for Primes p for which p - 1 has a large prime factor

2010-02-10T00:22:54Z

John A. Gordon introduced the notion of strong primes (try Wikipedia) which, beside other requirements, are primes $p$ satisfying $p \equiv 1 \mod r$ for some large prime $r$ of about the same size as $p$. In the following paper, it has been shown how to construct strong primes of arbitrary bit size efficiently and with high probability:

John A. Gordon: "Strong Primes Are Easy to Find", Proceedings of EUROCRYPT '84, LNCS 209, Springer, 1985.

Answer by Mark Lewko for Primes p for which p - 1 has a large prime factor

2010-02-12T08:33:51Z

See "On the number of primes p for which p+a has a large prime factor." (Goldfeld, Mathematika 16 1969 23--27.) Using Bombieri-Vinogradov he proves, for a fixed integer a, that

$\sum_{p \leq x} \sum_{ x^{1/2}< q \leq x : q | p+a} ln(q) = x/2 + O(x ln ln x / ln(x))$

where the summation is over p and q prime. Note that this implies that the number of primes $p$ less than $x$ such that $p-1$ has a prime factor greater than $p^{1/2}$ is asymptotically at least x/ 2ln(x).