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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.

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4 Answers 4

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).

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Strange - the density of integers $n$ that have a prime factor greater than $n^{1/2}$ is $\log 2$ by results on friable (or smooth) numbers (see the Dickman rho function). I would think that the density of such shifted primes is also $\log 2$. I have proved this subject to a version of the Hardy-Littlewood prime tuples conjecture (see my J. Number Th. paper of 2002). –  Greg Martin Mar 4 '11 at 19:48
    
Mark, your conclusion that the number of primes less than x such that p-1 has a prime factor larger than $\sqrt(p)$ is asymptotic to x/(2ln(x)) is not correct. Note that we have a log(q) factor in the result not log(p) and for the relevant range of q we have $\log(p)/2 \leq \log(q) \leq \log(p)$ and they are not asymptotically equal. Greg is correct that the density should be $\log 2$. –  Johan Andersson Nov 26 '12 at 13:26
    
Johan, thanks! I've adjusted the text. –  Mark Lewko Nov 29 '12 at 5:52

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)

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I didn't understand. Did you leave the statement incomplete? What is the conclusion about the relative density? –  Vipul Naik Feb 9 '10 at 22:56
    
..."is positive"? –  David Hansen Feb 9 '10 at 23:06
    
Did you mean $\alpha \le 0.6687$ above instead of $\alpha > 0.6687$? The larger the $\alpha$, the smaller the density, and the paper you've quoted also seems to suggest that. –  Vipul Naik Feb 10 '10 at 1:07
    
@Vipul: I'm sure that I mean $\alpha > 0.6687$ since we're interested in $p-1$ have a large prime factor. –  Victor Miller Feb 10 '10 at 3:31
1  
@DavidHansen: perhaps I didn't state it clearly enough. Fouvry showed that there is an $\alpha$ which is approximately 0.6687 so that for a fixed $a \ne 0$ the set $\{p \text{ prime}: P(p+a) \ge p^{\alpha} \}$ has positive density in the set of all primes, where $P(n)$ denotes the larges prime factor dividing $n$. Bigger values of $\alpha$ are better. The post alluding to a result of Goldfeld had $\alpha = \frac 12$. –  Victor Miller Feb 15 '10 at 3:50

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

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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:

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