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.


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

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

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 $\frac{x}{2\ln x}$.

  • $\begingroup$ 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). $\endgroup$ Mar 4, 2011 at 19:48
  • $\begingroup$ 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 $\log 2$. $\endgroup$ Nov 26, 2012 at 13:26
  • $\begingroup$ Johan, thanks! I've adjusted the text. $\endgroup$
    – Mark Lewko
    Nov 29, 2012 at 5:52

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


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, Théorème de Brun-Titchmarsh: application au théorème de Fermat, Invent. Math 79 (1985), 383–407. MR0778134 (86g:11052)

  • $\begingroup$ I didn't understand. Did you leave the statement incomplete? What is the conclusion about the relative density? $\endgroup$
    – Vipul Naik
    Feb 9, 2010 at 22:56
  • $\begingroup$ ..."is positive"? $\endgroup$ Feb 9, 2010 at 23:06
  • $\begingroup$ 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. $\endgroup$
    – Vipul Naik
    Feb 10, 2010 at 1:07
  • $\begingroup$ @Vipul: I'm sure that I mean $\alpha > 0.6687$ since we're interested in $p-1$ have a large prime factor. $\endgroup$ Feb 10, 2010 at 3:31
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    $\begingroup$ @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 $\alpha = \frac 12$. $\endgroup$ Feb 15, 2010 at 3:50

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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