# Reference Request for a result on divisors of $p-1$

I have seen this result in several places without an English reference:

There exist infinitely many primes $$p$$ such that $$p-1=2q_1q_2$$ where $$q_1$$ and $$q_2$$ are prime numbers with $$q_1,q_2>p^{1/4}$$.

There is a French reference (E. Bombieri. Le Grand Crible dans la Theorie Analytique des Nombres. Asterisque 18 Societe Mathematique de France 1974). However, I have not been able to find. I am wondering if someone knows an English reference for this claim or knows of similar results about small number of divisors of $$p-1$$.

Update: the french reference produced in the answer does not include this statement. I expected the result to appear there as stated by Murty in this article on page 14 he mentions this statement (with $$1/4$$ replaced by $$\theta>1/4$$) and asks the reader to consult the mentioned reference for 'technical details'.

• I don't know of a reference that mentions this exact result, but such results are corollaries of much more general statements about almost prime sieves. One should find such statements in say Opera de Cribro. Dec 2, 2018 at 21:01
• @YCor I added the french reference. Dec 2, 2018 at 21:06
• I don't think that the required result has been proved. Similarly, it is open that there are infinitely many primes $p$ with $p+2=q_1q_2$ for some primes $q_1,q_2$ though there are infinitely many (Chen) primes $p$ with $p+2$ a product of at most two primes. Dec 2, 2018 at 23:14
• Can you give a reference (or better: references) where this result is cited? Dec 3, 2018 at 0:12
• @GHfromMO I just added the citing article. Dec 3, 2018 at 0:20

If you allow $$(p-1)/2$$ to be prime, not just a product of two primes exceeding $$p^{1/4+\epsilon}$$, then the result is contained in somewhat stronger form in Heath-Brown: Artin's conjecture for primitive roots (Quart. J. Math. Oxford 37 (1986), 27-38). See Lemma 1 in that paper, and apply it with $$k=1$$, $$u=3$$, $$v=16$$. The result is based on the deep work of Bombieri-Friedlander-Iwaniec (1984). I believe this is the state-of-the-art. Heath-Brown also mentions that Gupta-Murty arrived at a similar conclusion but with at most three prime factors exceeding $$p^{1/10+\epsilon}$$.