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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'.

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  • $\begingroup$ 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. $\endgroup$ Dec 2 '18 at 21:01
  • $\begingroup$ @YCor I added the french reference. $\endgroup$
    – Marco
    Dec 2 '18 at 21:06
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    $\begingroup$ 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. $\endgroup$ Dec 2 '18 at 23:14
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    $\begingroup$ Can you give a reference (or better: references) where this result is cited? $\endgroup$
    – GH from MO
    Dec 3 '18 at 0:12
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    $\begingroup$ @GHfromMO I just added the citing article. $\endgroup$
    – Marco
    Dec 3 '18 at 0:20
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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 $p^{1/10+\epsilon}$.

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