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I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly.

I don't really know much about sieves, but I became intrigued when I read over the notes on Brun's Sieve. I remember hearing Chen's result that there are infinitely primes $p$ such that $p+2$ is either a near-prime (i.e. the product of two primes) or prime, but I've never pursued its proof. In these notes on Brun's Sieve, an exercise is assigned that is similar in spirit, though (almost certainly) much easier and direct (I paraphrase a little):

The Task

Use Brun's Sieve to show that there are infinitely many primes $p$ such that $p+2$ is the product of at most $20$ primes.

I couldn't manage to solve this exercise (in fact, I got a bit lost). So I opened up my Iwaniec-Kowalski, which I believe may be where this problem was sourced for MIT's OCW. It is assigned as exercise 2 in Chapter 6, with a bit more background (I wouldn't exactly say that Iwaniec-Kowalski ever provides much exposition, but it's certainly been a good reference for me in the past). But I still wasn't able to provide a proof.

I'm aware that it hinges around the idea of considering a sequence/arithmetic function

$$ a_n = \begin{cases} 1 & n + 2 \text{ is prime} \\ 0 & \text{else}\end{cases}$$

and then, apparently, sieving using Brun's Sieve on the interval $[1,x]$ excluding primes on $[1, x^{1/20}]$. When I try to perform the sieving, I get caught up in circular arguments concerning what MIT's notes and I-K both call $V(z)$, so I must be parsing things incorrectly. But I also highly suspect that this is a very straightforward exercise.

My immediate goal is twofold:

  1. Learn how to use Brun's Sieve to do the above task, and
  2. See how far we can push Brun's Sieve to get similar results for $p+2$ as the product of fewer than $20$ primes.

If you happen to be very familiar with these sieves, I would also be interested in knowing why one might use Brun's Sieve as opposed to, say, Selberg's Sieve (which was recently put to good use by Yitang Zhang). But I think this is perhaps a very different matter.

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    $\begingroup$ Section 1 of terrytao.wordpress.com/2012/03/01/… works through a slightly easier exercise: There are infinitely many $n$ so that $n$ and $n+2$ both have at most $20$ prime factors. I'm afraid that I don't understand the proof well enough to give a hint without giving an answer. $\endgroup$ – David E Speyer Oct 9 '13 at 13:21
  • $\begingroup$ Halberstam and Richert in Sieve Methods (Dover, 2011) prove using Brun's sieve that there are infinitely many p such that p+2 has at most 8 prime factors. Including some introductory material the exposition takes 67 pages. The key is their definition of the characteristic function on p. 58. The basic idea is simple enough but doesn't look like it lends itself to anything one could describe as a "straightforward exercise." $\endgroup$ – daniel Feb 7 '16 at 8:22

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