Weaker than twin prime conjecture

puuhikki · Yang-Mills Theory Offline Joined: 06 Jun 2005 Posts: 849

I was wondering if the following has been proved:

Do there exists $i\in\mathbb{N}$ such that there are infinitely many sets $A_i(n) = \{n,n + 1,\cdots ,n + i\}$ that contains at least two primes, as $n = 1,2,\ldots$ ?

**Posted:** Tue Jun 17, 2008 5:21 pm

Rust

It is open problem. $i=2$ problem of twins.

**Posted:** Tue Jun 17, 2008 7:40 pm

ZetaX

It is _not_ open. The case $i=16$ was shown by Goldston, Pintz and Yildirim.

**Posted:** Wed Jun 18, 2008 2:12 am

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Albanian Eagle

Doesn't their proof assume the Elliot-Halberstam conjecture? (for 16 at least?)
and as far as I know that hasn't been proven. It is "somehow" deeper than the Generalized Riemann Hypothesis..

**Posted:** Wed Jun 18, 2008 2:42 am

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maxal · Riemann Hypothesis Offline Joined: 09 Apr 2005 Posts: 373

They do assume the Elliot-Halberstam conjecture. See http://arxiv.org/abs/math.NT/0508185 for details.

**Posted:** Wed Jun 18, 2008 12:29 pm