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- Mar 22, 2013

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I don't know if such thread has been created, all I can find out is one mentioning Zhang's initial bound of $7 \times 10^7$. This has been greatly improved by now so I thought it is worthwhile to post it here as well as the resources which I somehow collected from here and there.

(1915) V. Brun showed that the sum of reciprocals of twin primes converges

(1940) Erdos showed

(2004-2005) Goldston-Pintz-Yildirim showed that the constant can be assumed to be arbitrarily small, i.e., $\lim \inf \frac{p_{n+1} - p_n}{\log p_n} = 0$ and further that there are infinitely many prime pairs with gap 16, assuming Elliot-Halberstam conjecture.

I might be missing something, but these are the most major improvements I can recall.

(2013) Y. Zhang showed that that there are infinitely many prime pairs with some gap smaller than $7 \times 10^7$.

After some explicit reduction on the bound by Tao and Morrison, Tao announced a proposal of polymath project

The current best known trustworthy result is 300 by Clark & Jarvis

**History; a glance through the past**(1915) V. Brun showed that the sum of reciprocals of twin primes converges

^{[1]}, quite the opposite of primes. This was a major result in twin prime history. The rough implication of the result was that there are not too many twin primes there up to some $N$. Indeed, a consequence of the result was that there are $k N/\log^2 N$ twin primes for some constant $k > 0$. The method uses some basic sieve theoretic (combinatorial) methods which at present is named upon him, i.e., Brun sieve.(1940) Erdos showed

^{[11]}that $\frac{p_{n+1} - p_n}{\log p_n} \leq k$ for some constant $k \leq 1$ and it was improved greatly by Goldston-Pintz-Yildirim by showing that $k \approx 0.08578$.(2004-2005) Goldston-Pintz-Yildirim showed that the constant can be assumed to be arbitrarily small, i.e., $\lim \inf \frac{p_{n+1} - p_n}{\log p_n} = 0$ and further that there are infinitely many prime pairs with gap 16, assuming Elliot-Halberstam conjecture.

^{[2],[3]}I might be missing something, but these are the most major improvements I can recall.

**Present; to be written in mathematical history**(2013) Y. Zhang showed that that there are infinitely many prime pairs with some gap smaller than $7 \times 10^7$.

^{[4]}This was the smallest unconditional ever obtained. In a similar fashion, although much explicit, Tao proves a more general result^{[9]}tightening the bound to 57554086.After some explicit reduction on the bound by Tao and Morrison, Tao announced a proposal of polymath project

^{[5],[6]}. The best unconditional result, upto July 5, was 5414 which was greatly improved by Maynard's works^{[7]}.The current best known trustworthy result is 300 by Clark & Jarvis

^{[8]}, which is a consequence of Nielson's upper bound of 59^{[10]}.**References**- Small sieves : Brun's sieve
- D.A. Goldston, Y. Motohashi, J. Pintz, & C.Y. Yıldırım, Small Gaps between Primes Exist
- D. A. Goldston, S.W. Graham, J. Pintz, & C. Y. Yildirim, Small gaps between primes or almost primes
- Yitang Zhang, Bounded gaps between primes
- Terence Tao, Polymath proposal : bounded gaps between primes
- Polymath project, Bounded gaps between primes
- James Maynard, Small gaps between primes
- David A. Clark, Norman C, Jarvis, Dense admissible sequences
- Terence Tao, The prime tuples conjecture, sieve theory, and the work of ... and Zhang
- Pace Nielsen, Comment : Polymath8b, III: Numerical optimization of the ... search for new sieves
- Jerri Li, Erdos and twin prime conjecture

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