# Twin Prime Conjecture : A Brief History of the Present

#### mathbalarka

##### Well-known member
MHB Math Helper
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.

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

1. Small sieves : Brun's sieve
2. D.A. Goldston, Y. Motohashi, J. Pintz, & C.Y. Yıldırım, Small Gaps between Primes Exist
3. D. A. Goldston, S.W. Graham, J. Pintz, & C. Y. Yildirim, Small gaps between primes or almost primes
4. Yitang Zhang, Bounded gaps between primes
5. Terence Tao, Polymath proposal : bounded gaps between primes
6. Polymath project, Bounded gaps between primes
7. James Maynard, Small gaps between primes
8. David A. Clark, Norman C, Jarvis, Dense admissible sequences
9. Terence Tao, The prime tuples conjecture, sieve theory, and the work of ... and Zhang
10. Pace Nielsen, Comment : Polymath8b, III: Numerical optimization of the ... search for new sieves
11. Jerri Li, Erdos and twin prime conjecture

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#### mathbalarka

##### Well-known member
MHB Math Helper
• (Dec 20) Nielson made it to $55$, without Deligne. That implies the bound of $272$, which is the tightest possible.
• (Dec 22) Sutherland pushes the 3-tuple bound to $395122$ unconditionally.
• (Dec 23) Sutherland gives $k_0 = 530000$ for 6-tuples, unconditionally and without Deligne's theorem. A major improvement.
• (Dec 24) Sutherland again. 5-tuple assuming Elliot Halberstam is now at $474320$.