# The twin prime conjecture and other curiosities regarding prime numbers

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## Abstract

The paper begins with a reference to Riemann’s hypothesis on the sequence of prime numbers, still unproven today, and goes on to illustrate the twin prime conjecture and the more general Polignac’s conjecture; we then recount the recent result by Yitang Zhang about it, and the improvements obtained thanks to the online mathematical collaboration called Polymath 8.

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## Notes

1. According to Heinrich Tietze [20], the term “twin primes” was coined by Paul Stäckel in 1916.

2. In his paper “Recherches nouvelles sur les nombres premiers” [15]; it is worth noting that at the address https://books.google.fr/books?id=O6EKAAAAYAAJ&hl=it a same-titled pamphlet by Polignac is available [16].

3. For more on these topics, see the papers by Bruno Martin [10,11,12].

4. Calculation attributed to T. Alm, M. Fleuren, and J. K. Andersen by http://mathworld.wolfram.com/CousinPrimes.html and http://mathworld.wolfram.com/SexyPrimes.html.

5. A partial answer: (1009, 1019), (1021, 1031), (1039, 1049), (1051, 1061), …

6. We also have (1301, 1307, 1309). Are there other ones of the form (10n + 1, 10n + 3, 10n + 7)?

7. In [17] the opinions of numerous researchers who participated in the project are collected.

8. The post that originated this adventure [6] contains a series of recommendations in this regard, in order to encourage collaboration and facilitate the work of those who coordinate it.

## References

1. Büthe, J.: An analytic method for bounding ψ(x). arXiv:1511.02032 [math.NT] (2015)

2. Chen, J.R.: On the representation of a large even integer as the sum of a prime and the product of at most two primes. Sci. Sinica. 16, 157–176 (1973)

3. Conway, J.H., Guy, R.K.: The book of numbers. Springer, New York (1996)

4. Cranshaw, J., Kittur, A.: The Polymath Project: Lessons from a Successful Online Collaboration in Mathematics. Proc. SIGCHI Conf. on Human Factors in Computing Systems, ACM, 1865–74: (2011), available at: http://www.cs.cmu.edu/~jcransh/papers/cranshaw_kittur.pdf. Accessed 31 Oct 2017

5. Gourdon, X., Sebah, P.: Introduction to twin primes and Brun’s constant (2002). Available at: http://numbers.computation.free.fr/Constants/Primes/twin.html. Accessed 31 Oct 2017

6. Gowers, T.: Is massively collaborative mathematics possible? Gowers’s Weblog: (2009), available at: https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/. Accessed 31 Oct 2017

7. Gowers, T., Nielsen, M.: Massively collaborative mathematics. Nature. 461, 879–881 (2009)

8. Lehman, R.S.: On the difference π(x)-Li(x). Acta Arith. 11, 397–410 (1966)

9. Littlewood, J.E.: Sur la distribution des nombres premiers. Comptes Rendus. 158, 1869–1872 (1914)

10. Martin, B.: Des jumeaux dans la famille des nombres premiers I. Images des mathématiques, 20 March: (2015), available at: http://images.math.cnrs.fr/Des-jumeaux-dans-la-famille-des-nombres-premiers-I. Accessed 31 Oct 2017

11. Martin, B.: Des jumeaux dans la famille des nombres premiers II. Images des mathématiques, 21 June: (2015), available at: http://images.math.cnrs.fr/Des-jumeaux-dans-la-famille-des-nombres-premiers-II. Accessed 31 Oct 2017

12. Martin, B.: Des jumeaux dans la famille des nombres premiers III. Images des mathématiques, 20 September: (2015). http://images.math.cnrs.fr/Des-jumeaux-dans-la-famille-des-nombres-premiers-III. Accessed 31 Oct 2017

13. Nicely, T.: Enumeration to 1014 of the twin primes and Brun’s constant. Virginia J. of Science 46, 195–204: (1996). http://www.trnicely.net/twins/twins.html; Accessed 31 Oct 2017

14. Nicely, T.: A new error analysis of Brun’s constant. Virginia J. of Science 52, 45–55: (2001). http://www.trnicely.net/twins/twins4.html. Accessed 31 Oct 2017

15. Polignac, A.: Recherches nouvelles sur les nombres premiers. Comptes Rendus Paris 29, 400 (1849) (738–39 [Rectification])

16. Polignac, A.: Recherches nouvelles sur les nombres premiers. Bachelier, Paris (1851)

17. Polymath, D.H.J.: The “bounded gaps between primes” Polymath project—a retrospective. arXiv:1409.8361 [math.HO] (2014)

18. Riemann, B.: Über die Anzahl der Primzahlen unter einer gegebenen Grosse. Monatsberichte der Berliner Akademie, 671–680 (1859)

19. Skewes, S.: On the difference π(x)–li(x). J. Lond. Math. Soc. 8, 277–283 (1933)

20. Tietze, H.: Famous problems of mathematics: solved and unsolved mathematics problems from antiquity to modern times. Graylock Press, Baltimore (1965)

21. Zhang, Y.: Bounded gaps between primes. Ann. Math. 179, 1121–1174 (2014)

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Correspondence to Renato Betti.

## Appendix

### 1.1 New forms of mathematical research?

In recent times, largely due to computing tools, email and so on, mathematical research is increasingly taking place in collaboration. It is easy, and sometimes even pleasant, to communicate our ideas to colleagues, ask for a suggestion and perhaps discover unforeseen and unexpected contributions. How far is it possible to extend this form of collaboration and to what extent can it be planned?

Of course, there are precedents: we all know the story of Bourbaki, a group of people who decided to work together to refound the principles and tools of mathematics; we are familiar with the quest that in the last century led to the classification of finite simple groups through the collaboration of many mathematicians; not to mention the research involving the massive, well-coordinated use of a large computer network, for instance in number theory—prime numbers, twin numbers etc., such as the efforts discussed in the article—or aimed at decrypting complex cryptographic systems.

Even in the most complex cases it had been possible to recognise that the problem could be naturally divided into a number of sub-problems, perhaps several ones, but in a way that implies a “hierarchical”, fairly clear organisation. In 2009, the mathematician Timothy Gowers, a Fields Medal winner in 1998, proposed a new model of a forum open to all, not in order to make useful and fine debates—ok, for this too—but especially to address well-defined mathematical research problems without such a structural character [7]. And the proposal obtained an unexpected success.

The first project, consisting in the search for an elementary proof of the Hales–Jewett theorem, a combinatorial result that was well known but hard to grasp, was successful after a few weeks, thanks to the collaboration of dozens of mathematicians, including graduate students. Subsequent projects have had varying degrees of success, on occasions with partial results, but sometimes excellent ones, as in the case of project no. 8, attempting to improve Zhang’s result on the difference between consecutive primes.Footnote 9 The idea is that anyone who has something to say about the problem should have no qualms about participating. Of course, everybody should make sure to express their comments—whatever form they take—in a reasonable time and in concise terms, because the aim is not to write a long and detailed paper but to encourage the debate. The project itself raised a number of issues, such as who gets credit for the results and the papers to which it gives rise, how to acknowledge those who participated only partially and so on. For now, the temporary solution is to sign the articles under the pseudonym D.H.J. Polymath and preserve the whole history so that the contribution of everyone is clear (the “name” D.H.J. comes from the initial project, Polymath 1, about the already mentioned “density” Hales-Jewett theorem).

But the real challenge—it seems to me—is that of organising work, starting with a problem that lends itself to be worked on collectively, to provide for the action of a moderator and methods to distinguish, since the beginning, valid contributions from useless ones or even deliberate obstruction. In short, how to organise a genuine collective research.Footnote 10

For now it seems that there have not been too many problems, thanks partly to the fact that the filtering of the various contributions was done by great mathematicians. The advantages of this approach to research are obvious. In the words of the man who proposed it: “if a large group of mathematicians could connect their brains efficiently, they could perhaps solve problems very efficiently as well” [6].

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Betti, R. The twin prime conjecture and other curiosities regarding prime numbers. Lett Mat Int 5, 297–303 (2017). https://doi.org/10.1007/s40329-017-0205-1