# 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.

### Keywords

Prime numbers Riemann’s hypothesis Twin primes Polignac’s Conjecture Bounded gaps between prime numbers### References

- 1.Büthe, J.: An analytic method for bounding ψ(x). arXiv:1511.02032 [math.NT] (2015)Google Scholar
- 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)MathSciNetMATHGoogle Scholar - 3.Conway, J.H., Guy, R.K.: The book of numbers. Springer, New York (1996)CrossRefMATHGoogle Scholar
- 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)CrossRefGoogle Scholar - 8.Lehman, R.S.: On the difference π(x)-Li(x). Acta Arith.
**11**, 397–410 (1966)MathSciNetCrossRefMATHGoogle Scholar - 9.Littlewood, J.E.: Sur la distribution des nombres premiers. Comptes Rendus.
**158**, 1869–1872 (1914)MATHGoogle Scholar - 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 10
^{14}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])Google Scholar - 16.Polignac, A.: Recherches nouvelles sur les nombres premiers. Bachelier, Paris (1851)Google Scholar
- 17.Polymath, D.H.J.: The “bounded gaps between primes” Polymath project—a retrospective. arXiv:1409.8361 [math.HO] (2014)Google Scholar
- 18.Riemann, B.: Über die Anzahl der Primzahlen unter einer gegebenen Grosse. Monatsberichte der Berliner Akademie, 671–680 (1859)Google Scholar
- 19.Skewes, S.: On the difference π(x)–li(x). J. Lond. Math. Soc.
**8**, 277–283 (1933)MathSciNetCrossRefMATHGoogle Scholar - 20.Tietze, H.: Famous problems of mathematics: solved and unsolved mathematics problems from antiquity to modern times. Graylock Press, Baltimore (1965)MATHGoogle Scholar
- 21.Zhang, Y.: Bounded gaps between primes. Ann. Math.
**179**, 1121–1174 (2014)MathSciNetCrossRefMATHGoogle Scholar

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© Centro P.RI.ST.EM, Università Commerciale Luigi Bocconi 2017