Bounded gaps between primes

World records

Date	[math]\displaystyle{ \varpi }[/math]	[math]\displaystyle{ k_0 }[/math]	[math]\displaystyle{ H }[/math]
14 May	1/1168 (Zhang)	3,500,000 (Zhang)	70,000,000 (Zhang)
21 May			63,374,611 (Lewko)
28 May			59,874,594 (Trudgian)
30 May			59,470,640 (Morrison) 59,093,364 (Morrison) 58,885,998 (Tao, conditional) 57,554,086 (Morrison)
31 May		2,618,607 (Morrison)	42,543,038 (Morrison) 42,342,946 (Morrison)
1 Jun			42,342,924 (Tao)
2 Jun		866,605 (Morrison)	13,008,612 (Morrison)
3 Jun		341,640 (Morrison)	4,982,086 (Morrison) 4,802,222 (Morrison)

Polymath threads

• I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart, Scott Morrison, 30 May 2013
• The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang, Terence Tao, 3 June 2013.
• Polymath proposal: bounded gaps between primes, Terence Tao, 4 June 2013.

Code and data

• Github, Scott Morrison
  • A mathematica notebook for finding k_0, Scott Morrison
• k-tuple pattern data, Thomas J Engelsma

Other relevant blog posts

• Marker lecture III: “Small gaps between primes”, Terence Tao, 19 Nov 2008.
• The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?, Emmanuel Kowalski, 22 Jan 2009.
• Number Theory News, Peter Woit, 12 May 2013.
• Bounded Gaps Between Primes, Emily Riehl, 14 May 2013.
• Bounded gaps between primes!, Emmanuel Kowalski, 21 May 2013.
• Bounded gaps between primes: some grittier details, Emmanuel Kowalski, 4 June 2013.
  • The slides from the talk mentioned in that post

MathOverflow

• Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture, 20 May 2013.
• A technical question related to Zhang’s result of bounded prime gaps, 25 May 2013.
• How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the [math]\displaystyle{ S_2 }[/math] sum, 31 May 2013.
• Tightening Zhang’s bound (closed), 3 June 2013.
  • Metathread for this post
• Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?, 3 June 2013.

Wikipedia

• Brun-Titchmarsh theorem
• Prime gap
• Second Hardy-Littlewood conjecture
• Twin prime conjecture

Recent papers and notes

• Bounded gaps between primes, Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
• Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture, Janos Pintz, 27 May 2013.
• A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million, T. S. Trudgian, 28 May 2013.
• The Friedlander-Iwaniec sum, É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
• Notes on Zhang's prime gaps paper, Terence Tao, 1 June 2013.
• Bounded prime gaps in short intervals, Johan Andersson, 3 June 2013.

Media

• First proof that infinitely many prime numbers come in pairs, Maggie McKee, Nature, 14 May 2013.
• Proof that an infinite number of primes are paired, Lisa Grossman, New Scientist, 14 May 2013.
• The Beauty of Bounded Gaps, Jordan Ellenberg, Slate, 22 May 2013.

Bibliography

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

• [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. MathSciNet
• [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. MathSciNet
• [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. MathSciNet
• [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. MathSciNet Article
• [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. JSTOR
• [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. arXiv MathSciNet
• [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. MathSciNet Article
• [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. MathSciNet
• [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. MathSciNet
• [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. MathSciNet
• [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. MathSciNet
• [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. MathSciNet Article
• [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. MathSciNet