Bounded gaps between primes

World records

Date	[math]\displaystyle{ \varpi }[/math]	[math]\displaystyle{ k_0 }[/math]	[math]\displaystyle{ H }[/math]	Comments
14 May	1/1168 (Zhang)	3,500,000 (Zhang)	70,000,000 (Zhang)	All subsequent work is based on Zhang's breakthrough paper.
21 May			63,374,611 (Lewko)	Optimises Zhang's condition [math]\displaystyle{ \pi(H)-\pi(k_0) \gt k_0 }[/math]; can be reduced by 1 by parity considerations
28 May			59,874,594 (Trudgian)	Uses [math]\displaystyle{ (p_{m+1},\ldots,p_{m+k_0}) }[/math] with [math]\displaystyle{ p_{m+1} \gt k_0 }[/math]
30 May			59,470,640 (Morrison) 58,885,998? (Tao) 59,093,364 (Morrison) 57,554,086 (Morrison)	Uses [math]\displaystyle{ (p_{m+1},\ldots,p_{m+k_0}) }[/math] and then [math]\displaystyle{ (\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1}) }[/math] following [HR1973], [HR1973b], [R1974] and optimises in m
31 May		2,947,442 (Morrison) 2,618,607 (Morrison)	48,112,378 (Morrison) 42,543,038 (Morrison) 42,342,946 (Morrison)	Optimising Zhang's condition [math]\displaystyle{ \omega\gt 0 }[/math], and then using an improved bound on [math]\displaystyle{ \delta_2 }[/math]
1 Jun			42,342,924 (Tao)	Tiny improvement using the parity of [math]\displaystyle{ k_0 }[/math]
2 Jun		866,605 (Morrison)	13,008,612 (Morrison)	Uses a further improvement on the quantity [math]\displaystyle{ \Sigma_2 }[/math] in Zhang's analysis (replacing the previous bounds on [math]\displaystyle{ \delta_2 }[/math])
3 Jun	1/1040? (v08ltu)	341,640 (Morrison)	4,982,086 (Morrison) 4,802,222 (Morrison)	Uses a different method to establish [math]\displaystyle{ DHL[k_0,2] }[/math] that removes most of the inefficiency from Zhang's method.
4 Jun	1/224?? (v08ltu) 1/240?? (v08ltu)		4,801,744? (Sutherland) 4,788,240 (Sutherland)	Uses asymmetric version of the Hensley-Richards tuples
5 Jun		34429? (Paldi/v08ltu)	4,725,021 (Elsholtz)	k_0 bound uses the optimal Bessel function cutoff. Only provisional so far because the kappa error is not yet analysed. H bound obtained by a hybrid Schinzel/greedy sieve

? - unconfirmed or conditional

?? - theoretical limit of an analysis, rather than a claimed record

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.
• Online reading seminar for Zhang’s “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, 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.
• Unheralded Mathematician Bridges the Prime Gap, Erica Klarreich, Simons science news, 20 May 2013.
  • The article also appeared on Wired as "Unknown Mathematician Proves Elusive Property of Prime Numbers".
• The Beauty of Bounded Gaps, Jordan Ellenberg, Slate, 22 May 2013.
• Game of proofs boosts prime pair result by millions, Jacob Aron, New Scientist, 4 June 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 Article
• [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 Article
• [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 MathSciNet Article
• [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). MathSciNet Article
• [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 Article
• [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. MathSciNet Article
• [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. arXiv 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
• [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. 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 Article arXiv