Bounded gaps between primes: Difference between revisions

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* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 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.  [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [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.  [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers''.  
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers''. Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf 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. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]
* [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. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Revision as of 22:30, 4 June 2013

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

? - unconfirmed or conditional

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

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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
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