Twin primes and their applications 

Page maintained by Jean-Claude Evard. Last update: June 17, 2003.

AMS classification numbers  Link  11A41  Link , 11N05  Link , 11Y35  Link 
Link .

Mathematical Review copyright:  Copies of reviews from Mathematical 
Review cannot be posted on Web pages, but they can be seen through links 
to MathSciNet.  These links work only in the networks of institutions or on 
the computers of users who are current subscribers to MathSciNet.            

 Main directions of research about twin primes 

1. The twin prime conjecture and extensions of Chen's theorem  Link .
2. Search for lower and upper bounds for the number of twin primes in intervals [1, n].
3. Search for the largest known twin primes.
4. Computation of Brun's constant.
5. Computation of the twin prime constant.
6. Generalizations to Gaussian primes.
7. Applications of twin primes.

 Twin primes on large Web sites 

1. List of 114 Web sites about twin primes, with summaries  Link 
    provided by Geometry, The Online Learning Center.

2. Web pages related to twin primes maintained by Eric Weisstein  Link 
    on the Web site MathWorld 
Link  at Wolfram Research  Link .
          Introduction to twin primes 
Link .  
The twin prime conjecture  Link . 
The twin prime constant  Link .
          The Brun's constant  Link . 

3. Web pages related to twin primes on the Web site of Chris Caldwell  Link 
    at the University of Tennessee at St. Martin:
          Introduction to twin primes 
Link .  
          The twin prime conjecture  Link . 
          First generalization of the twin prime conjecture: 
                    The prime k-uple conjecture  Link . 
          Second generalization of the twin prime conjecture: 
                    The Dickson's conjecture  Link .  
          The largest twenty known twin primes  Link .  
          The twin prime constant  Link .  
          The Brun's constant  Link . 

4. Web page on twin primes,  Link  
    on the Web site of Number Theory 
 Link ,
    maintained by Keith Matthews
    at the University of Queensland in Australia.

5. Web page on Brun's constant  Link  maintained by Steven Finch  Link  at MathSoft  Link .

6. Lists of twin primes  Link  provided by  Link .

 Web pages about twin primes 
 (In alphabetic order of the name of the first author). 

1. Rensselaer Scientists Test Famous Hypothesis, by Bruce Adams,  Review, 
      weekly newspaper published by Rensselaer Polytechnic Institute, 
      issue of February 13, 1998 

2. Web pages on twin primes  Link  constructed by Patrick Fry  Link  and  Link 
    Jeffrey Nesheiwat 
Link , and Boleslaw Szymanski  Link , when there were 
    graduate student in the Department of Computer Science of  Rensselaer 
    Polytechnic Institute

3. Twin Primes and Brun's Constant: A Distributed Approach, by Patrick Fry,  
    Jeffrey Nesheiwat,  and Boleslaw Szymanski, Seventh IEEE International 
    Symposium on High Performance Distributed  Computing, 
    Chicago, IL, July 1998 

4. A Characterization of Twin Prime Pairs  Link , by Jeffrey Gold  Link  and 
    Don H. Tucker,  Proceedings - Fifth National Conference on Undergraduate 
    Research, Volume I, pp. 362-366,  University of North Carolina Press, 
    University of North Carolina at Asheville (UNCA), 1991. 

5. Web pages on twin primes  Link  and  Link  
by Xavier Gourdon and Pascal Sebah  Link .

6. Web pages on twin primes  Link  and Brun's constant  Link  maintained 
     by Thomas Nicely 
Link , retired  Link  from Lynchburg College.

7. The twin prime constant  Link , Web page maintained 
    by Gerhard Niklasch 
Link  at ConSol Software  Link 
    and Pieter Moree
 Link  and  Link  at Korteweg-de Vries Institute 
    for Mathematics
 (KdV Institute), Amsterdam, the Netherlands  Link .

8. Recent history of twin primes: ScienceNews, Twin primes,
    by Ivars Peterson, June 2, 2001, Vol. 159, No 22  Link .

9. Web page entitled "Computing the number of twin primes up to 1014 Link ,
    maintained by Joerg Richstein 
 Link  at Dalhousie University in Canada.

10. Web page  Link  of Sebastian Martmn Ruiz and Azmy Ariff
      on a characterization of twin primes and generalization.

11. Parallel Computing, Counting Prime Numbers two by two, by 
      Boleslaw Szymanski, New&Ideas, monthly research newspaper 
      published by Rensselaer Polytechnic Institute, issue of April 1998 
Link .

12. Preprints of Marek Wolf on twin primes, cousin primes, and Brun's constant 
      to download from 
Link  and  Link .

 Publications related to twin primes and their applications 

 Published in 2002 
J. V. Leyendekkers and A. G. Shannon, A note on twin primes 
and the modular ring $\Bbb Z\sb 6$, Internat. J. Math. Ed. Sci. 
33 (2002), no. 2, 303--306.

V. Yegnanarayanan, On a question concerning prime distance graphs, 
Discrete Math.
245 (2002), no. 1-3, 293--298. Summary
  Link .

 Published in 2001 
Abdullah Aziz Ergin, A relation between prime numbers and twin prime numbers, 
Math. Comput. Appl.
6 (2001), no. 2, 153--154.

José Luis Gómez Pardo,  Computational aspects of primes. I. (Spanish) 
Gac. R. Soc. Mat. Esp. 4 (2001), no. 3, 649--673. 
 This paper contains a survey of computational aspects of analytic number theory 
 with programs in Mathematica. 
Review by Pedro J. Berrizbeitia  Link .

A. I. Medyanik, A regular simplex inscribed in a cube, half-circulant Hadamard 
matrices, and Gaussian sums. (Russian), Mat. Fiz. Anal. Geom. 8 (2001), 
no. 1, 58--81. Review by Matti Jutila  Link .

Hugh Montgomery, Harmonic analysis as found in analytic number theory, 
Twentieth century harmonic analysis---a celebration, NATO Sci. Ser. II, 
Math., Phys., Chem., 33, Kluwer Acad., 2001, 271--293.
 This paper shows that a strong form of the Riemann Hypothesis would follow 
 from the d-twin prime conjecture of Hardy and Littlewood. 
Review by George Greaves 
Link .

Thomas R. Nicely, A new error analysis for Brun's constant, 
Virginia J. Sci.
52 (1): 45--55, 2001.
Summary  Link .
Mathematical Review number: 2003d:11184 
AMS Classification numbers: 11Y60 (11A41 11Y35 11Y70)

John Renze, Stan Wagon, and Brian Wick, The Gaussian zoo, Experiment. 
10 (2001), no. 2, 161--173. Review by Igor Rivin  Link .

 Published in 2000 
P. D. T. A. Elliott, The multiplicative group of rationals generated by the shifted 
primes, II, J. Reine Angew. Math. 519 (2000), 59--71. 
Review by Gérald Tenenbaum  Link .

D. A. Goldston, S. M. Gonek, A. E. Özlük, C. Snyder, On the pair correlation 
of zeros of the Riemann zeta-function. Proc. London Math. Soc. (3) 80 (2000), 
no. 1, 31--49.  Leads to new questions about twin primes. 
Review by
D. R. Heath-Brown  Link .

Pieter Moree, Approximation of singular series and automata, with an appendix 
by Gerhard Niklasch, Manuscripta Math. 101 (2000), no. 3, 385--399.
 In the appendix, Gerhard Niklasch indicates how one can apply the theorem of 
 this paper to obtain 1000 decimal-place accuracy in the computation of certain 
 constants such as the Artin and the twin prime constants.
Review by Maruti Ram Murty  Link .

Noel Patson, Interesting property observed in the prime numbers, Austral. 
Math. Soc. Gaz.
27, 2000, no. 5, 232--236.  Contains an important 
 extension of a new promising approach to Goldbach's conjecture started 
 by Daniel Zwillinger in 1979. 
Review by Vilius Stakenas  Link .

Laurentiu Panaitopol, Properties of the series of differences of prime numbers, 
Publications du Centre de Recherches en Mathématiques Pures. Sér. I, Fasc. 31, 
21--28, Publ. Cent. Rech.Math. Pures, Sér. I, 31, Univ. Neuchâtel, 2000.
 Contains some results similar to the convergence of the series of reciprocals of 
 twin primes (Brun's constant). 
Review by Cécile Dartyge  Link .

Derong Qiu and Xianke Zhang,  Elliptic curves of twin-primes over Gauss field 
and Diophantine equations, Adv. Math. (China) 29 (2000), no. 3, 279--281.
Mathematical Review 1 789 429  Link .

Derong Qiu and Xianke Zhang, Selmer groups and Mordell-Weil groups of 
two types of elliptic curves, Progr. Natur. Sci. 10 (2000), no. 12, 946--949.
 With some applications of twin primes to  elliptic curves.  
Review by Andrew Bremner  Link .

Peter Sarnak, Some problems in number theory, analysis and mathematical 
physics, Mathematics: frontiers and perspectives, 261--269, Amer. Math. Soc.,  
2000. Review by John B. Friedlander  Link .

Maria Suzuki, Alternative formulations of the twin prime problem, Amer. Math. 
107 (2000), no. 1, 55--56. 
Review by H. L. Abbott  Link  Contains 
 an interesting relation with the paper published by
Pinhas Naor in 1955. 

Shin-ichi Yoshida, On the equation y2 = x3 + pqx, Comment. Math. Univ. 
St. Paul.
49 (2000), no. 1, 23--42.  With some applications of twin primes to 
 elliptic curves. 
Review by Andrew Bremner  Link .

 Published in 1999 
Krassimir T. Atanassov, On some of the Smarandache's problems, American 
Research Press, Lupton, AZ, 1999.  Problem 117 contains a new 
 characterization of  twin primes. 
Review by J. Sándor   Link .

Richard Blecksmith, Paul Erdös, and J. Selfridge, L. Cluster 
primes, Amer. Math. Monthly 106 (1999), no. 1, 43--48. 
Review by Michael Filaseta   Link .

H. Gadiyar and R. Gopalkrishna, Padma,Ramanujan-Fourier series, 
the Wiener-Khintchine formula and the distribution of prime pairs, 
Phys. A 269 (1999), no. 2-4, 503--510. 
Review by  D. R. Heath-Brown  Link .

Solomon W. Golomb, Cyclic Hadamard difference sets---constructions 
and applications, Sequences and their applications, 
Ser. Discrete Math. Theor. Comput. Sci., Springer, 1999, 39--48. 
Review by Mathematical Review  Link .

Karl-Heinz Indlekofer and Antal Járai, Largest known twin primes and 
Sophie Germain primes, Math. Comp. 68 (1999), no. 227, 1317--1324. 
Review by
Oliver Schirokauer  Link .

Enrico Laeng, "Elementary" proofs and distribution of primes (Italian), Lett. Mat., 
Pristem No. 33-34, (1999), 28--35. Review by Giovanni Coppola
  Link .

M. B. S. Laporta, A short intervals result for 2n-twin primes in arithmetic progressions, 
Tsukuba J. Math. 23 (1999), no. 2, 201--214. Review by
A. Perelli  Link .

Peter Swinnerton-Dyer, Some applications of Schinzel's hypothesis to 
Diophantine equations, Number theory in progress, Vol. 1, 503--530, 
de Gruyter, 1999.  (The Schinzel's hypothesis is a generalization of the 
twin prime conjecture). Review by Tamás Szamuely  Link .

 Published in 1998 
Takashi Agoh, Additive representations of integers, Manuscripta 
95 (1998), no. 3, 311--321.  Contains sufficient conditions 
 for the Golbach's and the twin primes conjectures to be true. 
Review by Alessandro Languasco  Link . 

P. Berrizbeitia and P. D. T. A. Elliott, On products of shifted primes, Paul Erdös 
(1913--1996). Ramanujan J. 2 (1998), no. 1-2, 219--223.  Contains an 
 analogue of the twin prime conjecture. See continuation of this work published 
 by the second author in 2000. 
Review by Gérald Tenenbaum  Link .

Marco Buratti, Small quasimultiple affine and projective planes: some improved 
bounds, J. Combin. Des. 6 (1998), no. 5, 337--345.  Contains an application 
 of twin primes to design theory. 
Summary  Link .

Andrej Dujella, A problem of Diophantus and Dickson's conjecture, 
Number theory
, 147--156, de Gruyter, 1998. Summary  Link .

D. A. Goldston, and C. Y. Yildirim, Primes in short segments of arithmetic 
progressions, Canad. J. Math. 50 (1998), no. 3, 563--580. 
Review by John B. Friedlander  Link . 

J. Nesheiwat and B. K. Szymanski, Computing Twin Primes and Brun's Constant: 
A Distributed Approach," Proceedings of the Seventh IEEE International Symposium 
on High Performance Distributed Computing (HPDC7), pages 42-49, 1998. 

K. Ramachandra, Many famous conjectures on primes; meagre but precious 
progress of a deep nature, Proc. Indian Nat. Sci. Acad., Part A 64 (1998), 
no. 5, 643--650.  Short survey paper. See below reference.  
Review by R. C. Baker  Link .

K. Ramachandra, Many famous conjectures on primes; meagre but precious 
progress of a deep nature, Math. Student 67 (1998), no. 1-4, 187--199. 
 Short survey paper. See above reference.  Review by Wen-Bin Zhang  Link .

Simon Wong, Concentration function of additive functions on shifted twin 
primes, Acta Arith. 84 (1998), no. 3, 193--224. 
Review by Matti Jutila  Link .

 Published in 1997 
Tony Forbes, A large pair of twin primes, Math. Comp., 66 (1997), no. 217, 451-455.      

Y. K. Huen, The twin prime problem revisited, Internat. J. Math. Ed. Sci. 
28 (1997), no. 6, 825--834. Review by Marc Deléglise  Link .

Jerzy Kaczorowski, Hilbert's Eighth Problem. (Polish. English summary), 
Hilbert's Problems, 85--118, Polsk. Akad. Nauk, Warsaw, 1997. 
Review by Andrzej Madrecki  Link .

Jörg Richstein, Unendliche Geschwisterliebe unter Zahlen oder Die Suche nach 
den Primzahlzwillingen (German) [Infinite sibling love among numbers or The 
search for the twin primes], Überblicke Mathematik
[Surveys of mathematics], 
Edited by Albrecht Beutelspacher, Norbert Henze, Ulrich Kulisch and Hans 
Wußing,Vieweg, Braunschweig, 1997, 135--143 .  Expository paper on the 
 history of prime number theory, with an emphasis on the twin prime problem 
Review by Matti Jutila 
Link .

 Published in 1996 
K. Indlekofer and A. Járai, "Largest known twin primes," Math. Comp., 65 (1996) 427-428.        

T. Nicely, Enumeration to 1014 of the Twin Primes and Brun's Constant, 
Virginia J. Sci., (1996), vol. 46, p. 195-204.

Aldo Peretti, The twin prime problem, according to Hardy-Littlewood, 
Notes Number Theory Discrete Math.
2 (1996), no. 3, 33--41. 

 Published in 1995 
Joseph B. Dence and Thomas P.Dence,  A necessary and sufficient condition for twin primes, 
Missouri J. Math. Sci. 7 (1995), no. 3, 129--131.

Thomas R. Nicely, Enumeration to $10\sp {14}$ of the twin primes and Brun's constant, 
Virginia J. Sci.
46 (1995), no. 3, 195--204. 

 Published in 1994 
Mojtaba Moniri, Recursive models of open induction of prescribed finite 
transcendence degree $>1$ with cofinal twin primes, C. R. Acad. Sci. Paris 
Sér. I Math.
319 (1994), no. 9, 903--908.        

 Published in 1992 
T. Aaron Gulliver and Vijay K. Bhargava, Self-dual codes based on the 
twin prime product $35$, Appl. Math. Lett. 5 (1992), no. 3, 95--98.

Alberto Perelli  and János Pintz, On the exceptional set for the $2k$-twin primes 
problem, Compositio Math. 82 (1992), no. 3, 355--372. 

 Published in 1991 
Ding Zhu Du and D. Frank Hsu, Partitionable starters for twin prime power type, 
Discrete Math.
87 (1991), no. 1, 23--28. 

 Published in 1990   
Jean-Pierre Bauer, Sur un problème equivalent au problème des nombres premiers 
jumeaux (French) [On a problem equivalent to that of twin primes], Ann. Sci. Univ. 
Franche-Comté Besançon Math. (4)
No. 2 (1990), 3--7. 

Liu Hong-Quan, On the prime twins problem, Sci. China Ser. A., 33 (1990), 281-298.

Bodo K. Parady, Joel F. Smith, and Sergio E. Zarantonello, 
Largest known twin primes, Math. Comp., 55, no. 191, 1990, pp. 381-382.            

 Jie Wu, Sur la suite des nombres premiers jumeaux (French) [On the 
sequence of twin primes], Acta Arith. 55 (1990), no. 4, 365--394. 
 See important review by Wen-Bin Zhang 

 Published in 1986   
Norberto Cuesta Dutari, Arithmetic of the sequences $6n-1,\;6n+1$ and 
of twin primes, (Spanish), Collect. Math. 37 (1986), no. 3, 211--227. 

A. Peretti, On the twin prime problem, Bull. Number Theory 
Related Topics
10 (1986), no. 2, 48--60. 

 Published in 1983  
E. B. Belyakov and V. I. Martcyanov, V. I. 
The universal theory of the integers and the extended twin prime conjecture 
(Russian), Algebra i Logika 22 (1983), no. 1, 26--34. 

 Published in 1982  
Ding Zhu Du and F. K. Hwang, Balanced Howell rotations of the twin prime 
power type, Trans. Amer. Math. Soc. 271 (1982), no. 2, 415--421. 

Rudolf Ondrejka, More very large twin primes, 
J. Recreational Math.
15 (1982/83), no. 1, 7.

 Published in 1980  
R. F. Faiziev, Two criteria for the theory of $k$-twin primes (Russian), 
Izv. Akad. Nauk Tadzhik. SSR Otdel. Fiz.-Mat. Khim. 
i Geol. Nauk
1980, no. 1(75), 76--77.

 Published in 1979  
A. O. L. Atkin and N. W. Rickert, On a larger pair of twin primes, 
Abstract 79T-A132, Notices Amer. Math. Soc., 26 (1979), A-373

Crandall, R. E.; Penk, M. A. A search for large twin prime pairs. 
Math. Comp.
33 (1979), no. 145, 383--388. 

Daniel Zwillinger, A Goldbach conjecture using twin primes, Math. Comp. 33 
(1979), no. 147, 1071.  Contains a promising new approach to Goldbach's 
 conjecture. This approach has been extended by Noel Patson in 2000. 

Link .

 Published in 1976 
Richard P. Brent, Tables Concerning Irregularities in the Distribution of Primes 
and Twin Primes Up to 1011Math. Comput., (1976), vol. 30, p. 379.

Richard P. Brent, Correction to: "Irregularities in the distribution of primes 
and twin primes", (Math. Comp. 29 (1975), 43--56), 
Math. Comput.
30 (1976), no. 133, 198. 

J. Mlcek , J. Twin prime problem in an arithmetic without induction,
Comment. Math. Univ. Carolinae 17 (1976), no. 3, 543--555. 

 Published in 1975 
Richard P. Brent, Irregularities in the distribution of primes and twin primes, 
Collection of articles dedicated to Derrick Henry Lehmer on the occasion 
of his seventieth birthday, Math. Comp. 29 (1975), 43--56. 

Werner Neudecker, On twin "primes" and gaps between successive "primes" 
for the Hawkins random sieve, 
Math. Proc. Cambridge Philos. Soc.
77 (1975), 365--367. 

 Published in 1974 
D. Shanks and J. W Wrench, Brun's Constant, 
Math. Comput.
, 28 (1974), 293-299.

 Published in 1972 
A. Hedayat, E. Seiden, and W. T. Federer, Some families of designs 
for multistage experiments: mutually balanced Youden designs 
when the number of treatments is prime power or twin primes, 
I. Ann. Math. Statist. 43 (1972), 1517--1527. 

 Published in 1968 
C. A. Holben and J. H. Jordan, J. H., The twin prime problem and Goldbach's 
conjecture in the Gaussian integers, Fibonacci Quart. 6 1968 no. 5, 81--85, 92.

 Published in 1967 
P. Turán, On the twin-prime problem III, Acta Arith. 14, 1967/1968, 399--407.

P. Turán, On the twin-prime problem II, Acta Arith. 13, 1967/1968, 61--89.

 Published in 1964 
P. Turán, On the twin-prime problem I, Magyar Tud. Akad. 
Mat. Kutató Int. Közl.
9, 1964, 247--261.

 Published in 1961 
C.E. Fröberg, On the sum of inverses of primes and twin primes, 
Nordisk Tidskr, Informationsbehandling (BIT), (1961), vol. 1, p. 15-20.

A. F. Lavrik, The number of $k$-twin primes lying on an interval 
of a given length, Dokl. Akad. Nauk SSSR 136, 281--283 (Russian); 
translated as Soviet Math. Dokl. 2, 1961, 52--55.

John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, 
Math. Comp.
15, 1961, 396--398.

 Published in 1960 
S. W. Golomb, The twin prime constant, 
Amer. Math. Monthly
67 (1960), 767-769.

A. F. Lavrik, On the distribution of $k$-twin primes (Russian), 
Dokl. Akad. Nauk SSSR
132, 1960, 1258--1260.

A. F. Lavrik, On the twin prime hypothesis of the theory of primes 
by the method of I. M. Vinogradov, Dokl. Akad. Nauk SSSR 132, 1013--1015 
(Russian); translated as Soviet Math. Dokl. 1, 1960, 700--702.

Daniel Shanks, A note on Gaussian twin primes, 
Math. Comput.
14, 1960, 201--203.

 Published in 1955 
Pinhas Naor, A property of prime twins (Hebrew. English summary), 
Riveon Lematematika 9 (1955), 36--40. Review by E. G. Straus  Link .
 A translation into English would be most welcome. Contains an interesting 
 isomorphism which sends ordinary twin primes to primes in a special 
 semigroup. Also contains an interesting relation with the paper published 
 by Maria Suzuki in 2000. 

 Published in 1923 
G.H. Hardy and J.E Littlewood, Some problems of 'Partitio Numerorum' III , 
On the expression of a number as a sum of primes, Acta Mathematica, (1923), 
vol. 44, p. 1-70.  Correction of the date, 1923 instead of 1922, communicated 
by Noel Patson. The error is wide spread in the mathematical litterature. 

 Published in 1919 
V. Brun, La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., 
où les dénominateurs sont nombres premiers jumeaux est convergente ou finie, 
Bulletin des sciences mathématiques, (1919), vol. 43, p. 100-104 and p. 124-128.

 Published in 1878 
J.W.L. Glaisher, An enumeration of prime-pairs, 
Messenger of Mathematics
, (1878), vol. 8, p. 28-33.

 Related Books  (In reverse chronological order of publication)

Richard Crandall and Carl Pomerance 
Prime numbers. A computational perspective.
Springer-Verlag, New York,  2001 . xvi+545 pp. 
Review by: Samuel S. Wagstaff, Jr.  Link 

George Greaves
Sieves in number theory 
Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 43
Springer-Verlag 2001 , xii+304 pp.
Review by H. G. Diamond  Link .

Melvyn B. Nathanson, 
Additive number theory, The classical bases. 
Graduate Texts in Mathematics, 164
Springer-Verlag, xiv+342 pp.,  1996 .
Review by D. R. Heath-Brown  Link .

Paulo Ribenboim
The new Book of Prime Number Records
Springer-Verlag, xxiv+541 pp.,  1996 .
No review available for this edition.  
Review of the first edition by Don Redmond  Link .

Godfrey Harold Hardy and Edward Maitland Wright
An Introduction to the Theory of Numbers, fifth edition
The Clarendon Press, Oxford University Press, xvi+426 pp., 1979 .
Review by T. M. Apostol  Link .

William J. LeVeque
Fundamentals of Number Theory
Reprint of the  1977  original.
Dover Publications, Inc., viii+280 pp., 1996 . 
Review by R. F. Churchhouse  Link .

H. Halberstam and H.-E. Richert
Sieve methods
London Mathematical Society Monographs, No. 4
Academic Press
, xiv+364 pp., 1974

 The most cited book in publications about twin primes, which is well deserved. 
Review by J. W. Porter  Link .

Exercise about twin primes

Problem 10844 on Fibonacci twin primes, The American Mathematical Monthly, Volume 108, 
Number 1, page 77, Proposed by Nick MacKinnon
at Winchester College, UK, January 2001.
Solution by Stephen Gagola, The American Mathematical Monthly, Volume 109, Number 1, 
page 78, January 2002.

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