Page 012

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 ,
11Y60 Link .

Mathematical Review copyright: Copies of reviews from Mathematical
Review cannot be posted on Web pages, but they can be seen through links
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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 Link
    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 mathematical.com 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 Link

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 Link

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
maintained 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.
Tech. 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.
Math. 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.
Monthly 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
Math. 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.
Tech. 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
conjecture. 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 Link

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.
Summary Link .

Published in 1976
Richard P. Brent, Tables Concerning Irregularities in the Distribution of Primes
and Twin Primes Up to 10¹¹, Math. 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.

Back to my home page Link .