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
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 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 .