21. Prime Numbers Historical topics about prime numbers.Category Science Math Number Theory Prime Numbers...... to prime numbers. Some unsolved problems The twin primes conjecturethat there are infinitely many pairs of primes only 2 apart. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html

22. Ìàòåìàòè÷åñêèé æóðíàë â Èíòåðíåò wwwgap.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html =cut= 1. Thetwin primes conjecture that there are infinitely many pairs of primes only 2 http://mathmag.spbu.ru/conference/fido7.ru.math/12909/

Derive Subscribe.Ru HTML TEXT KOI LAT WIN fido7.ru.math open problems related to primes OS/2 Hi, All ! www-gap.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html ===cut=== 1. The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart. 2. Goldbach's Conjecture (made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes. 3. Are there infinitely many primes of the form n^2 + 1 ? coprime contains infinitely many primes.) 4. Is there always a prime between n^2 and (n + 1)^2 ? (The fact that there is always a prime between n and 2n was called Bertrand's conjecture and was proved by Chebyshev.) 5. Are there infinitely many prime Fermat numbers? Indeed, are there any prime Fermat numbers after the fourth one? 6. Are there infinitely long arithmetic progressions of consecutive primes? e.g. 251, 257, 263, 269 has length 4. The largest example known has length 10.

23. Mathsoft: Mathsoft Unsolved Problems: Unsolved Problems On Other Sites MathPro Press Unsolved Math Problem of the Week column and General References;twin primes conjecture and Goldbach's Conjecture, discussed in Prime Numbers http://www.mathsoft.com/mathresources/problems/article/0,,1999,00.html

search site map about us  + news  + ... Unsolved Problems Unsolved Problems Links On a Generalized Fermat-Wiles Equation Zero Divisor Structure in Real Algebras Sleeping Habits of Armadillos Engineering Standards ... Math Resources Unsolved Problems on Other Sites Jeff Lagarias' 3x+1 problem and related problems The Generalized 3x+1 Mapping (University of Queensland) and 1999 Conference on the Collatz Problem Proceedings Alex Lopez-Ortiz's sci.math FAQ on Famous Problems in Mathematics (University of New Brunswick) Chris Caldwell's Riemann Hypothesis (University of Tennessee at Martin); also Daniel Bump's Riemann Hypothesis (Stanford University) and Barry Cipra's A Prime Case of Chaos MathPro Press Unsolved Math Problem of the Week column and General References Twin Primes Conjecture and Goldbach's Conjecture, discussed in Prime Numbers: Some Unsolved Problems , part of MacTutor History of Mathematics (University of St Andrews); also Chris Caldwell's Prime Conjectures and Open Questions (University of Tennessee at Martin) Jan Otto Munch Pedersen's Known Amicable Pairs (Vejle Business College, Denmark) and Chris Caldwell's

24. Www.prism.uvsq.fr/~dedu/math/unsolvedPbs.txt (SOLVED RECENTLY, see http//www.spiegel.de/spiegel/0,1518,203235,00.html) UnsolvedProblem 2 (twin primes conjecture) Are there an infinite number of twin http://www.prism.uvsq.fr/~dedu/math/unsolvedPbs.txt

25. ABCNEWS.com : Prove A Theorem, Win $1,000,000! The twin primes conjecture is another There are an infinite number of prime pairs,prime numbers that differ by 2. Examples are 5 and 7, 11 and 13, 17 and 19 http://abcnews.go.com/sections/science/WhosCounting/whoscounting000401.html

December 20, 2000 Good Morning America World News Tonight Downtown Primetime ... ABCNEWS.com var flash = 0; var ShockMode = 0; var Flash_File_Path = "http://akaads-abc.starwave.com/ad/sponsors/compaq/comp-log0302/comp-log0302.swf"; var default_image = "http://akaads-abc.starwave.com/ad/sponsors/compaq/comp-log0302/comp-log0302.gif"; var default_alttext = "visit hp.com"; var ad_width = "95"; var ad_height = "30"; on error resume next FlashInstalled = (IsObject(CreateObject("ShockwaveFlash.ShockwaveFlash.4"))) If FlashInstalled = "True" then flash = 1 End If GO TO: Select a Topic Sci/Tech Index HOMEPAGE SCIENCE WHO'S COUNTING? FEATURE Prove This, Win $1,000,000! Who Wants to Be a Millionaire Mathematician? Greek author Apostolos Doxiadis offers $1 million to anyone who can meet his mathematical challenge. (Amanda Shepherd/ABCNEWS.com) By John Allen Paulos Special to ABCNEWS.com April 1 an engaging first novel by Greek author Apostolos Doxiadis. The Story Behind the Math The cover of (Bloomsbury Publishing) Slowly, Uncle Petros is revealed to be a character of complexity and nuance, having devoted his considerable mathematical talents and much of his life to a futile effort to prove a classic unsolved problem. His solitary efforts give one a taste of the delight and the despair of mathematical research.

26. Unsolved Unsolved Problem 2 (twin primes conjecture) Are there an infinitenumber of twin primes? A prime number is an integer larger than http://mcraefamily.com/MathHelp/PuzzleUnsolved.htm

Unsolved Questions Unsolved Problem 1: (Catalan's Conjecture) Are 8 and 9 the only consecutive perfect powers? An integer is a perfect power if it is of the form m^n where m and n are integers and n>1. It is conjectured that 8=2^3 and 9=3^2 are the only consecutive integers that are perfect powers. Unsolved Problem 2: (Twin Primes Conjecture) Are there an infinite number of twin primes? A prime number is an integer larger than 1 that has no divisors other than 1 and itself. Twin primes are two prime numbers that differ by 2. For example, 17 and 19 are twin primes. Unsolved Problem 3: (The Rational Box) Does there exist a rectangular box all of whose edges and diagonals are integers? By a rectangular box, we mean a solid with six rectangular faces. This common figure is also known as a rectangular parallelepiped. The diagonals of a box include the face diagonals and the main diagonals. A face diagonal joins opposite vertices of a face. A main diagonal (or space diagonal) joins opposite vertices of the box.

27. Twin Primes Brun's Constant, de Polignac's conjecture Prime Constellation, Sexy primes, twin Prime conjecture, twin primes Constant http://mathworld.pdox.net/math/t/t437.htm

Twin Primes Twin primes are Primes ) such that . The first few twin primes are for , 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (Sloane's ). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (Sloane's and Let be the number of twin primes and such that . It is not known if there are an infinite number of such Primes (Shanks 1993), but all twin primes except (3, 5) are of the form . J. R. Chen has shown there exists an Infinite number of Primes such that has at most two factors (Le Lionnais 1983, p. 49). Bruns proved that there exists a computable Integer such that if , then (Ribenboim 1989, p. 201). It has been shown that where has been reduced to (Fouvry and Iwaniec 1983), (Fouvry 1984), 7 (Bombieri et al. 1986), 6.9075 (Fouvry and Grupp 1986), and 6.8354 (Wu 1990). The bound on is further reduced to 6.8325 (Haugland 1999). This calculation involved evaluation of 7-fold integrals and fitting of three different parameters. Hardy and Littlewood conjectured that (Ribenboim 1989, p. 202).

28. Prime Conjectures And Open Question consecutive primes with difference 2n. twin Prime conjecture Thereare infinitely many twin primes. In 1919 Brun proved that the http://www.utm.edu/research/primes/notes/conjectures/

Prime Conjectures and Open Questions (Another of the Prime Pages ' resources Home Search Site Largest Finding ... Submit primes Below are just a few of the many conjectures concerning primes. Goldbach's Conjecture: Every even n Goldbach wrote a letter to Euler in 1742 suggesting that . Euler replied that this is equivalent to this is now know as Goldbach's conjecture. Schnizel showed that Goldbach's conjecture is equivalent to distinct primes It has been proven that every even integer is the sum of at most six primes [ ] (Goldbach's conjecture suggests two) and in 1966 Chen proved every sufficiently large even integers is the sum of a prime plus a number with no more than two prime factors (a P ). In 1993 Sinisalo verified Goldbach's conjecture for all integers less than 4 ]. More recently Jean-Marc Deshouillers, Yannick Saouter and Herman te Riele have verified this up to 10 with the help, of a Cray C90 and various workstations. In July 1998, Joerg Richstein completed a verification to 4

29. The Prime Glossary: Twin Prime Conjecture twin primes are pairs of primes of the form (p, ). The term "twin prime" was coined by Paul Stäckel (18921919; Tietze 1965, p. 19). Proof of this conjecture would also imply the existence an infinite number of twin primes. http://primes.utm.edu/glossary/page.php/TwinPrimeConjecture.html

twin prime conjecture (another Prime Pages ' Glossary entries) Glossary: Index Search Home Previous ... Mail Editor Prime Pages: Home Search Index The (weak) twin prime conjecture is that there are infinitely many twin primes There is also a strong form of this conjecture [HL23] which states that there are about twin primes less than or equal to x . The constant written above as an infinite product is the twin primes constant. See the page "a simple heuristic " linked below for information on how this conjecture is formed. See Also: Brun's constant twin prime constant Related pages (outside of this work) Twin primes (research and records) A simple heuristic (draft article about construcitng such conjectures) References: 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 Math. (1923) 1-70. Reprinted in "Collected Papers of G. H. Hardy," Vol. I, pp. 561-630, Clarendon Press, Oxford, 1966.

30. Twin Primes -- From MathWorld for x n of is given by, (6). Proof of this conjecture would alsoimply the existence an infinite number of twin primes. Define, (7). http://mathworld.wolfram.com/TwinPrimes.html

Foundations of Mathematics Mathematical Problems Unsolved Problems Number Theory ... Prime Numbers Twin Primes Twin primes are pairs of primes of the form p ). The term "twin prime" was coined by Paul Stäckel (1892-1919; Tietze 1965, p. 19). The first few twin primes are for n = 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (Sloane's ). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (Sloane's and The following table gives the first few p for the twin primes ( p cousin primes p sexy primes p ), etc. Triplet Sloane First Member p Sloane's p Sloane's p Sloane's p Sloane's p Sloane's p Sloane's Let be the number of twin primes p and such that . It is not known if there are an infinite number of such primes (Wells 1986, p. 41; Shanks 1993), but it seems almost certain to be true (Hardy and Wright 1979, p. 5). All twin primes except (3, 5) are of the form . J. R. Chen has shown there exists an infinite number of primes p such that has at most two factors (Le Lionnais 1983, p. 49). Bruns proved that there exists a computable integer such that if , then (Ribenboim 1996, p. 261). It has been shown that

31. Re: Twin Primes By Antreas P. Hatzipolakis (when n=1, the twin Prime conjecture) Source http// www. utm. edu/ research/ primes/ notes/ conject. http://mathforum.com/epigone/math-history-list/thahtwecha/v01540B00AF941276EFD7@

Re: Twin Primes by Antreas P. Hatzipolakis reply to this message post a message on a new topic Back to messages on this topic Back to math-history-list Subject: Re: Twin Primes Author: xpolakis@hol.gr Date: http://www.utm.edu/research/primes/notes/conject.html http://www.mathsoft.com/asolve/constant/brun/brun.html http://www.astro.virginia.edu/~eww6n/math/Brun http://users.hol.gr/~xpolakis/ ... The Math Forum

32. Twin Prime Conjecture -- From MathWorld sometimes called the strong twin prime conjecture (Shanks 1993, p. 30) or first HardyLittlewoodconjecture, states that the number of twin primes less than or http://mathworld.wolfram.com/TwinPrimeConjecture.html

Foundations of Mathematics Mathematical Problems Unsolved Problems Number Theory ... Prime Numbers Twin Prime Conjecture There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes (Guy 1994, p. 19). It is not known if there are an infinite number of such primes (Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to be true (Hardy and Wright 1979, p. 5). In the words of Shanks (1993, p. 219), "the evidence is overwhelming." The conjecture that there are infinitely many integers n such that is prime and n is twice a prime is very closely related (Shanks 1993, p. 30). A second twin prime conjecture states that adding a correction proportional to to a computation of Brun's constant ending with will give an estimate with error less than . An extended form of this conjecture, sometimes called the strong twin prime conjecture (Shanks 1993, p. 30) or first Hardy-Littlewood conjecture , states that the number of twin primes less than or equal to x is asymptotically equal to where is the so-called twin primes constant (Hardy and Littlewood 1922). The value of

33. Introduction To Twin Primes And Brun's Constant Computation An article by Pascal Sebah with the results of computation of the twin primes up to 5.10^15.Category Science Math Number Theory Prime Numbers...... According to this conjecture the density of twin primes is equivalent to the densityof cousin primes. For example, the exact computed values up to 10 12 are http://numbers.computation.free.fr/Constants/Primes/twin.html

Introduction to twin primes and Brun's constant computation (Click here for a Postscript version of this page and here for a pdf version) Introduction It's a very old fact (Euclid 325-265 B.C., in Book IX of the Elements ) that the set of primes is infinite and a much more recent and famous result (by Jacques Hadamard (1865-1963) and Charles-Jean de la Vallee Poussin (1866-1962)) that the density of primes is ruled by the law p (n) n log(n) where the prime counting function p (n) is the number of prime numbers less than a given integer n. This result proved in 1896 is the celebrated prime numbers theorem and was conjectured earlier, in 1792, by young Carl Friedrich Gauss (1777-1855) and by Adrien-Marie Legendre (1752-1833) who studied the repartition of those numbers in published tables of primes. This approximation may be usefully replaced by the more accurate logarithmic integral Li(n): p (n) Li(n)= n dt log(t) However among the deeply studied set of primes there is a famous and fascinating subset for which very little is known and has generated some famous conjectures: the twin primes (the term prime pairs was used before [ Definition 1 A couple of primes (p,q) are said to be twins if q=p+2. Except for the couple (2,3), this is clearly the smallest possible distance between two primes.

34. Re: Twin Primes By Julio Gonzalez Cabillon primes which differ by k (which here I am calling "Polignac's conjecture") The mentioned article does not contain EXPLICITLY written the twin http://mathforum.com/epigone/math-history-list/thahtwecha/1.5.4.32.1997050804063

Re: Twin Primes by Julio Gonzalez Cabillon reply to this message post a message on a new topic Back to messages on this topic Back to math-history-list Subject: Re: Twin Primes Author: jgc@adinet.com.uy Date: The Math Forum

35. Twin Prime Conjecture - Wikipedia form of this conjecture, the so called Hardy Littlewood conjecture, which is concernedwith the distribution of twin primes, in analogy to the prime number http://www.wikipedia.org/wiki/Twin_Prime_Conjecture

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Twin prime conjecture (Redirected from Twin Prime Conjecture The twin prime conjecture is a famous unsolved problem in number theory that involves prime numbers . Its weak form states: There are an infinite number of primes p such that p + 2 is also prime. Such a pair of prime numbers is called a twin prime . The conjecture has been researched by many number theorists. The majority of mathematicians believe that the conjecture is true, based on numerical evidence and heuristic reasoning involving the probabilistic distribution of primes. In Alphonse de Polignac made the more general conjecture that for every natural number k , there are infinitely many prime pairs which have a distance of 2 k . The case k =1 is the twin prime conjecture.

36. Twin Prime - Wikipedia A strong form of the twin Prime conjecture, the HardyLittlewood conjecture, postulatesa distribution law for twin primes akin to the prime number theorem. http://www.wikipedia.org/wiki/Twin_prime

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Twin prime From Wikipedia, the free encyclopedia. A couple of two prime numbers p q ) are said to be twin primes if q p +2. This is the smallest possible distance between two primes. The first twin primes are: It is unknown whether there exist infinitely many twin primes, but most number theorists believe this to be true. This is the content of the Twin Prime Conjecture . A strong form of the Twin Prime Conjecture, the Hardy Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem It is known that the sum of the reciprocals of all twin primes converges (see Brun's constant ). This is in stark contrast to the sum of the reciprocals of all primes, which diverges.

37. The Prime Glossary: Twin Prime 19}. It has been conjectured that there are infinitely many twin primes(see the twin prime conjecture for further information). http://primes.utm.edu/glossary/page.php?sort=TwinPrime

38. The Prime Glossary: Twin Prime Conjecture There is also a strong form of this conjecture HL23 which states that there areabout twin primes less than or equal to x. The constant written above as an http://primes.utm.edu/glossary/page.php?next=twin prime

39. Project Proposals, Math 413 (Number Theory), Spring 2003, UMBC twin Prime conjecture If d n is the difference between the n th and the ( n +1)st primes, then d n =2 for an infinite number of values of n . Guy81, A8 http://www.math.umbc.edu/~campbell/Math413Spr03/projects.html

Project Proposals Math 413, Spring 2003 Projects are not (generally) expected to contain extensive original work but can be expository or can apply existing methods. The project will be prosented both as a paper and orally. Expected as part of the paper is an abstract and references. Carmichael Numbers N composite with a N =1 (mod N for all a with gcd( a N N N Related Conjecture: There is no composite N N N Annals of Math. , v139 (1994), pp 703-722 Math Comp. , 65 (1996), pp 823-836 Carmichael's Conjecture If N n k k n n other than n n Carmichael's "Empirical Theorem" by S. Wagon, Math Intelligencer , v8 (1986), pp 61-63 x n College Math. J Goldbach's Conjecture If N p q we have N p q A Reformulation of the Goldbach Conjecture by Gerstein, Math. Mag. Uncle Petros and Goldbach's Conjecture (novel) by Apostolos Doxiades, Bloomsbury, 2000 Twin Prime Conjecture If d n is the difference between the n th and the ( n st primes, then d n =2 for an infinite number of values of n Related conjectures apply to "Constellations" of more than two primes. Twin Primes and their Applications , J-C Evard http://www.math.utoledo.edu/~jevard/Page012.htm

40. Page 012 Introduction to twin primes Link . The twin prime conjecture Link . Daniel Zwillinger,A Goldbach conjecture using twin primes, Math. Comp. 33 (1979), no. http://www.math.utoledo.edu/~jevard/Page012.htm

Twin primes and their applications Page maintained by Jean-Claude Evard. Last update: February 7, 2003. Mathematics subject classification numbers: 11A41: Primes 11N05: Distribution of primes Mathematical Review 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 . List of 114 Web sites about twin primes, with summaries Link provided by Geometry, The Online Learning Center. . 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

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