1. Twin Primes Conjecture Are there an infinite number of twin primes? A prime number than 1 and itself. Twin primes are two prime numbers that For example, 17 and 19 are twin primes. http://cage.rug.ac.be/~hvernaev/problems/Problem2.html

8-Jan-1995 Unsolved Problem 2: 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. Reference: [Sierpinski 1964] Waclaw Sierpinski, A Selection of Problems in the Theory of Numbers. Pergamon Press. New York: 1994. Pages 30-31. Each week, for your edification, we publish a well-known unsolved mathematics problem. These postings are intended to inform you of some of the difficult, yet interesting, problems that mathematicians are investigating. We do not suggest that you tackle these problems, since mathematicians have been unsuccessfully working on these problems for many years. general references previous problem problem archive next problem ... electronic publications

2. Conjecture 3. Twin Prime's Conjecture Therefore, if this the Mr Liu's argument is correct then also the twin primes conjecture has been proved. http://www.primepuzzles.net/conjectures/conj_003.htm

Conjectures Conjecture 3. Twin Prime's Conjecture If we define d n as : d n = p n+1 - p n , is easy to see that d =1 and d n Now, that " for n>1, dn=2 infinitely often" (Ref. 2, p. 19). This is the "Twin prime Conjecture", which can be paraphrased this way : "There are infinite consecutive primes differing by 2". SOLUTION Mr Liu Fengsui has sent (3/9/01) an argument that proves - according to him - the well known and named " k-tuple conjecture " This conjecture can be expressed the following way (see " Any admissible constellation of primes occurs infinitely often ". Therefore, if this the Mr Liu's argument is correct then also the Twin Primes conjecture has been proved. As you soon will discover this argument is close related to the Liu's approach to the prime numbers definition, approach that has been exposed in detail in the Problem 37 of these pages. What follows is Mr Liu's argument. I should strongly point out that the most that Mr. Liu

3. Enumeration To 1.6e15 Of The Twin Primes And Brun's Constant as the infinitude of K2 remains undecidedthe famous "twin primes conjecture " a topic of discussion even in a recent http://www.trnicely.net/twins/twins2.html

of the twin primes and Brun's constant Thomas R. Nicely Date of first release 16 July 1999. Last modified 22 March 2002. Abstract . An improved estimate is obtained for Brun's constant, B 1991 Mathematics Subject Classification Primary: 11A41. Secondary: 11-04, 11Y70, 11Y60. Key words and phrases Twin primes, Brun's constant, prime numbers. 1. Introduction The set K The present study results from the continuation of a project initiated in 1993, with results to 10 described in the author's 1995 paper [13]. A detailed description of the general problem, the computational methods employed, and the incidental discovery of the P*ntium FDIV flaw may be found there, with additional details given in [14] and [15]; only a brief summary will be included here. The prime numbers themselves continue to retain most of their secrets, but still less is known about the twin primes. A matter as fundamental as the infinitude of K remains undecided-the famous "twin primes conjecture," a topic of discussion even in a recent popular motion picture [21]. Nonetheless, Brun [3] proved in 1919 that in any event the sum of the reciprocals

4. Enumeration To 1e14 Of The Twin Primes And Brun's Constant This is hardly unexpected when dealing with Brun's constant, the twin primes conjecture,and the HardyLittlewood approximation, all of which have proved http://www.trnicely.net/twins/twins.html

Enumeration to 10 of the twin primes and Brun's constant Thomas R. Nicely This document may be reproduced and distributed for educational and non-profit purposes. Document history Last modified....................22 March 2002. Mathematical Reviews.............MR 97e:11014. Date of public issue.............March 1996. Journal citation.................Virginia Journal of Science 46:3 (Fall, 1995) 195-204. Accepted for publication.........15 November 1995. Acknowledgment of submission.....31 August 1995. Original submission..............28 August 1995. Abstract The count pi , at intervals of 10 . An algorithm for estimating the standard deviation of such calculations is described. In the course of the computation a flaw was discovered in the hardware divider of the floating point unit of Int*l Corporation's P*ntium microprocessor. 1991 Mathematics Subject Classification Primary: 11A41. Secondary: 11-04, 11N36, 11Y60, 11Y70, 11Y35, 68M15. Key words and phrases Twin primes, Brun's constant, primes, P*ntium flaw, computer errors.

5. Twin Primes Conjecture twin primes conjecture. There exist an infinite number of positive integersp with p and p+2 both prime. See the largest known twin prime section. http://db.uwaterloo.ca/~alopez-o/math-faq/node63.html

Next: Mathematical Games Up: Unsolved Problems Previous: Goldbach's conjecture Twin primes conjecture There exist an infinite number of positive integers p with p and p +2 both prime. See the largest known twin prime section. There are some results on the estimated density of twin primes. Alex Lopez-Ortiz Mon Feb 23 16:26:48 EST 1998

6. Unsolved Problem Of The Week Archive A list of unsolved problems published by MathPro Press during 1995.Category Science Math Research Open Problems...... Conjecture 22Jan-1995 Problem 4 Equichordal Points 15-Jan-1995 Problem 3 TheRational Box 8-Jan-1995 Problem 2 twin primes conjecture 1-Jan-1995 Problem 1 http://cage.rug.ac.be/~hvernaev/problems/archive.html

Unsolved Problem of the Week Archive Welcome to the archive for the Unsolved Math Problem of the Week Each week, for your edification, we publish a well-known unsolved mathematics problem. These postings are intended to inform you of some of the difficult, yet interesting, problems that mathematicians are investigating. We give a reference so that you can get more information about the topic. These problems can be understood by the average person. Nevertheless, we do not suggest that you tackle these problems, since mathematicians have been unsuccessfully working on these problems for many years. Should you wish to discuss aspects of these problems with others, one of the newsgroups, such as sci.math , might be the appropriate forum. 3-Sep-1995 Problem 36 : Primes of the form n^n+1 27-Aug-1995 Problem 35 : Must one of n points lie on n/3 lines? 20-Aug-1995 Problem 34 : Squares with Two Different Decimal Digits 13-Aug-1995 Problem 33 : Unit Triangles in a Given Area 6-Aug-1995 Problem 32 : Can the Cube of a Sum Equal their Product 30-Jul-1995 Problem 31 : Different Number of Distances 23-Jul-1995 Problem 30 : Sum of Four Cubes 16-Jul-1995 Problem 29 : Fitting One Triangle Inside Another 9-Jul-1995 Problem 28 : Expressing 3 as the Sum of Three Cubes 2-Jul-1995 Problem 27 : Factorial that are one less than a Square 25-Jun-1995 Problem 26 : Inscribing a Square in a Curve 18-Jun-1995 Problem 25 : The Collatz Conjecture 11-Jun-1995 Problem 24 : Primes Between Consecutive Squares 4-Jun-1995 Problem 23 : Thirteen Points on a Sphere 28-May-1995

7. NewsPro Archive Giuliano has found the asked Langoford Numbers. See Puzzle 144. Posted onSaturday, September 22, 2001. Alan Tyte and the twin primes conjecture. http://www.primepuzzles.net/new/arc8-2001.html

Home Melancholia References Links ... Puzzlers News Archive: September 2001 Jud McCranie confirms up to certain limit the conjecture 25 Posted on Sunday, September 30, 2001 Posted on Thursday, September 27, 2001 Several Contributions F. Marchant and Leadhyena Inrandomtan add enthusiastic criticisms to Mr. Alan Tyte's proof (Conjecture 3) Walter Schnider found a new record sequence, 7 members, for the autodescriptive primes (Puzzle 36) Jean Brette adds some interesting contributions to Puzzle 150 Posted on Wednesday, September 26, 2001 Walter Schnider and the Puzzle 36 Walter Schnider found another sequence of length 6 for the Puzzle 36 Posted on Sunday, September 23, 2001 Giuliano Dadario and the Puzzle 144 Giuliano has found the asked Langoford Numbers. See Puzzle 144 Posted on Saturday, September 22, 2001 Alan Tyte and the Twin Primes Conjecture Alan has constructed an analytical proof that shows that there are more twins in the range (q, q^2) than in the range (p, p^2) being q the following prime to p. But if this is rigorously true then it's also true than in each new range we have at least one more twin prime. Being the quantity of primes infinite, this also means that the number of twin primes is infinite.See Posted on Wednesday, September 19, 2001

8. F. Conjectures (Math 413, Number Theory) A collection of easily stated conjectures which are still open. Each conjecture is stated along with Category Science Math Number Theory Open Problems...... The twin primes conjecture Prime Gaps. Def Twin primes are a pair ofprimes of the form n , n +1. Examples are n = 3, 5, 11, 17, 29, http://www.math.umbc.edu/~campbell/Math413Fall98/Conjectures.html

F. Conjectures Number Theory, Math 413, Fall 1998 A collection of easily stated number theory conjectures which are still open. Each conjecture is stated along with a collection of accessible references. The Riemann Hypothesis Fermat Numbers Goldbach's Conjecture Catalan's Conjecture ... The Collatz Problem The Riemann Hypothesis Def: Riemann's Zeta function, Z(s), is defined as the analytic extension of sum n infty n s Thm: Z( s )=prod i infty p i s , where p i is the i th prime. Conj: The only zeros of Z( s ) are at s s Thm: The Riemann Conjecture is equivalent to the conjecture that for some constant c x )-li( x c sqrt( x )ln( x where pi( x ) is the prime counting function. Refs: Unsolved Problems Elementary Number Theory http://www.utm.edu/research/primes/notes/rh.html (notes in Chris Caldwell's Prime Pages) http://match.stanford.edu/rh/ (Dan Bump's notes) Def: n is perfect if it is equal to the sum of its divisors (except itself). Examples are 6=1+2+3, 28, 496, 8128, ... Def: The n th Mersenne Number, M n , is defined by M n n Thm: M n is prime implies that n n is perfect. (Euclid)

9. The Anti-Divisor 1)+' ')script /head /html . Click here for the output Infinitenumber of twin primes conjecture. The base number x for http://www.users.globalnet.co.uk/~perry/maths/primegen.htm

The Anti-Divisor [Home] Generating Primes Anti-divisor theory can be used to generate the prime numbers. The theory goes like this: A number is prime iff cad[k,k+1]=0 (cad represents Common Anti-Divisors). This can be seen from the derivation of anti-divisors from 2n-1, 2n and 2n+1. So, if a number y can be written as y-a=(2a+1)b, then it follows that y+1 shares a CAD with y (e.g. 22 is 3.7+1, so 23 is 3.7+2, and both cad[22,23] contains 3. Or, 17=3.5+2, 18=3.5+3, so cad[17,18] contains 5.) A number therefore is prime iff it is not expressible as (2ab+a+b)+(2ab+a+b+1). This is fairly obvious, this equals 4ab+2a+2b+1, which equals (2a+1)(2b+1), and so represents the set of all odd composites. An odd number not in the composites is prime. And so, if a number y is not generated in an exhaustive mapping of 2ab+a+b, then 2y+1 is prime. Here is the program: Click [here] for the output: Twin Primes This can also be used to derive the twin primes, and also offers insight to prove that there are an infinite number of twin primes. An number x with only even anti-divisors forms a twin prime pair (2x-1,2x+1). This is obvious from the fact that both 2x-1 and 2x+1 must be prime, otherwise the number would contain an odd anti-divisor.

10. Sci.math FAQ: Unsolved Problems Collatz Problem * Goldbach's conjecture * twin primes conjecture _Names of large numbers http://isc.faqs.org/faqs/sci-math-faq/unsolvedproblems/

sci.math FAQ: Unsolved Problems Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76LD.Fnt@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math hv@cix.compulink.co.uk (Hugo van der Sanden): To the best of my knowledge, the House of Commons decided to adopt the US definition of billion quite a while ago - around 1970? - since which it has been official government policy. dik@cwi.nl (Dik T. Winter): The interesting thing about all this is that originally the French used billion to indicate 10^9, while much of the remainder of Europe used billion to indicate 10^12. I think the Americans have their usage from the French. And the French switched to common European usage in 1948. gonzo@ing.puc.cl alopez-o@barrow.uwaterloo.ca By Archive-name By Author ... Help Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update March 05 2003 @ 01:20 AM

11. Sci.math FAQ: Table Of Contents Collatz Problem + Goldbach's conjecture + twin primes conjecture * Symbolic ComputationPackages * Fields Medal + Historical Introduction + Table of Awardees http://isc.faqs.org/faqs/sci-math-faq/tableofcontents/

sci.math FAQ: Table of Contents Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76Iz.1ww@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math alopez-o@barrow.uwaterloo.ca Thu Dec 08 20:49:28 EST 1994 By Archive-name By Author By Category By Newsgroup ... Help Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update March 05 2003 @ 01:20 AM

12. Prime Numbers Conjectures. Conjecture(twin primes conjecture) The number of twin primes(two primes that differ by 2 ) is infinite. Conjecture(Goldbach). http://ppatten.ngc.peachnet.edu/math4310/numthy1_2.html

Prime Numbers Paul R. Patten Dept. of Mathematics December 21, 2000 Objectives After studying this section you should be able to: identify a positive integer as a unit, prime number, or a composite number; prove that there are an infinite number of primes and identify the only even prime number; use the sieve of Eratosthenes to construct a list of prime numbers; state the prime number theorem (this is a statement about estimating the number of primes, , less than a given real number); identify special types of primes such as twin primes, Mersenne primes, and Fermat primes. state famous conjectures about primes such as the Golbach conjecture, twin primes conjecture, etc. explain the difference between a conjecture and a theorem. Some Definitions Prime Number An integer is said to be a prime number if and only if its only positive divisors are and Composite Number An integer is said to be a composite number if and only if there is an integer such that and Unit The integers and are considered to be units of the integers. These integers are neither prime nor composite.

13. Untitled Fermat primes. \item state famous conjectures about primes such as theGolbach conjecture, twin primes conjecture, etc. \item explain http://ppatten.ngc.peachnet.edu/math4310/numthy1_2.tex

14. Www.math.niu.edu/~rusin/known-math/99/brun_const I am devoting most my life to battling the twin primes conjecture and I need asmany theorems in my arsenal as I can get. any help wound be appreciated. http://www.math.niu.edu/~rusin/known-math/99/brun_const

From: Andreas Homrighausen Subject: Re: Twin Primes Date: Wed, 10 Feb 1999 13:13:54 +0100 Newsgroups: sci.math Keywords: Brun's constant Janus xPuN wrote: > > Hey, what are some already proven theorems about twin primes? I am devoting > most my life to battling the Twin Primes Conjecture and I need as many theorems > > in my arsenal as I can get. any help wound be appreciated. of course, > anything about just normal primes would also be quite handy. Thanks. Hello all! This is one of the most impressive theorem about twin primes: The sum of the reciprocals of the twin primes is convergent. B=(1/3+1/5)+(1/5+1/7)+... is called Brun's constant. The sum of the reciprocals of primes is divergent. Greetings, Andreas

15. Some Assertions About Primes Assertion 3 is the ``twin primes conjecture''. According to http//perso.wanadoo.fr/yves.gallot/primes/chrrcds.html twinon May http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/lecture3/html/node2.htm

Next: Some Tools for Computing Up: Lecture 3: Introduction to Previous: Introduction Some Assertions About Primes A computer can quickly ``convince'' you that many assertions about prime numbers are true. Here are three. The polynomial takes on infinitely many prime values. Let and is prime With a computer, we quickly find that Surely is unbounded! The PARI code to compute is very simple: ? f(n) = s=0; for(x=1,n,if(isprime(x^2+1),s++)); s ? f(100) %1 = 19 ? f(1000) %2 = 112 ? f(10000) %3 = 841 ? f(100000) %4 = 6656 Every even integer is a sum of two primes. With a computer we find that this seems true ... and much further. In practice, it's easy to write an even number as a sum of two primes. Why should there be any weird even numbers out there for which this can't be done? PARI code to find and There are infinitely many primes such that is also prime. Let and is prime . Using a computer we quickly find that The PARI code to compute is very simple: ? t(n) = s=0; forprime(p=2,n,if(isprime(p+2),s++)); s ? t(10^2) %12 = 8 ? t(10^3) %13 = 35 ? t(10^4) %14 = 205 ? t(10^5) %15 = 1224 Surely keeps getting bigger!!

16. Www.faqs.org/ftp/faqs/sci-math-faq/unsolvedproblems * Collatz Problem * Goldbach's conjecture * twin primes conjecture _Names of large numbers http://www.faqs.org/ftp/faqs/sci-math-faq/unsolvedproblems

Newsgroups: sci.math,sci.answers,news.answers Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!spool.mu.edu!torn!watserv3.uwaterloo.ca!undergrad.math.uwaterloo.ca!neumann.uwaterloo.ca!alopez-o From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math FAQ: Unsolved Problems Summary: Part 18 of many, New version, Originator: alopez-o@neumann.uwaterloo.ca Message-ID:

17. Sci.math FAQ: Unsolved Problems twin primes conjecture There exist an infinite number of positive integersp with p and p+2 both prime. See the largest known twin prime section. http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/unsolved.html

Note from archivist@cs.uu.nl : This page is part of a big collection of Usenet postings, archived here for your convenience. For matters concerning the content of this page , please contact its author(s); use the source , if all else fails. For matters concerning the archive as a whole, please refer to the archive description or contact the archivist. Subject: sci.math FAQ: Unsolved Problems This article was archived around: 17 Feb 2000 22:55:51 GMT All FAQs in Directory: sci-math-faq All FAQs posted in: sci.math Source: Usenet Version Archive-name: sci-math-faq/unsolved Last-modified: February 20, 1998 Version: 7.5 http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick

18. Maths Thesaurus Turing, Turn, Turtle, Twelfth, Twelve. Twenties, Twentieth, Twenty, Twice, Twinprime conjecture. Twin primes, twin primes conjecture, Two, Two dimensional, Twothirds. http://thesaurus.maths.org/dictionary/map/indices/T

T (257 terms) t test t-distribution t-table t-test ... Type II error

19. Math 314 property. The twin primes conjecture. There are infinitely many primenumbers p such that p+2 is also a prime. The n 2 + 1 Conjecture. http://mathserv.monmouth.edu/coursenotes/kuntz/math314/m31412.htm

Math 314 - Number Theory Lecture Notes #12 Chapters 13 Counting Primes Remark: Some preliminary ideas regarding a "counting" concept. (ev(x)/x) = 1/2 n and n 2 (mod 5) f(x)/x = 1/5. This is exercise 13.1. 3. Exercise 13.2 (b). Hint: observe that sq(x) = int(sqrt(x)) then show that lim sq(x)/x = There's a remarkable theorem that was eventually proven by Hadamard and Poussin (apromiately 1896) which states that the number of primes less than or equal to a given number is approximately x/log e x . You should read the historical account on pp. 83 and 84. More formally the theorem is: Theorem 52 (Prime Number Theorem): Let x ) denote the of primes x . Then There are many unsolved problems involving prime numbers. The author lists a few of the famous conjectures. For your convenience, they are listed below. Be sure to read pp. 84 and 86 which provide some historical commentary on them. Goldbach's Conjecture . Every even number greater than 3 is the sum of two primes. Note: This is an interesting conjecture because the concept of prime involves a multiplication property while the conjecture involves an addition property. The Twin Primes Conjecture.

20. Citations: Unsolved Problems In Number Theory - Guy (ResearchIndex) 3,5) 5,7) 11, 13) and (41,43) are all twins. The twin primes conjectureproposes that twin primes occur infinitely often13. Although http://citeseer.nj.nec.com/context/9372/0

52 citations found. Retrieving documents... R. K. GuY, Unsolved Problems in Number Theory , Springer-Verlag, 2 ed., 1994. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: First 50 documents Next 50 Experiences With Distributed Computation Of Twin - Primes Distribution Patrick (Correct) ....We will also describe the first application of our farmer worker system: computing twin primes distribution. This application was chosen both because it is a good representative of a class of applications which lends itself well to the farmer worker paradigm and its importance to number theory We have already collected this information for an interval an order of magnitude larger than any previously computed and will have another order of magnitude on completion of the project. A prime number is a positive integer that is evenly divisible by exactly two positive integers: itself and .... ....that is evenly divisible by exactly two positive integers: itself and one. Two subsequent odd numbers that are both primes are called twin primes or twins for short. 3,5) 5,7) 11, 13) and (41,43) are all twins.

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