41. 11N: Multiplicative Number Theory Large departures of pi(x) from Li(x); Review of selected literatureon twin primes; Numerical data for the twinPrime conjecture. http://www.math.niu.edu/~rusin/known-math/index/11NXX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 11N: Multiplicative number theory Introduction History Applications and related fields Subfields Distribution of primes Primes in progressions [See also 11B25] Distribution of integers with specified multiplicative constraints Primes represented by polynomials; other multiplicative structure of polynomial values Sieves Applications of sieve methods Asymptotic results on arithmetic functions Asymptotic results on counting functions for algebraic and topological structures Rate of growth of arithmetic functions Distribution functions associated with additive and positive multiplicative functions Other results on the distribution of values or the characterization of arithmetic functions Distribution of integers in special residue classes Applications of automorphic functions and forms to multiplicative problems [See also 11Fxx] Generalized primes and integers None of the above, but in this section Parent field: 11: Number Theory Browse all (old) classifications for this area at the AMS.

42. From Hlm@math.lsa.umich.edu (Hugh Montgomery) Subject Re De Hardy Littlewood put the twin prime conjecture in a quantitative form The numberof 2ktwin pairs of primes not exceeding x is asymptotic to c(k)x/(log x)^2 http://www.math.niu.edu/~rusin/known-math/00_incoming/polignac

<= x such that p + 2k is also prime but p and p+2k are NOT consecutive primes is O(x/(log x)^3). Thus the stronger H-L conjecture implies de Polignac's conjecture. My expectation is that the twin prime conjecture will be proved by proving the H-L conjecture, and so I regard de Polignac's conjecture as being virtually the same as the twin prime conjecture. Hugh Montgomery

43. ThinkQuest : Library : A Taste Of Mathematic is given by, (5). Proof of this conjecture would also imply the existencean infinite number of twin primes. Define, (6). If there are http://library.thinkquest.org/C006364/ENGLISH/problem/Twin.htm

Index Math A Taste of Mathematic Welcome to A Taste of Mathematics.You will find the taste of mathematics here.The history of Mathematics,famous mathematicians,cxciting knowledge,the world difficult problems and also mathematics in our life... Browsing,thinking,enjoying,and have a good time here! Visit Site 2000 ThinkQuest Internet Challenge Languages English Chinese Students fangfei Beijing No.4 High School, Beijing, China ziyan Beijing No.4 High School, Beijing, China Coaches Tife Zesps3 Szks3 Ogslnokszta3c9cych Numer 1, Beijing, China xueshun Beijing No.4 High School, Beijing, China Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site. Privacy Policy

44. Primarily Primes Every even number can be expressed as the difference of two primes. Can youcheck this conjecture for the even numbers from 2 to 50? twin primes. http://www.dlk.com.au/beingmathematical/numbers/primarily_primes.html

Primarily Primes Prime numbers have been the source of fascination for mathematicians for centuries. Nothing has changed! It has been known for over 2000 years that there are an infinite number of primes. Euclid's proof is claimed to be one of the most beautiful proofs ever written. Euclid's Proof of the existence of an infinite number of prime numbers Every number which is not a prime (called a composite number) is itself divisible by at least one prime. To prove there are an infinite number of primes, let us assume there are not. That is, let's assume P is the largest Prime. We can then prove this is impossible. The primes are - for our sake - 2, 3, 5, 7, 11 ...... P Let us then define Q as: Q = (2 x 3 x 5 x 7 x 11 x ..... x P) + 1. If Q is divided by any of the prime numbers below it, then the remainder will be 1. So it is not divisible by any number less than it other than 1. Hence Q is prime. But Q is bigger than our largest prime P. Hence there cannot ever be a P which is the largest Prime. There are many theories which have been tested without an exception found. But that doesn't mean there is a proof. Here's some:

45. Math 300 Lesson 4 twin Prime conjecture. The number of pairs of twin primes less than the number Xis approximately 1.32X/(1+1/2+1/3+ +1/X) 2; twin Prime conjecture was Stated. http://www.math.odu.edu/~noren/math300/m300sp04.html

The Primes The second of four lessons in Chapter 2. Adjacent Primes Adjacent, or consecutive primes, have no primes between them. 13 and 17 are an example of adjacent primes because no prime lies between them. 17 and 23 are not adjacent primes; 19 lies between them. Large Gaps Among Primes There are gaps as large as we please between adjacent primes. recall 3!=(3)(2)(1), in general, n!=(n)(n-1)...(1) For instance, we may form 200 consecutive non-primes; 201!+2, 201!+3, 201!+4,..., 201!+201. 2 divides 201!+2 3 divides 201!+3 etc., 201 divides 201!+201 In general, for n consecutive non-primes, form (n+1)!+2, (n+1)!+3,..., (n+1)!+(n+1); 2, 3, ... , n+1, respectively, divide these numbers. Twin Primes Twin primes are consecutive odd numbers that are prime. Some examples: 3 and 5, 29 and 31, 71 and 73. Some consecutive odds that are not: 7 and 9, 31 and 33. Are there finitely many or not? Prime Number Theorem Using the notation Pn for the "nth" prime, P1=2, P2=3, P3=5, and so on, then Pn is 'approximately' (n)(1+1/2+1/3+...+1/n). More precisely, if we denote (n)(1+1/2+1/3+...+1/n) by A(n)

46. "The Mathematical Experience" By Philip J Davis & Reuben Hersh No one knows; this is the notorious Goldbach conjecture 1; 3 or 17;19 or 10,006,427;10,006,429which differ by 2? This is the problem of the twin primes, and no http://www.fortunecity.com/emachines/e11/86/mathex5.html

web hosting domain names email addresses The Mathematical Experience 5.The Prime Number Theorem (p209) THE THEORY of numbers is simultaneously one of the most elementary branches of mathematics in that it deals, essentially, with the arithmetic properties of the integers 1, 2, 3,. . . and one of the most difficult branches insofar as it is laden with difficult problems and difficult technique. Among the advanced topics in theory of numbers, three may be selected as particularly noteworthy: the theory of partitions, Fermat's "Last Theorem," and the prime number theorem. The theory of partitions concerns itself with the number of ways in which a number may be broken up into smaller numbers. Thus, including the "null" partition, two may be broken up as 2 or 1 + 1. Three may be broken up as 3, 2 + 1, 1 + 1 + 1, four may be broken up as 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. The number of ways that a given number may be broken up is far from a simple matter, and has been the object of study since the mid-seventeen hundreds. The reader might like to experiment and see whether he can systematize the process and verify that the number 10 can be broken up in 42 different ways. Pierre de Fermat n + y n = z n cannot be solved in integers x, y, z, with xyz

47. Professeur Badih GHUSAYNI The Goldbach and twin Prime conjectures. Zeta of 3. Maple Explorations. Abstract.The twin prime conjecture states that the number of twin primes is infinite. http://www.ul.edu.lb/francais/publ/ghus.htm

Professeur Badih GHUSAYNI Name : Dr. Badih Ghusayni Email : bgou@ul.edu.lb Faculty of Science -1, Department of Mathematics, Lebanese University Research Interests and Specialties Complex and Harmonic Analysis : Entire functions and Fourier Transforms, Representation of Entire Functions by Series and Integrals. Approximation Theory : Approximation by a Nonfundamental Sequence of translates Analytic Number Theory : Tauberian Theorems, Distribution of Primes, Twin Primes, Perfect Numbers, The Zeta Function at Odd Arguments, Factorization and Primality. Computerized Instruction : Maple Publications Books Number Theory from an analytic point of view ISBN 9953-0-0282-7 : Badih Ghusayni Paperback 198 pages Contents Overview of Complex Numbers and Functions. Hadamard Factorization Theorem and Entire Functions of Order One and Infinite Type. The Goldbach and Twin Prime Conjectures. Zeta of 3. Maple Explorations. Function Characterizations. Exploring New Identities with Maple as a Tool. Mersenne Primes, Perfect Numbers, and Friendly Numbers. The Prime Number Theorem from an Analytic Point of View.

48. Crazy News - Strange But True - Weird News - Bizarre And Odd Facts While no one has proved the twin prime conjecture itself, Goldston and Yikdirim tackleda related question Can you find an infinite number of primes that may http://www.crazynews.net/dp/1-135.htm

49. Goldbach Conjecture Verification twin p odd prime (p1)^2. is the twin primes constant. 2 log(x) log(nx). The numericalevidence supporting this conjectured asymptotic formula is very strong. http://www.ieeta.pt/~tos/goldbach.html

Goldbach conjecture verification Introduction Results Acknowledgements References ... [Up] Introduction The Goldbach conjecture is one of the oldest unsolved problems in number theory [1, problem C1] . In its modern form, it states that every even number larger than two can be expressed as a sum of two prime numbers. Let n be an even number larger than two, and let n=p+q , with p and q prime numbers, , be a Goldbach partition of n . Let r(n) be the number of Goldbach partitions of n . The number of ways of writing n as a sum of two prime numbers, when the order of the two primes is important, is thus R(n)=2r(n) when n/2 is not a prime and is R(n)=2r(n)-1 when n/2 is a prime. The Goldbach conjecture states that , or, equivalently, that , for every even n larger than two. In their famous memoir [2, conjecture A] , Hardy and Littlewood conjectured that when n tends to infinity, R(n) tends asymptotically to (i.e., the ratio of the two functions tends to one) n p-1 N2(n) = 2 C PRODUCT - , twin (log n)(log n-2) p odd prime p-2 divisor of n where p(p-2) C = PRODUCT - = 0.6601618158... twin p odd prime (p-1)^2

50. Primes We give a couple of examples of this idea. conjecture 1.5.4 (twin primesconjecture) There are infinitely many primes for which and are prime. http://web.usna.navy.mil/~wdj/book/node16.html

Next: Pascal's triangle revisited Up: Some elementary number theory Previous: Euler's phi function Contents Index Primes Recall an integer is prime if and the only positive integers dividing are and itself. The first few primes are The primes form ``building blocks'' for the integers in some sense (made more precise by the Fundamental Theorem of Arithmetic and by Goldbach's conjecture). We will later see how primes occur in the encryption of information passed over the internet. It has been know since the times of the Greeks that there are infinitely many primes. The following result is one of the oldest and best known results in mathematics! Theorem 1.5.1 (Euclid's Second Theorem) There are infinitely many primes. proof : If denote a sequence of primes then is not divisible by any of these primes. Therefore, the set cannot be all the primes. If there is no finite list of primes then they must be infinite in number. In spite of their basic nature and importance, many questions about primes remain unknown. Question : Given a ``random'' integer is there a ``fast'' method of determining if is a prime or not?

51. Fine Distribution Of Primes of twin primes. The twin prime conjecture is that infinitely manypairs of prime twins exist. This is still unproved today. It is http://www.math.okstate.edu/~wrightd/4713/nt_essay/node18.html

Next: Problems involving congruences Up: Multiplicative Number Theory and Previous: Distribution of primes Fine distribution of primes Besides the basic problem of counting primes, there are many interesting questions about what kinds of special primes exist. For instance, when looking over the list of primes, occasionally we will see pairs like (11,13), (17,19), (71,73), (1031,1033). No matter how far we extend the list, there always seems to appear another prime pair of this kind. A pair of primes of the form p p +2 is called a pair of twin primes. The twin prime conjecture is that infinitely many pairs of prime twins exist. This is still unproved today. It is also unknown whether or not there exist infinitely many primes of the form p n +1, although the list in this case also appears unending, e.g. 5=2 is a sum of three primes. Computers large enough to check all the integers less than or equal to 10 unfortunately do not exist yet. Next: Problems involving congruences Up: Multiplicative Number Theory and Previous: Distribution of primes David J. Wright

52. FOM: Twin Primes Vs. Goldbach Conjecture FOM twin primes vs. Goldbach conjecture. Peter Schuster pschust@rz.mathematik.unimuenchen.deMon, 19 Jun 2000 163004 +0200 (MET DST) http://www.cs.nyu.edu/pipermail/fom/2000-June/004160.html

FOM: twin primes vs. Goldbach conjecture Peter Schuster pschust@rz.mathematik.uni-muenchen.de Mon, 19 Jun 2000 16:30:04 +0200 (MET DST) Previous message: FOM: Re: science and constructive mathematics Next message: FOM: Some thought on "Realism" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] The problem with using Goldbach's conjecture as an example of a possibly indeterminate statement is that it is hard to imagine how it could be both false and unknowable, because a counterexample can be finitely verified. This asymmetry obscures the relationship between "unknowable" and "indeterminate" that I was trying to illustrate. Couldn't also the falsehood of "there are infinitely many twin primes" be finitely veryfied by exhibiting the greatest pair and by giving a proof that it is so? Such a proof might even be simpler than all the calculations necessary for demonstrating that some large even integer is not sum of two prime numbers. Peter Schuster. Previous message: FOM: Re: science and constructive mathematics Next message: FOM: Some thought on "Realism"

53. Twin Primes: An Introduction To Number Theory Answer The number of twin primes is suspected to be infinite, butthat conjecture has not been proven. The cousin primes, 37 http://web.mit.edu/esp/www/Pro/OldPrograms/HSSPS2000/Classes/OSM/ooze/twinPrimeN

Twin Primes: An Introduction to Number Theory Requires: Prime Numbers Twin Primes Let's start by taking an unusual fact, then exploring it. Here is the fact I propose: Isn't is unusual that if you take a pair of twin primes other than 3 and 5, multiply them together, and add one, you get a number that is both A multiple of 36 and a square number 5 x 7 + 1 = 36 = 36 x 1 11 x 13 + 1 = 144 = 36 x 4 17 x 19 + 1 = 324 = 36 x 9 29 x 31 + 1 = 900 = 36 x 25 41 x 43 + 1 = 4164 = 36 x 49 59 x 61 + 1 = 3600 = 36 x 100 etc. First: Why is it always a square number? The Hint... How could you algebraically express the twin primes? How do they relate to the number in between them? How does their product relate to the number in between? The Answer... One way to look at these numbers is that the larger one (for example, 61) is n +1, the smaller (for example, 59) is n -1, and the non-prime in between (for example, 60) is n . Thus, the product of the twin primes is: n +1) x ( n n Thus, when you add one, you get n ^2, which is always a square number. Second: Why is it always a multiple of 36?

54. Science News: Uncovering A Prime Failure - Mathematics - Brief Article few additional lines that there are infinitely many pairs of primes differing by12 or lessa finding almost as strong as the elusive twinprimes conjecture. http://articles.findarticles.com/p/articles/mi_m1200/is_22_163/ai_103565255

@import url(/css/us/style.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); Advanced Search Home Help IN all publications this publication Reference Automotive Business Computing Entertainment Health News Reference Sports YOU ARE HERE Articles Science News May 31, 2003 Content provided in partnership with Print friendly Tell a friend Find subscription deals Uncovering a prime failure - Mathematics - Brief Article Science News May 31, 2003 Mathematicians have returned to the drawing board after what looked like a dramatic step forward in understanding prime numbersthose whole numbers divisible only by themselves and 1. Daniel A. Goldston of San Jose (Calif.) State University and Cem Y. Yildirim of Bogazici University in Istanbul have acknowledged a flaw in work they announced in March, which appeared to say that tight clusters of primes show up among whole numbers no matter how large the numbers are (SN: 3/29/03, p. 195). For more than a century, mathematicians have speculated that there are infinitely many pairs of "twin" primes, such as 11 and 13, which differ only by two. Goldston and Yildirim had created much excitement among number theorists when it appeared that they had come much closer to proving the twin-prime conjecture than others had managed to do in previous attempts. Mathematicians Andrew Granville of the University of Montreal and Kannan Soundararajan of the University of Michigan in Ann Arbor discovered the error in Goldston and Yildirim's work after realizing, to their surprise, that they could adapt the new result to prove in just a few additional lines that there are infinitely many pairs of primes differing by 12 or lessa finding almost as strong as the elusive twin-primes conjecture.

55. Media Advisory: SJSU Math Professor Makes Breatkthrough Discovery Therefore, while the twin prime conjecture remains open, one can now provethat there are many primes that are unusually close together. http://www2.sjsu.edu/news_and_info/releases/032403.htm

Mon., Mar. 24, 2003 Contact: Nancy L. Stake at 408-924-1166 SJSU MATH PROFESSOR MAKES BREAKTHROUGH DISCOVERY The result proved by Goldston and Yildirim is that one can replace 1/4 by any fraction, no matter how small, and further one can prove that there are not just two, but as many primes as you wish within this distance. Therefore, while the twin prime conjecture remains open, one can now prove that there are many primes that are unusually close together. The Problem The problem is related to the famous twin prime conjecture that there are infinitely many prime numbers differing by two (for example 29 and 31). The twin prime conjecture remains unsolved, but mathematicians asked the easier question: whether one could always find prime numbers that are much closer together than the average difference between consecutive primes. The Prime Number Theorem, proved in 1896, showed that for primes of size around x, this average difference is the natural logarithm of x. Many mathematicians have worked on this problem since the famous results of Hardy and Littlewood in 1923, but the best result until now is that there exist infinitely many prime pairs whose difference is less than 1/4 of the average.

56. Homework 5 following open problem in mathematics The twinPrime conjecture There are an infinitenumber of twin primes (two primes are twins if they differ by 2). http://www1.cs.columbia.edu/~tdiament/cs3261/hw5.html

Homework #5 Do problem 3.19 in your text. Prove that the base-2 logarithm of 7 is irrational (Hint: Recall that y = log(7) means that 2^y = 7. Assume that y is rational and arrive at a contradiction). Write (short) computer programs in Java (or any other programming language with which you are comfortable) to perform each of the following tasks (a) List all integers. Let R be the set of real numbers. Let Z be the set of integers. For each of the following functions, draw a graph of the function and state whether it is "one-one" and whether it is "onto". Explain your reasoning for each with reference to the graph. (a) f: R > R, f(x) = x^3 - 12x (b) f: R > R, f(x) = 2^(-x) (d) f: R > Z, f(x) = ceiling(x) (note ceiling(x) = the smallest integer greater than or equal to x). Suppose you had access to a magical black box (also known as an oracle). The oracle is a device that takes two input strings. The first input string is a specification of a Turing Machine M (i.e., a representation of the state diagram of a Turing machine). The second input string x is supposed to be an input to that Turing machine. Once you provide the two inputs and hit a the button on top of the device, the magic box responds instantaneously telling you whether or not the Turing Machine you provided accepts the input string you provided (i.e., if the oracle responds "YES" then that means that M(x) accepts, and if it responds "NO" then that means that the M(x) does not accept). Explain how you would make use of the device to resolve the following open questions in mathematics:

57. Tim Melrose : Problems With Primes whether there are infinitely many of these twin primes. However most mathematiciansbelieve the answer is ‘yes’. A more famous conjecture regarding primes http://www.maths.adelaide.edu.au/people/pscott/history/tim/tmp6.html

Problems with Primes Other Facts About Primes Unproved Conjectures References U nproved Conjectures Primes have a tendency to arrange themselves in pairs of the form ( p p +2): for example 3 and 5, 5 and 7, 17 and 19. This is also evident among much larger numbers such as 29,879 and 29,881. Such primes are called twin primes or prime pairs, A more famous conjecture regarding primes is the Goldbach Conjecture , named after Christian Goldbach (1) Every even number greater than or equal to 4 is the sum of two primes; for example (It is easy to verify that this conjecture fails for odd numbers, 11 or example.) In the letter Goldbach also expressed the following belief: (2) Every integer n greater than or equal to 5 is the sum of three primes. As far as is known, Euler did not prove (1), but neither Euler nor anyone else has been able to find a counter-example. This conjecture has since been tested for all even numbers up to at least 10 and found to be true. This still remains one of the great unsolved conjectures of mathematics. Pierre de Fermat conjectured that is prime for any non-negative integer n . The conjecture was proven to be incorrect by Euler in 1732 who showed that F More recently analysis of these so-called Fermat numbers have found no other primes above F . However no-one has yet proved that F is the largest Fermat prime.

58. The Top Twenty: Twin Primes Littlewood Constants; Two HardyLittlewood Conjectures; The Prime Glossary stwin primes; A table of the number of twin primes to 10 14; The http://primes.utm.edu/top20/page.php?id=1

59. The Top Twenty: Twin Primes twin primes. Home Search Index This Page definition(s) records references related pages. As part of the Prime Pages and its list of the Largest Known primes, we keep a list of the 5000 http://www.utm.edu/research/primes/lists/top20/twin.html

Twin Primes Home Search Index This Page: definition(s) records references related pages As part of the Prime Pages and its list of the Largest Known Primes , we keep a list of the 5000 largest known primes (currently those with 49310 digits or more) plus twenty each of certain selected forms . This page is about one of those forms. Comments and suggestions requested Definitions and Notes Twin primes are pairs of primes which differ by two. The first twin primes conjectured (but never proven) that there are infinitely many twin primes. If the probability of a random integer n and the integer n +2 being prime were statistically independent events, then it would follow from the prime number theorem that there are about n log n twin primes less than or equal to n . These probabilities are not independent, so Hardy and Littlewood conjectured that the correct estimate should be the following. Here the infinite product is the twin prime constant (estimated by Wrench and others to be approximately 0.6601618158...), and we introduce an integral to improve the quality of the estimate. This estimate works quite well! For example: The number of twin primes less than N N actual estimate There is a longer table by Kutnib and Richstein available online.

60. Twin Prime Conjecture -- From MathWorld twin Prime conjecture. There are two related conjectures, each calledthe twin prime conjecture. The first version states that there http://mathworld.wolfram.com/TwinPrimeConjecture.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Mathematical Problems Unsolved Problems ... Prime Clusters 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

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