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
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
Extractions: <= 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
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
Extractions: Index Math 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
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
Extractions: 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.
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
Extractions: 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. 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 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? 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)
"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
Extractions: 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.
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
Extractions: 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.
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
Extractions: Introduction Results Acknowledgements References ... [Up] 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
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
Extractions: Next: Pascal's triangle revisited Up: Some elementary number theory Previous: Euler's phi function Contents Index 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?
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
Extractions: Next: Problems involving congruences Up: Multiplicative Number Theory and Previous: 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.
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
Extractions: Mon, 19 Jun 2000 16:30:04 +0200 (MET DST) 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"
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
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
Extractions: 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.
Extractions: 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.
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
Extractions: 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:
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
Extractions: Problems with Primes Other Facts About Primes Unproved Conjectures References 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.
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
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
Extractions: 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 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
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
Extractions: 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
