1. Several Proofs Of The Twin Primes Conjecture Goldbach's Conjecture proves and extends the twin primes conjecture as probable. http://www.coolissues.com/mathematics/Tprimes/tprimes.htm

SEVERAL PROOFS OF THE TWIN PRIMES AND GOLDBACH CONJECTURES James Constant math@coolissues.com Proof of Goldbach's Conjecture, the Prime Number Theorem, and Euclid's Logic Provide Proofs of the Twin Primes Conjecture. Proof of the Twin Primes Conjecture Provides Proof of Goldbach's Conjecture Theorem There are infinitely many twin primes. Proof of the Twin Primes Conjecture Using Proofs of Goldbach's Conjecture or Using the Prime Number Theorem The twin primes conjecture (TPC) suggests that there is an infinite number of primes a and b with a difference , i.e., a - b = 2. Goldbach's conjecture (GC) suggests that every even number greater than is the sum s of two prime numbers a and b , i.e., a + b = s where s is even GC is proved by the author herein below and elsewhere For prime numbers a,b,c a - b = (a + c) - (b + c) even integer and thus, generally, a - b = 2k k = integer and since a + b is an even number a + b = 2n Now, using (2) and (3) results in a = n + k and b = n - k which say that for every single value of k primes a and b are separated by an interval and occur as numbers n + k and n - k . Suppose that n ,n ,n , . . . ,n

2. The Top Twenty Twin Primes Related Pages. Twin Primes from the World of Mathematics. Brun's Constant. HardyLittlewood Constants. Two Hardy-Littlewood Conjectures http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

3. The Prime Glossary Goldbach's Conjecture Goldbach's conjecture Every even integer n greater than two is the sum of two primes. of two primes, G(n), is asymptotic to twice the twin http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

4. Introduction To Twin Primes And Brun's Constant Computation a more general conjecture on the primes separated by a gap of d. A natural generalization of the twin primes is to search for primes distant http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. MercuryNews.com 03/26/2003 SJSU Math Professor Breaks Barrier with something called the twin prime conjecture. top 20 twin primes can be found on the Web at www.utm.edu/research/ primes/lists/top20/ http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. Several Proofs Of The Twin Primes Conjecture Goldbach's Conjecture proves and extends the twin primes conjecture as probable. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Sci.math Message All Discussions sci.math Topic Message previous next Message Re Svara that the twin primes conjecture is Unprovable http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. Sci.math Message Subject Re Another twin primes conjecture Author Gib Bogle bogle@ihug.too.much.spam.co.nz Date Posted Jun 19 2004 62401000PM http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. Major Advance On The Twin Primes Conjecture Major Advance on the twin primes conjecture By Keith Devlin http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Major Advance On The Twin Primes Conjecture The twin prime conjecture says that there are infinitely many socalled twin-primepairs, that is, pairs of primes of the form p and p + 2 (such as 3 and 5 http://www.maa.org/news/052505twinprimes.html

Search MAA Online MAA Home Major Advance on the Twin Primes Conjecture By Keith Devlin One way to approach this problem is to look at the gaps between successive primes. For example, if p(1), p(2), p(3), ... denotes the sequence of all primes, are there infinitely many values of n for which p(n+1) - p(n) is less than 10, say, or less than 10 ? If you can find a gap for which there are infinitely many pairs of successive primes that differ by no more than that gap, maybe you can start to bring the gap down. If you get the gap down to 2, you will have solved the twin prime conjecture. A variation of this more general question is to forget the idea of looking for a fixed gap, and instead compare the gaps between pairs of successive primes with the average size of p(n+1) - p(n), which the prime number theorem tells us is asymptotically equal to log p(n). This comes down to looking at the number D = lim inf It was back to the drawing board, as Goldston and Yildrim sought a different approach. Finally, in December of last year, working with Janos Pintz of the Hungarian Academy of Sciences, their joint efforts seemed to have paid off. They found a new proof that worked. Mindful of the events of two years previously, they sent their new proof to various experts, including Granville. Only when the word came back that the new proof was correct did they "go public", publishing an abstract on the arXiv website . (Reference number math.NT/0505300.) Goldston lectured on the new proof at a meeting in New York last week and formally announced the proof in a lecture at the American Institute of Mathematics research center in Palo Alto.

11. Twin Prime Conjecture as the HardyLittlewood conjecture (after G. H. Hardy and John Littlewood), which is concerned with the distribution of twin primes, in http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

12. Approaching A Twin Primes Conjecture Proof Approaching a twin primes conjecture proof http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. The Anti-Divisor Infinite number of twin primes conjecture. The base number x for a twin prime Therefore the twin primes conjecture is equivalent to proving that there 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.

14. *Concerns Of Young Mathematicians* || Comments || Twin Prime Conjecture Proven? Twin Prime Conjecture Proven? 4 comments (3 topical, 1 editorial, 0 hidden) Post A Comment. new twin primes conjecture (none / 0) ( 2) http://concerns.youngmath.net/comments/2004/5/29/131042/934/2

The Young Mathematicians' Network Serving the Community of Young Mathematicians Sections: Front Page News Grad Life Undergrad Life ... All Display: Threaded Minimal Nested Flat Sort: Unrated, then Highest Highest Rated First Lowest Rated First Ignore Ratings Newest First Oldest First Twin Prime Conjecture Proven? comments (3 topical, 1 editorial, hidden) [new] TWIN PRIMES CONJECTURE none / 0 by Anonymous Hero on Mon May 16th, 2005 at 08:54:54 PDT There are intuitive arguments for Goldbach ... by overconvergent, 05/18/2005 15:19:36 PDT ( none / Twin Prime Conjecture Proven? comments (3 topical, 1 editorial, hidden) Display: Threaded Minimal Nested Flat Sort: Unrated, then Highest Highest Rated First Lowest Rated First Ignore Ratings Newest First Oldest First Menu Submit a Story create account FAQ Search ... Recent Comments Login Make a new account Username: Password: submit story create account faq search

15. Twin Primes Conjecture twin primes conjecture. twin primes conjecture. There exist an infinite numberof positive integers p with p and p+2 both prime. 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

16. Mersenneforum.org - Twin Primes Conjecture published a manuscript with a proof of the twin prime conjecture. In fact,no proof of the twin primes conjecture has been constructed despite the http://www.mersenneforum.org/archive/index.php/t-2550.html

mersenneforum.org Great Internet Mersenne Prime Search Math PDA View Full Version : Twin Primes Conjecture Bob Silverman May 28th, 2004, 03:25 PM In case noone has heard, Richard Arenstorf (emeritus, Vanderbilt) has published a manuscript with a proof of the twin prime conjecture. It is too early to say whether the paper might contain errors (I have skimmed it and probably lack the skills to determine any subtle errors). However, the basic approach looks sound to me and represents some serious analytic number theory. Basically: Let L(n) be the VonMangoldt function. Arenstorf considers the Dirichlet that T(s) -B/(s-1) has an analytic continuation onto Re(s) = 1. This allows Indeed, it allows showing that the density of twin primes (the constant B) I do not know the details of the Tauberian Thm, nor how it is applied. [note for the uninitiated. A Tauberian Thm allows one to deduce local behavior of a function from its long term ergodic behavior] I will be trying to read the paper in depth this weekend. I will need to do so with references at my side.

17. Twin Primes Conjecture / Prime Sieve twin primes conjecture / Proof via Prime Sieve Method. An almost proof of theThe twin primes conjecture About John Send email to John http://members.aol.com/SciRealm/TwinPrimes.html

Twin Primes Conjecture / Proof via Prime Sieve Method An almost proof of the The Twin Primes Conjecture... About John Send email to John John's Science, Math, Philosophy Stuff: The Science Realm: John's Virtual Sci-Tech Universe 4-Vectors Antipodes Cyrillic Projector ... Turkish Grammar FHS Business Websites: FHS Supply, Inc. FHS Racing Oils FHS Red Max Model Fuels FHS SmokeLess Oil Welcome to Quantum Reality: Virtual worlds of imaginary particles: The dreams stuff is made of: Life, the eternal ghost in the machine... This site is dedicated to the quest for knowledge and wisdom, through science, mathematics, philosophy, invention, and technology. May the treasures buried herein spark the fires of imagination like the twinkling jewels of celestial light crowning the midnight sky... ***Learn, Discover, Explore*** Site last modified: 2005-Feb-17-7pm Note: This work is not finished, but if you notice errors or have comments, please let me know. Twin Primes Conjecture / Proof via Prime Sieve Method The Prime Sieve is a constructive method or algorithm for finding prime numbers. This document will analyze the method in some detail, hopefully adding to our mathematical knowledge. Summary of Twin Primes Conjecture Proof.

18. Crooked Timber » » Riemann Hypothesis Proved ? Let’s put the twin primes conjecture another way Consider the How often isit a product of two primes? If the twin primes conjecture is true, http://crookedtimber.org/2004/06/11/riemann-hypothesis-proved

Crooked Timber Main Operation Bagration Riemann hypothesis proved ? Posted by John Quiggin According to this report It is also, I think, the last of the big and well-known unsolved problems in mathematics, and it would be nice to see the search ending in success. Some of the other big problems have been closed, rather than solved. The classic problems of the Greeks such as squaring the circle were shown to be insoluble in the 19th century, and the Hilbert program of formalisation was shown by Godel to be infeasible. And the four-colour problem (not a really important problem, but a big one because it was easily described, interesting and very tough) was dealt with by a brute-force computer enumeration. Almost instant update Commenter Eric on my blog points to Mathworld posted on Friday, June 11th, 2004 at 3:54 am comments Posted by Jason Kuznicki June 11th, 2004 at 4:34 am Concerning the purported proof of the twin prime conjecture, the following is from http://www.youngmath.net/concerns/story/2004/5/29/131042/934 Posted by Michael Kremer · June 11th, 2004 at 4:45 am

19. F. Conjectures (Math 413, Number Theory) The twin primes conjecture Prime Gaps. Def Twin primes are a pair of primesof the form { n , n +2}. Examples are { n , n +2} = {3, 5}, {5, 7}, {11, 13}, 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. Thm: The only zeros of Z( s ) are at s s Conj: The only zeros of Z( s ) are at s =-2, -4, -6, ... and on the line Re( 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

20. ABC Online Forum Subject re twin primes conjecture, post id 1104411. What value of pi did he use?(nice result )). From Chris W (Avatar), 29/05/2004 93615 AM http://www2b.abc.net.au/science/k2/stn/archives/archive79/newposts/1104/topic110

From: Chris W (Avatar) 29/05/2004 9:30:28 AM Subject: Twin Primes Conjecture post id: 1104408 Apologies if this is a repost... Following text from slashdot.org: "R. F. Arenstorf [1] from Vanderbilt University has presented a 38-page possible proof of the twin-prime conjecture [2] using methods from classical analytic number theory. The paper [3] is on arxiv.org and is freely available to the public. Twin primes are pairs of primes where both p and p + 2 are prime. "It is conjectured that there are an infinite number of twin primes ... but proving this remains one of the most elusive open problems in number theory." More information about twin primes can be found on Mathworld [4]." [1] http://www.math.vanderbilt.edu/faculty/Arenstorf.html [2] http://mathworld.wolfram.com/TwinPrimeConjecture.html [3] http://arxiv.org/abs/math.NT/0405509 [4] http://mathworld.wolfram.com/TwinPrimes.html From: Cusp 29/05/2004 9:34:31 AM Subject: re: Twin Primes Conjecture post id: 1104411 What value of pi did he use? (nice result :)) From: Chris W (Avatar) 29/05/2004 9:36:15 AM Subject: re: Twin Primes Conjecture post id: 1104412 Be nice to see it stand up to scrutiny. I won't be the one to knock holes in it though :)

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