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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. Conjecture 3. Twin Prime's Conjecture
Problems Puzzles Conjectures. Conjecture 3. Twin Prime's Conjecture. If we define dn as dn = pn+1 pn, is easy to see that d1=1 and dn=even for n 1. Now, its believed that "for n 1, dn=2 infinitely often" ( Ref. 2, p. 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. Goldbach Conjecture And Twin Primes Conjecture Proved
Goldbach Conjecture and twin primes conjecture proved. post a message on this topic. post a message on a new topic. 18 Sep 1999 Goldbach Conjecture and twin primes conjecture proved, by Paul Bruckman. 18 Sep 1999
http://mathforum.com/epigone/sci.math.research/spouphumzhah
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Goldbach Conjecture and Twin Primes Conjecture proved
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18 Sep 1999 Goldbach Conjecture and Twin Primes Conjecture proved , by Paul Bruckman
18 Sep 1999 Re: Goldbach Conjecture and Twin Primes Conjecture proved , by G. A. Edgar
19 Sep 1999 Re: Goldbach Conjecture and Twin Primes Conjecture proved , by Math
19 Sep 1999 Re: Goldbach Conjecture and Twin Primes Conjecture proved , by Greg Kuperberg
20 Sep 1999 Re: Goldbach Conjecture and Twin Primes Conjecture proved , by Math
The Math Forum

 4. The Proof Of Goldbach Conjecture, Twin Primes Conjecture And Other The Proof of Goldbach Conjecture, twin primes conjecture and Other Two Propositions By creating a new method, the author proved the wellknown world's baffling problems Goldbach conjecture, twinhttp://rdre1.inktomi.com/click?u=http://citebase.eprints.org/cgi-bin/citations?i

5. Brun's Constant For Twin Primes - Encyclopedia Article About Brun's Constant For
This is the content of the Twin Prime Conjecture. Had this seriesdiverged, we would have a proof of the twin primes conjecture
http://encyclopedia.thefreedictionary.com/Brun's constant for twin primes
Dictionaries: General Computing Medical Legal Encyclopedia
Brun's constant for twin primes
Word: Word Starts with Ends with Definition In 1919 Viggo Brun Viggo Brun (October 13, 1882 - August 15, 1978) was a Norwegian mathematician. He was born in Lier and died in DrÃ¸bak. His best known contribution is that he showed that the sum of reciprocals of twin primes converges to a sum now called Brun's constant. He proved that by means of devising an improvement over Legendre's version the sieve of Eratosthenes.
Click the link for more information. showed that the sum of the reciprocals of the twin primes A twin prime is a prime number that differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are 5 and 7, 11 and 13, and 821 and 823. (Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin 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.

6. Problem 3.- Erdos Conjecture
**. Roberto Botrugno has detected (17/10/2000) a relationship between theErdÃ¶s conjecture and the twin primes conjecture. In his own words
http://www.primepuzzles.net/problems/prob_003.htm
 Problems Problem 3.- Erdos Conjecture N = n - 2 k k Similar to the Pomerance observation, ErdÃ¶s observation got a simple relation "crowded" with primes. This conjecture has been empirically verified until 2 by Uchiyama Yorinaga Do you wan to try to extend the range ? Or to prove that there is no other numbers ? Can you construct a similar interesting question like or related with this of ErdÃ¶s (Ref. 2, p. 42) Solution At last, a first strong hit on this problem. Imran Ghory has produced (15/9/2000) an almost-solution to this conjecture showing that any further solution larger than 105 - if any - must end in 5, using an extremely simple reasoning. So the Conjecture is now reduced to deal with only one type of numbers. This is his e-mail: "I've managed to almost to prove Problem 3 (Erdos Conjecture), but I'm stuck at the final hurdle; in developing the proof I've managed to show that any further solutions have to be multiples of 165. By the same technique lad(n)=9, 9-(2^2) and for lad(n)=1, 1-(2^4) will mean that in the formula n-2^k, k has a value which means that the formula results in a multiple of 5.

7. 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

 8. TwinPrimeConjecture Twin prime conjecture twin primes conjecture (English). Definition level2. The conjecture that there are infinitely many sets of twin primes.http://thesaurus.maths.org/mmkb/entry.html?action=entryById&id=891

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. Twin Primes Conjecture / Prime Sieve
twin primes conjecture / Proof via Prime Sieve Method. Welcome to TwinPrimes Conjecture / Proof via Prime Sieve Method The Prime
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Twin Primes Conjecture / Proof via Prime Sieve Method
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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.

11. References About The Twin Primes Conjecture By Pbruckman1496
references about the twin primes conjecture by pbruckman1496. reply tothis message post a message on a new topic Back to sci.math.research
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references about the Twin Primes Conjecture by pbruckman1496
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Subject: references about the Twin Primes Conjecture Author: pbruckman1496@my-deja.com Organization: Deja.com - Share what you know. Learn what you don't. Date: http://www.deja.com/ Share what you know. Learn what you don't. The Math Forum

12. Goldbach Conjecture And Twin Primes Conjecture Proved
a topic from sci.math.research Goldbach Conjecture and twin primes conjectureproved. post a message on this topic post a message on a new topic
http://mathforum.org/epigone/sci.math.research/spouphumzhah
a topic from sci.math.research
Goldbach Conjecture and Twin Primes Conjecture proved
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18 Sep 1999 Goldbach Conjecture and Twin Primes Conjecture proved , by Paul Bruckman
18 Sep 1999 Re: Goldbach Conjecture and Twin Primes Conjecture proved , by G. A. Edgar
19 Sep 1999 Re: Goldbach Conjecture and Twin Primes Conjecture proved , by Math
19 Sep 1999 Re: Goldbach Conjecture and Twin Primes Conjecture proved , by Greg Kuperberg
20 Sep 1999 Re: Goldbach Conjecture and Twin Primes Conjecture proved , by Math
The Math Forum

13. Progress In Number Theory. No.1
to be solved of the form prove there are an infinite number of primes of the form These include The Infinite Number of twin primes conjecture and The
http://www.fidn.org/pint/lres/lres.html
 F.I.D.N Progress in Number Theory. No.1 Logical Resonance. (Mills) 2001. Logical Resonance (LR) is the name for an application of the Chinese Remainder Theorem (of Sun Tsu). Discovered by Dr. Paul Mills Ph.D (F.I.D.N Director) in 2001. Notes by Paul Mills: 'LR was the fruit of my research on a specific number theory problem. The solution of Goldbach's Conjecture. '2n can always be expressed as the sum of 2 primes.' (Not, 'the sum of 2 odd primes is always an even integer' which is obvious!). Yet these different logical statements can be described by the same algebraic equation 2n = p+q. This gives a clue to the method. It is possible to uncover mathematical truth by a process of logical deduction and not just algebraic manipulation of equations. My goal was to try and discover an already existing mathematical mechanism that helped this process of logical deduction. It was the unique solution from the Chinese Remainder theorem that proved to be the solution. If my 'hypothesis' could be inserted into a CRT 'machine' so that it formed a valid inference using the CRT, then the hypothesis would be the unique solution and therefore logically true. It is the fact that the CRT admits a unique solution that enables LR to work. If the 'hypothesis' was just one solution among many to the inference then it could not be deduced as actually existing as true in reality. It would remain a 'hypothetical' solution.

14. [math/0309103] The Proof Of Goldbach Conjecture, Twin Primes Conjecture And Othe
From Kaida Shi view email Date Fri, 5 Sep 2003 173833 GMT (221kb) The Proofof Goldbach Conjecture, twin primes conjecture and Other Two Propositions.
http://arxiv.org/abs/math.GM/0309103
Mathematics, abstract math.GM/0309103
From: Kaida Shi [ view email ] Date: Fri, 5 Sep 2003 17:38:33 GMT (221kb)
The Proof of Goldbach Conjecture, Twin Primes Conjecture and Other Two Propositions
Authors: Kaida Shi
Subj-class: General Mathematics
MSC-class:
By creating a new method, the author proved the well-known world's baffling problems Goldbach conjecture, twin primes conjecture, the Proposition (C) and the Proposition \$n^2+1\$.
Full-text: PDF only
References and citations for this submission:
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Links to: arXiv math find abs

15. F. Conjectures (Math 413, Number Theory)
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
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.
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)

16. Brun's Constant
of all primes is divergent. Had this series diverged, we would havea proof of the twin primes conjecture. But since it converges
http://www.fact-index.com/b/br/brun_s_constant.html
Main Page See live article Alphabetical index
Brun's constant
In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by two) converges to a sum now called Brun's constant for twin primes and usually denoted by B Sloane 's in stark opposite contrast to the fact that the sum of the reciprocals of all primes is divergent. Had this series diverged, we would have a proof of the twin primes conjecture . But since it converges, we do not yet know if there are infinitely many twin primes. His sieve was refined by J.B. Rosser, G. Ricci and others. By calculating the twin primes up to 10 (and discovering the infamous Pentium FDIV bug along the way), Thomas R. Nicely heuristically estimated Brun's constant to be 1.902160578. The best estimate to date was given by Pascal Sebah in , using all twin primes up to 10
B
There is also a Brun's constant for prime quadruplets . A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by B , is the sum or the reciprocals of all prime quadruplets: with value:
B
This constant should not be confused with the Brun's constant for cousin primes , prime pairs of the form ( p p + 4), which is also written as

17. The Music Of Primes
Bridges explains one of the great prime number mysteries in his first date withEnglish Literature professor, Barbara Streisand The twin primes conjecture.
http://www.musicoftheprimes.com/films.htm
 The art of maths Maths and the creative arts :: Films back to creative arts index Contact contact-themovie.warnerbros.com Pi Max Cohen, the hero of this film, is a mathematician who spends his life tracking patterns in the decimal expansion of pi. Convinced these numbers hold the key to the stock market and the Jewish kabbalah, he eventually goes crazy. www.pithemovie.com top A beautiful mind http://www.abeautifulmind.com top The cube Six characters wake up inside a complicated system of interconnected cubes. Some of the rooms are trapped. They soon discover that if the number of the room is a prime number then the room contains a deadly trap. The primes are the key to their survival [link to prime number cicadas] http://www.cubethemovie.com

18. 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
Prime numbers
Number theory index History Topics Index
Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians. The mathematicians of Pythagoras 's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
You can see more about these numbers in the History topics article Perfect numbers By the time Euclid 's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

19. Sci.math FAQ: Unsolved Problems
Collatz Problem * Goldbach s conjecture * twin primes conjecture _Names of large numbers
http://www.faqs.org/faqs/sci-math-faq/unsolvedproblems/
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sci.math FAQ: Unsolved Problems
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Help others by sharing your knowledge
Newsgroups: sci.math alopez-o@neumann.uwaterloo.ca 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 Rate this FAQ N/A Worst Weak OK Good Great
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20. John Quiggin: What I'm Reading, And More
The twin primes conjecture doesn t count as a big maths question? Or has it beensolved? So I think RH is in a class above the twin primes conjecture.
http://www.johnquiggin.com/archives/001384.html
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