Page 015 two primes, Kexue Tongbao 17 (1966), 385386. In this paper, Jing Run Chen stateshis famous theorem saying that both Goldbach's conjecture and the twin prime http://www.math.utoledo.edu/~jevard/Page015.htm
Twin Prime Conjecture twin Prime conjecture. The twin Prime conjecture states that thereare an infinite number of twin primes. A twin prime is defined http://www.users.globalnet.co.uk/~perry/maths/twinprimeconjecture/twinprimeconje
Extractions: Twin Prime conjecture The Twin Prime conjecture states that there are an infinite number of twin primes. A twin prime is defined as a pair of numbers, 6k-1 and 6k+1, such that both are prime. Proof i.e. the TPC is equivalent to the conjecture that there are an infinite number of integers with only even anti-divisors. As 3 as an anti-divisor leaves only multiples of 3 as a candidate, then we only need consider prime anti-divisors greater than or equal to 5. We only need consider prime anti-divisors, as numbers with odd composite anti-divisors also have the prime factors of these composites as anti-divisors. A number with an odd anti-divisor can be written as kp+(p-1)/2 or as kp+(p+1)/2. But we only need to consider integers 0mod3, and this allows us to eliminate some possibilities. To do this, consider the two forms of primes, 6k-1 and 6k+1. Note that these are not twin primes, but that all primes after 3 are of one of these forms. If we look at 6k-1, then the integers we can create are j(6k-1) + 3k - 1 and j(6k-1) + 3k. In both of these cases, if j=1mod3, then neither are divisible by 3, and so we can ignore these possibilites.
Twin Prime Conjecture Proof The six wide array further helps to demonstrate the otherwise still unproven conjecturethat there must be infinitely many twin primes, that is, pairs of http://www.recoveredscience.com/primes1ebook02.htm
Extractions: recoveredscience .com We offer surprises about in our e-book Prime Passages to Paradise by H. PeterAleff Site Contents PRIME PATTERNS Table of Contents Rectangular arrays Twin prime proof Prime facts Prime problems Polygonal numbers Number pyramids ... Reader responses Visit our Sections: Constants Board Games Astronomy Medicine
Extractions: Recently on MathTrek: First Digits 6/27/98 Prime Listening 6/20/98 Coins, Art, and Math in North Bay 6/13/98 July Prime Talent W hole numbers have all sorts of curious properties. Consider, for example, the integer 1998. It turns out that 1998 is equal to the sum of its digits plus the cubes of those digits (1 + 9 + 9 + 8 + 1 ). What's the largest number for which such a relationship holds? The answer is 1998. What about the smallest integer? That was one of the playful challenges presented by number theorist Carl Pomerance of the University of Georgia in Athens to an audience that included the eight winners of the 27th U.S.A. Mathematical Olympiad, their parents, assorted mathematicians, and others. The occasion was the U.S.A.M.O. awards ceremony on June 8 at the National Academy of Sciences in Washington, D.C. Cubes of Perfection ). What's the smallest prime the sum of whose digits is perfect? The answer is 1999. The prime 1999 also comes up in another context: 1999 = 2 . Pomerance calls any positive integer
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/HSSP2000/Classes/osm/ooze/twinPrimeNumberTheory4.
Number Theory remarks is to give a tight characterization of twin primes greater than three. Itis hoped that this might lead to a decision on the conjecture that infinitely http://www.math.utah.edu/~gold/numbertheory.html
Extractions: Jeffrey Frederick Gold Mathematical Interests: Twin Primes, Experimental Number Theory, Elementary Number Theory, Chinese Remainder Theorem, Covering Sets, Linear Congruences, Prime Numbers (of course), abundant numbers, odd perfect numbers, group theory, Galois theory, vectors, and more. Don H. Tucker and I have been working on the Twin Prime Conjecture for about six or seven years now. We have developed a mathematical algorithm which, when tested using a computer analogue, correctly predicted the twin primes in ascending order up to 5,000,000. Of course, the computer is never a proof (except maybe by intimidation), so we have been working on the induction argument for quite some time. It always seems to be within grasp, and just when I'm about to say, "Oh, to hell with it," I stare back down onto the page and the numbers give me something, they always give me something, something to come back and work on the problem again. Damn! I thought I'd get away!!!! A Characterization of Twin Prime Pairs, (with Don H. Tucker). Proceedings - Fifth National Conference on Undergraduate Research, Volume I, pp. 362-366, University of North Carolina Press, University of North Carolina at Asheville (UNCA), 1991. Abstract The basic idea of these remarks is to give a tight characterization of twin primes greater than three. It is hoped that this might lead to a decision on the conjecture that infinitely many twin prime pairs exist; that is, number pairs
Jeffrey Gold's Curriculum Vitae On a conjecture of Erdös , (with Don H. Tucker). Broken Symmetry inprimes and twin primes. (with Don H. Tucker). In preparation. http://www.math.utah.edu/~gold/vitae.html
Extractions: University of Utah, Minor, Mathematics, June 1996. University of Cambridge (United Kingdom). Fitzwilliam College , Department of Physics, Cavendish Laboratories, Microelectronic Research Centre, 1996-97. Have lived in Moline, Illinois; Castroville, California; Heidelberg, Germany Las Cruces, New Mexico ; Bangor, Maine; Honolulu, Hawaii; Annapolis, Maryland ; Cambridge, England; Albi, France; and am currently living in Salt Lake City, Utah. Fluent in German and English. Hobbies include fishing, softball, soccer, sailplane gliding, and number theory. A Characterization of Twin Prime Pairs , (with Don H. Tucker). Proceedings - Fifth National Conference on Undergraduate Research, Volume I, pp. 362-366, University of North Carolina Press, University of North Carolina at Asheville (UNCA), 1991.
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.
11N: Multiplicative Number Theory Numerical data for the twinPrime conjecture. Brun's constant (sum of reciprocalsof all twin primes; Brun's constant counting twin primes. http://www.math.niu.edu/~rusin/known-math/index/11NXX.html
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"
FOM: Re: Twin Primes Again again; Next message FOM Re twin primes again; Peter Schuster wrote I understandfrom your contributions that the twin prime conjecture is something http://www.cs.nyu.edu/pipermail/fom/2000-June/004172.html
Extractions: Wed, 21 Jun 2000 12:40:36 -0400 I understand from your contributions that the twin prime conjecture is something different from Goldbach's conjecture or Fermat's last theorem. Do I correctly understand that, according to your opinion, no position is possible which simultaneously (a) does not assume that the truth-value of such "highly infinitary" statements as the twin prime conjecture is determinated from the outset; (b) does not deny the whole set of integers as a "completed whole", as something "to quantify over"; (c) does not distinguish between statements like "for each integer ..." and the corresponding "universally quantified" formula? Note that (a) is a crucial point for every constructive philosophy, if not for any pragmatic view of mathematics in general; (b) is just what I tend to assign to (Bishop's) constructive mathematics, although Bishop possibly would not agree;
Goldbach Conjecture Verification Computational results and graphics by Tomás Oliveira e Silva.Category Science Math Open Problems Goldbach conjecture In their famous memoir 2, conjecture A, Hardy and Littlewood conjectured that whenn tends to infinity R(n twin p odd prime (p1)^2 is the twin primes constant 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 n p-1 N2(n) = 2 C - PRODUCT - , twin (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
Extractions: We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like fixed probability random events. As the sequence of primes grows, the probability decreases as the reciprocal of the count of primes to that point. The manner of the decrease is consistent with the HardyLittlewood Conjecture, the Prime Number Theorem, and the Twin Prime Conjecture. Furthermore, our probabilistic model, is simply parameterized. We discuss a simple test which indicates the consistency of the model extrapolated outside of the range in which it was constructed. References and citations for this submission:
Maths Thesaurus: Twin Prime Conjecture Home twin prime conjecture The conjecture that there are infinitely manysets of twin primes. This has never been proved. (Find similar words). http://thesaurus.maths.org/dictionary/map/word/976
Math 300 Lesson 4 large. twin Prime conjecture. The number of pairs of twin primes lessthan the number X is approximately 1.32X/(1+1/2+1/3+ +1/X) 2; 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)
More Math News with a prize of $300 Poincaré conjecture Purported Proof Perforated (MathWorld)Two Gigantic primes with Prime A New Pair of twin primes (Science News http://math.smsu.edu/moremathnews.html
Mathematics For Elementary Teachers - "Math Activity 3.3" Find the numbers of twin primes in intervals of 100 (1 to 100, 100 to 200, etc.).Form a conjecture about the occurrence of twin primes for such intervals. http://www.rscs.net/~gb2570/Math_Investigations/MA_4.1.html
Extractions: 2. Pairs of numbers such as 3 and 5, 5 and 7, 11 and 13, whose difference is 2 are called twin primes. It is not known whether or not there are an infinite number of such primes. Find the numbers of twin primes in intervals of 100 (1 to 100, 100 to 200, etc.). Form a conjecture about the occurrence of twin primes for such intervals. 3. In #2 you found consecutive primes whose difference is 2. There are also consecutive primes whose difference is 4, such 7 and 3, 17 and 13, 23 and 19. Check intervals of 100 to see if such primes exist in each interval. Do the number of such primes appear to be increasing or decreasing from interval to interval? 4. Are there consecutive primes whose difference is any even number? For example, consecutive primes whose difference is 6? 8? 10? Are there consecutive primes whose difference is any odd number (1, 3, 5, etc.)? Investigate these questions and form a conjecture based on your evidence. (You may be interested to know that there are conjectures in mathematics involving even numbered differences of primes which as yet are unproved.)
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/pure/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.
Pictures Of Primes, II This is in good agreement with a conjecture of Hardy and Littlewood, which givesthe density of twin primes as 1.32/log(x)^2. The expected number of twin http://www.mathematik.uni-muenchen.de/~forster/primes2.html
Extractions: In the above 128 x 128 matrix the n-th square is black iff the number 10^50 + 2n + 1 is prime. By the prime number theorem, the density of primes in this range is approximatively 1/log(10^50) = 1/115, so we would expect about 2^15/115 = 285 primes in this picture. The actual number is 269. The first prime after 10^50 is 10^50 + 151. There are three twin primes (10^50 + x_i, 10^50 + x_i + 2) in this picture, with x_i = 18307, 19891, 29749. This is in good agreement with a conjecture of Hardy and Littlewood, which gives the density of twin primes as 1.32/log(x)^2. The expected number of twin primes in our interval of length 2^15 = 32768 according to this conjecture is 3.26.. . data file data file data file data file ... Otto Forster (forster@rz.mathematik.uni-muenchen.de), 95-05-29/97-04-30
Mathematics Until these conjecture were tied to FLT, FLT had been regarded by most Largest knowntwin primes The largest known twin primes are 1706595 * 2 ^ 11235 + 1 http://sciboard.louisville.edu/math.html
Extractions: Ans. The status of FLT has remained remarkably constant. Every few years, someone claims to have a proof ... but oh, wait, not quite. Meanwhile, it is proved true for ever greater values of the exponent (but not all of them), and ties are shown between it and other conjectures (if only we could prove one of them), so it has been for quite some time. It has been proved that for each exponent, there are at most a finite number of counter-examples to FLT.
