MATHEWS: Lucky Numbers the density of the twin luckies and the twin primes. a lot of conjectures about primesseem also of the most famous ones, the Goldbach conjecture, stating that http://www.wschnei.de/number-theory/lucky-numbers.html
Extractions: (last updated 24/12/2002) Lucky numbers are defined by a variation of the well-known sieve of Eratosthenes . Beginning with the natural numbers strike out all even ones , leaving the odd numbers 1, 3, 5, 7, 9, 11, 13, ... The second number is 3, next strike out every third number , leaving 1, 3, 7, 9, 13, ... The third number is 7, next strike out every seventh number a.s.o. The numbers surviving are called lucky numbers . The first lucky numbers are (Sloane's A000959): A list of the first 1000 lucky numbers is available here. What's most interesting about lucky numbers is the fact that they share a lot of properties with primes . As can be seen from the next table the density of the lucky numbers is close to the density of the primes. This seems also be true for the density of the twin luckies and the twin primes. In addition a lot of conjectures about primes seem also to be true for the luckies. For example one of the most famous ones, the Goldbach conjecture , stating that each even integer is the sum of at most two primes seems also to be true Because of the many similarities between primes and luckies it seems that a lot of the properties of the primes are just a result of the sieving process!
Ivars Peterson's MathTrek - Prime Listening a certain number demonstrates the scarcity of twin primeseven though after the initialset of primes starts with NJ, has verified that the conjecture is true http://www.maa.org/mathland/mathtrek_7_6_98.html
Extractions: Ivars Peterson's MathTrek July 6, 1998 Whole 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. Each year, the Olympiad competition includes a problem involving the year in which it is held. Looking ahead, Pomerance pondered 1999a prime number, evenly divisible only by itself and 1. In this case, the digits of 1999 add up to 28, which happens to be a perfect number. A perfect number is equal to the sum of all its divisors (see Cubes of Perfection ). What's the smallest prime whose sum of digits is perfect? The answer is 1999.
Ulearn Today - Magazine finding two primes, the largest known twin primes are substantially 2 p 1. The studyof Mersenne primes has been 1588-1648), made a famous conjecture on which http://www.ulearntoday.com/magazine/physics_article1.jsp?FILE=primestory
Goldbach Conjecture Research Information on research and computations on the Goldbach conjecture. By Mark Herkommer.Category Science Math Open Problems Goldbach conjecture Because we know that twin primes exist (two primes whose difference is 2 Thereforehas Goldbach's conjecture been proved TRUE?. well no, not really. http://www.flash.net/~mherk/goldbach.htm
Extractions: June 23, 2002 This conjecture dates from 1742 and was discovered in correspondence between Goldbach and Euler. It falls under the general heading of partitioning problems in additive number theory. Goldbach made the conjecture that every odd number > 6 is equal to the sum of three primes. Euler replied that Goldbach's conjecture was equivalent to the statement that every even number > 4 is equal to the sum of two primes. Because proving the second implies the first, but not the converse, most attention has been focused on the second representation. The smallest numbers can be verified easily by hand: This would suggest that the likelihood of finding that exceptional even number that is not the sum of two primes diminishes as one searches in ever larger even numbers. Euler was convinced that Goldbach's conjecture was true but was unable to find any proof (Ore, 1948). The first conjecture has been proved for sufficiently large odd numbers by Hardy and Littlewood (1923) using an "asymptotic" proof. They proved that there exists an n0 such that every odd number n > n0 is the sum of three primes. In 1937 the Russian mathematician Vingradov (1937, 1954) again proved the first conjecture for a sufficiently large, (but indeterminate) odd numbers using analytic methods. Calculations of n0 suggest a value of 3^3^15, a number having 6,846,169 digits (Ribenboim, 1988, 1995a).
Goldbach's Sequence And Goldbach's Conjecture but if the global Goldbach's sequence is contiguous, then Goldbach's conjecture willbe can pack will consist of one isolated prime and a pair of twinprimes. http://web.singnet.com.sg/~huens/paper43.htm
Extractions: A very efficient way of weeding out unnecessary tests for noncontiguities in Goldbach's sequences, i.e. Goldbach(z), is to test only the high ends of Prime(z). This comes from a theorem on the contiguity of Odd(z)^2 in which it was proved that if the second largest odd integer is removed from Odd(z) before squaring, the resultant even integer sequence is never contiguous [11]. Since Prime(z) is a subset of Odd(z), we know that if Odd(z)^2 is not conitiguous then Prime(z)^2 of the same integer range will not be contiguous. This method is used here to extend the range of search for noncontiguous Goldbach(z) above 10^9. The method is determinstic on noncontiguities only. To determine contiguities, we still need to perform the full contiguity tests.
Fine Distribution Of Primes A pair of primes of the form p, p+2 is called a pair of twin primes. The twinprime conjecture is that infinitely many pairs of prime twins exist. 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.
Professeur Badih GHUSAYNI Abstract. The twin prime conjecture states that the number of twin primes isinfinite. Many attempts to prove or disprove the conjecture have failed. http://www.ul.edu.lb/francais/publ/ghus.htm
Www.mathworks.com/company/pentium/Nicely_3.txt a binary search, the discrepancy was isolated to the pair of twin primes 824633702441and My first conjecture was that the error was again an artifact of the http://www.mathworks.com/company/pentium/Nicely_3.txt
Extractions: TO: Whom it may concern FROM: Dr. Thomas R. Nicely Professor of Mathematics Lynchburg College Lynchburg, Virginia 24501-3199 USA Phone: 804-522-8374 Fax: 804-522-8499 Internet: nicely@acavax.lynchburg.edu RE: Pentium FPU Bug DATE: 94.12.09.2115 EST Enumerated below are some questions that have frequently been posed to me. Each question is followed by my response. Many of these questions were submitted by Dr. Denis Delbecq of the Paris based computer periodical "Science et Vie Micro." Feel free to transmit unmodified copies of this document as you wish. /*************************************************************/ Q1: How can a user check a Pentium machine for the presence of the bug? /**************************************************************/ Perform Coe's calculation (see Question 5 below). That is, carry out the following division problem: 4195835.0/3145727.0 = 1.333 820 449 136 241 00 (Correct value) 4195835.0/3145727.0 = 1.333 739 068 902 037 59 (Flawed Pentium) The division can be done in BASIC, in a spreadsheet (such as Quattro Pro, Excel, or Microsoft Works), in the Microsoft Windows calculator, or in some other programming language such as Pascal, C, or Fortran. Make sure that the FPU has not been disabled (this usually has to be done intentionally through some specific action). /*************************************************************/ Q2: Could you summarize how you discovered the problem? Were you doing research calculations or were you studying the problem of accuracy with computers? /**************************************************************/ RESPONSE: I was pursuing a research project in an area of pure mathematics called computational number theory. Specifically, I have written a code which enumerates the primes, twin primes, prime triplets, and prime quadruplets for all positive integers up to an extremely large limit (currently to about 6e12). The totals are written to a file at intervals of 1e9. Also computed are the sums of the reciprocals of the twin primes, the triplets, and the quadruplets; each of these can be proved to converge to a limit, but the limit of the sum of the reciprocals of the twin primes is known imprecisely, and the others have not been previously computed. My intent is to publish the results in a research journal at such time as I have carried the computation to an extremely large limit (perhaps 20e12) and confirmed the results. The code is written so that the computation can be distributed over a large number of independent systems, with the final results synthesized upon completion. The calculation has run for over a year simultaneously on half a dozen systems; most are 486s, but one Pentium was added in March, 1994. Simultaneously with the calculation of the unknown quantities, a number of checks are maintained by calculating previously published values (such as pi(x), the number of primes 1e15 simulated divisions and reciprocals and have observed zero errors. The critical cases, such as my original example and Tim Coe's example, have also been tested individually. /***************************************************************/ Q8: What about the so-called "workarounds" for the bug? /***************************************************************/ RESPONSE: The workaround suggested by Cleve Moler of MathWorks consists of replacing each division by a function call. The function call first performs the division directly, then tests the answer for correctness (e. g., by comparing x*(y/x) to y). If the result is in error due to the Pentium bug, the numerator and denominator are each multiplied by 3/4 (which destroys the 0xBFFF denominator mask causing the problem) and the division is repeated. This process is continued in a loop until the result checks correctly. I use a similar workaround in my sample code, but use a multiplier of 3 rather than 3/4, which would appear to be two clocks faster. Of course, the workaround only works for applications whose code has been rewritten, recompiled, and reshipped since the bug appeared. Previously existing binaries can avoid the bug only by locking out the FPU (e. g., by setting 87=NO and NO87=NO87 in DOS, or by resetting the emulation bit in the machine status word of CR0 otherwise). The workaround slows the machine down slightly, perhaps 30 % (this is application dependent). Locking out the FPU may slow the machine down by a factor of five or ten, depending on the application. A separate workaround is required if the floating-point remainder instructions, such as fmod or fmodl in C, are used. /***************************************************************/ Q9: Why do you think this particular bug has received an inordinate amount of publicity, making it such a public relations nightmare for Intel? /***************************************************************/ I believe several factors contributed to this phenomenon. * Intel's initial failure to publicize the problem, even in a listing of errata to their OEMs and most valued customers, was in retrospect a mistake which alienated these constituencies. * Intel's subsequent response, once the bug had been detected independently, was considered unsatisfactory by nearly everyone outside the company. * The Pentium CPU has been the subject of a high-profile advertising campaign by Intel. * In contrast to most previous errors found in CPUs, this one occurs in an elementary, frequently-used operation which is easy to demonstrate to the non-specialist, even those who have little or no computer training. * The bug was found late in the life cycle of the chip, after millions of them were already distributed or in production. * The existence of the Internet, and its current widespread availability, caused the news and the reaction to Intel's response to spread much more rapidly than for previous bugs. /***************************************************************/ Q10: Can you tell us something of your own background? /***************************************************************/ I was born 6 February 1943, in Wareham, Massachusetts, but grew up in the coal mining town of Amherstdale, Logan County, West Virginia. My father and most of my male relatives were coal miners; my father died in 1973 due to heart disease caused by black lung disease. I graduated from Man High School in Logan County in 1959; earned a B. S. degree in physics from West Virginia University, Morgantown, West Virginia, in 1963; an M. S. degree in theoretical physics from WVU in 1965; and earned the Ph. D. in applied mathematics from the School of Engineering, University of Virginia, Charlottesville, Virginia, in August, 1971. I have spent nearly all of my professional career as a professor of mathematics at Lynchburg College, Lynchburg, Virginia, beginning in 1968. Lynchburg College is a small (full time undergraduate enrollment about 1420), private, non-profit, coeducational liberal arts college, most generally noted for its excellent programs in the fine arts (dramatic arts, art, music) and its success in Division III (non-scholarship) athletics. The College was founded in 1903 by Dr. Josephus Hopwood, and is an ecumenical, non- sectarian institution affiliated with the Christian Church (Disciples of Christ). I did take a leave of absence in 1985-86 to work as a staff member in X Division (nuclear weapon and nuclear reactor design and analysis) at Los Alamos National Laboratory, Los Alamos, New Mexico, but decided I preferred the academic environment. I also do consulting work for the Avalon Hill Game Company, Baltimore, Maryland, producing the team charts and rules each year for the "Paydirt" tabletop football game originally developed by Sports Illustrated Enterprises, and also the team charts and rules for "Bowlbound," the college football edition of the game. My wife of 21 years is a practicing HVAC mechanical engineer and consultant, Linda Carol Taylor Nicely, a graduate of the School of Engineering at the University of Tennessee. We have no children, but have the good fortune to enjoy the company of six cats. Sincerely, Dr. Thomas R. Nicely
References primes/glossary/PrimeKtupleconjecture.html prime ktuple conjecture. www.utm.edu/research/primes/glossary/WoodallNumber tabpi2.htmlCounts of twin prime pairs http://dmod.digitalrice.com/Report/References.htm
Introduction primes in the region 1 y. The truth of this conjecture is investigated Using C++programs, the twin primes, prime triplets and prime quadruplets in the first http://dmod.digitalrice.com/Report/Introduction.htm
Re: Twin Primes By Antreas P. Hatzipolakis that For every even number 2n are there infinitely many pairs of consecutive primeswhich differ by 2n. (when n=1, the twin Prime conjecture) Source http http://mathforum.org/epigone/math-history-list/thahtwecha/v01540B00AF941276EFD7@
Re: Twin Primes By Robert Redfield There is a nice section on twin primes on pages 145 148 of THE LITTLE BOOK OFBIG primes by Paulo who (and/or when) originated the conjecture of the http://mathforum.org/epigone/math-history-list/thahtwecha/v01540b00af93f5004a8c@
The Mathematical Tourist different from saying that a conjecture can never be proved, for a single breakthroughcould put such suppositions as the number of twin primes suddenly within http://www.fortunecity.com/emachines/e11/86/tourist2b.html
Extractions: The study of prime numbers has long been a central part of number theory, a field traditionally pursued for its own sake and for the beauty of its results . Once thought to be the purest of pure mathematics, this ancient pastime now figures prominently in modern computer science. The security of modern cryptosystems depends very strongly on the twin questions of how easy it is to identify primes and how hard it is to factor a large, random number. Neither question has a clear answer yet. Divisible evenly only by themselves and the number 1, the primes stand at the center of number theory. Like chemical elements in chemistry or fundamental particles in physics, they are building blocks in the mathematics of whole numbers. All other whole numbers, known as composites, can be written as the product of smaller prime numbers. In fact, according to the fundamental theorem of arithmetic , each composite number has a unique set of prime factors. Hence, the composite number 20 can be broken down into the prime factors 2, 2, and 5. No other composite number has the same set of factors. The number 1 is considered to be neither prime nor composite.
"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 related sites 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.
Susan Goldstine's Research Interests It is conjectured but not known that there are infinitely many twin primes. Anotherfamous unsolved conjecture about primes is the Golbach conjecture, which http://www.math.ohio-state.edu/~goldstin/research/nontechnical.html
Extractions: In the Elements, written 23 centuries ago, Euclid gives a famous proof that there are infinitely many primes. The Fundamental Theorem of Arithmetic states that every integer greater than 1 factors uniquely into prime numbers. twin primes . It is conjectured but not known that there are infinitely many twin primes. Another famous unsolved conjecture about primes is the Golbach Conjecture, which states that every even number greater than 4 is a sum of two primes. For instance, A Diophantine problem , named after the ancient Greek mathematician Diophantus, is an equation in one or more variables for which we seek either integer or rational solutions. The most famous of these problems is Fermat's Last Theorem , conjectured by Fermat in the seventeenth century and finally proven in 1994, which states that
Mersenne Prime Search - German Mirror twin primes and SophieGermain primes, Cunningham chains, reserach into the Sierpinskiproblem, find Keller primes, and more! Proving Catalan's conjecture is http://www7.brinkster.com/haugh/prime/projects.asp
Extractions: If in doubt, go to the real GIMPS Home Page This domain was created as a home for the Great Internet Mersenne Prime Search (GIMPS). Mersenne primes are named after the French monk Marin Mersenne . In his day, Marin Mersenne acted as great facilitator among the mathematicians and scientists of his day. In his honor, I have collected links to other distributed math and science projects that you can participate in. You do not need to be a math or science whiz to join in the fun. Entropia.com is running several projects in medical, environmental, economics, and scientific
PGIS News Volume 1 No.4, Volume 2 No.1 March 2001 Riemann hypothesis. The twin prime conjecture is the statement thatthere are infinitely many ?twin primes?. (Two prime numbers http://www.pgis.lk/newsletter/news3/
Extractions: Dr. N C Bandara (Editor) This is the inaugural issue of PGIS News published by the Postgraduate Institute of Science. The first issue reports the events of PGIS since its establishment in 1996. In the forthcoming issues, we intend to publish articles and short notes of academic nature. We shall be pleased to receive your comments, suggestions and contributions with a view to improving its quality. Correspondence and requests for copies of PGIS News should be addressed to Dr. N C Bandara - Editor:
The Top Twenty Twin Primes This pages, discussing twin primes, is one of a series of pages listing the 20 largest known primes of selected forms. This page provides definitions, theorems, records and references. http://www.utm.edu/research/primes/lists/top20/twin.html
Extractions: Twin Primes Select a top twenty page Primes in Arithmetic Progression Consecutive Primes in Arithmetic Progression Cullen Primes Cunningham Chain (1st kind) Cunningham Chain (2nd kind) Euler Irregular Fermat Divisors Generalized Fermats Generalized Lucas numbers Generalized repunits Generalized Fermat Divisors (base=10) Generalized Fermat Divisors (base=12) Generalized Fermat Divisors (base=6) Irregular Primes Largest Known Primes Lucas Aurifeuillian primitive part Mersenne Primes Near-repdigit Primes NSW primes Lucas primitive parts Primorial and Factorial Primes Sophie Germain Primes Twin Primes Woodall Primes 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 32223 digits or more) plus twenty each of certain selected forms . This page is about one of those forms. Comments and suggestions requested . This page last updated: 17 March 2003, 10:59am. Twin primes are pairs of 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
Extractions: Week of June 2, 2001; Vol. 159, No. 22 Ivars Peterson Number theory offers a host of problems that are remarkably easy to state but fiendishly difficult to solve. Many of these questions and conjectures feature prime numbersintegers evenly divisible only by themselves and 1. For instance, primes often occur as pairs of consecutive odd integers: 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. So-called twin primes are scattered throughout the list of all prime numbers. There are 16 twin prime pairs among the first 50 primes. The largest known twin prime is the 32,220-digit pair 318032361 x 2 +/1, found recently by David Underbakke and Phil Carmody. Although most mathematicians believe that there are infinitely many twin primes, no one has yet proved this conjecture to be true. Indeed, the twin prime conjecture is considered one of the major unsolved problems in number theory. It was even mentioned in the 1996 movie A Mirror Has Two Faces , which starred Barbra Streisand.
