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Multiply Perfect Numbers Contributor SAMIEL@FASTLANE.NET { Samiel (samiel@fastlane.net) wrote Here's my fast and elegant code snarf it and add it to SWAG perfect numbers http://www.uni-bielefeld.de/~achim/mpn.html
Extractions: Let o(n) be the number theoretic function which denotes the sum of all divisors of a natural number n . If o(n) is an integral multiply of n , then n is denoted as a multiply perfect number or k-fold perfect number (also called multiperfect number or pluperfect number ). Call o(n)/n abundancy (also called index or multiplicity ) of n . A multiply perfect number is called proper 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120=o(120)=o(2^3*3*5)=o(2^3)*o(3)*o(5)=(1+2+4+8)*(1+3)*(1+5)=15*4*6=360=3*120 Hence 120 is a 3-fold perfect number. Abundancy Count When last number was discovered Which was last? Are all discovered? Rough total number yes and proved no, there are infinitely many oo yes yes yes yes, with probability about 1 - 10^-9 probable to 0.999, perhaps single misses no no no no In column "Which was last?" the identifier ln(ln(MPN)) is given. Richard Schroeppel's archive of 2094 MPNs built 1995-12-13
Prime Numbers Historical topics about prime numbers.Category Science Math Number Theory prime numbers 1 is prime then the number 2 n1 (2 n - 1) is a perfect number. The mathematicianEuler (much later in 1747) was able to show that all even perfect numbers are http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html
Perfect Numbers 11 12 13 1 3 7 15 31 63 127 255 511 1023 2047 4095 8191 let them be called the radicalsof perfect numbers, since whenever they are prime, they produce them. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html
Prime Numbers - History Topics Check out a resource which dedicates itself to the historical aspects of prime number calculation and discovery. Authors include recommended reading. prime then the number 2n1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html
Math Forum - Ask Dr. Math Archives: Elementary Prime Numbers perfect numbers Basics, History 11/3/1996 What is the next perfect number after28? prime and Composite numbers, Sieve of Eratosthenes 01/28/1997 I need http://mathforum.org/library/drmath/sets/elem_prime_numbers.html
Perfect, Amicable, And Sociable Numbers Resource explains "how to have fun summing up divisors" and includes an introduction to perfect, amicable, and sociable numbers and Aliquot sequences. If 2p1 is prime, then 2p-1(2p-1) is even and perfect, and conversely, all even perfect numbers have this form. http://xraysgi.ims.uconn.edu:8080/amicable.html
Extractions: HTTP 200 Document follows Date: Tue, 18 Mar 2003 10:44:46 GMT Server: NCSA/1.5.2 Last-modified: Sun, 23 Jun 2002 01:44:17 GMT Content-type: text/html Content-length: 29239 Introduction Perfect numbers Amicable numbers Sociable numbers ... Technical appendix For a number n , we define s(n) to be the sum of the aliquot parts of n, i.e., the sum of the positive divisors of n, excluding n itself: so, for example, s(8)=1+2+4=7, and s(12)=1+2+3+4+6=16. If we start at some number and apply s repeatedly, we will form a sequence: s(15)=1+3+5=9, s(9)=1+3=4, s(4)=1+2=3, s(3)=1, s(1)=0. If we ever reach 0, we must stop, since all integers divide 0. There are three obvious possibilities for the behavior of this aliquot sequence It can terminate at like the example above. It can fall into an aliquot cycle , of length 1 (a fixed point of s) , or greater It can grow without bound and approach infinity A perfect number is a cycle of length 1 of s , i.e., a number whose positive divisors (except for itself) sum to itself. For example, 6 is perfect (1+2+3=6), and in fact 6 is the smallest perfect number. The next two perfect numbers are 28 (1+2+4+7+14=28) and 496 (1+2+4+8+16+31+62+124+248=496).
Perfect Security - Prime Numbers prime numbers, Definition. A number is called prime, if andonly if its only positive divisors are itself and one. http://www.perfect-security.net/en/prime.html
The Prime Glossary: Perfect Number Welcome to the prime Glossary a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'perfect number.' Come explore a new prime term today! out that for 2k1 to be prime, k must also be primeso the search for perfect numbers is the same as the search for http://www.utm.edu/research/primes/glossary/PerfectNumber.html
Extractions: (another Prime Pages ' Glossary entries) Glossary: Prime Pages: Many ancient cultures endowed certain integers with special religious and magical significance. One example is the perfect numbers, those integers which are the sum of their positive proper divisors . The first three perfect numbers are The ancient Christian scholar Augustine explained that God could have created the world in an instant but chose to do it in a perfect number of days, 6. Early Jewish commentators felt that the perfection of the universe was shown by the moons period of 28 days. Whatever significance ascribed to them, these three perfect numbers above, and 8128, were known to be "perfect" by the ancient Greeks, and the search for perfect numbers was behind some of the greatest discoveries in number theory. For example, in Book IX of Euclid 's elements we find the first part of the following theorem (completed by Euler some 2000 years later).
Landon Curt Noll's Prime Pages Landon Curt Noll Landon Curt Noll's picture. prime numbers, Mersenneprimes, perfect numbers, etc. Mersenne prime Digits and Names http://www.isthe.com/chongo/tech/math/prime/
Perfect Security - Prime Numbers Naohiro Nomoto Home Page 12 nice sequences A037160 A037159 A060448 A048055 A038528 A038527 A037201 A037264 A058524 A059998 A058241 A060030 William Nelles (wnelles@flashmail.com) A057720 ( Decimal expansion of muonto-electron mass ratio. ) http://www.onetimepad.net/en/prime.html
The Prime Glossary: Perfect Number prime Glossary a collection of definitions, information and facts all related toprime numbers. This pages contains the entry titled 'perfect number.' Come http://primes.utm.edu/glossary/page.php?sort=PerfectNumber
Computing Perfect(prime) Numbers {* perfectnumbers The definition of a perfect number can where p is prime and 2^p http://www.piclist.com/techref/language/delphi/swag/MATH0108.html
Prime Numbers 1) x 2 is the world's largest known perfect number at and so records for the highestknown prime can go For more facts about numbers get the book numbers Facts http://www.nottingham.ac.uk/education/number/gl/prime.html
Extractions: - 3 and 7 are called factors of 21. But some numbers cannot be made in this way and these are called prime numbers. For example, 23 is a prime number because it cannot be made by multiplying together smaller numbers. Numbers like 21 which are not prime are sometimes called composite numbers. All prime numbers, apart from 2, are odd numbers. The Mersenne primes are a special type of prime number. The first five are - For a mathematician, the equivalent of breaking the 100 metres world record is to find the highest known prime number. Every year or so, someone discovers a higher one and it gets reported in the newspapers. These record-breaking numbers are always Mersenne primes. At the time of writing the highest known prime is 2 - 1. To write it out you would use 258,716 digits and probably get through quite a few pencils. The record was broken in February 1994 by Slowinski and Gage.
Mathematics Enrichment Workshop: The Perfect Number Journey A Mathematics Enrichment Workshop introducing perfect numbers. Lessons and exercises extend over several Category Science Math Number Theory Factoring perfect numbers Beginning with the number 1, and keep adding the powers of 2 (ie doubling the numbers),until you get a sum which is a prime number. A perfect number is then http://home.pacific.net.sg/~novelway/MEW2/lesson1.html
Extractions: by Heng O.K. What are perfect numbers? Mathematicians and nonmathematicians have been fascinated for centuries by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). The smallest such example is , since = 1 + 2 + 3. Such numbers are called perfect numbers The search for perfect numbers began in ancient times. The first three perfect numbers: and were known to the ancient mathematicians since the time of Pythagoras (circa 500 BC). How to find perfect numbers? Euclid (circa 300 BC), the famous Greek mathematician, devised a simple method for computing perfect numbers. Beginning with the number 1, and keep adding the powers of 2 (i.e. doubling the numbers), until you get a sum which is a prime number . A perfect number is then obtained by multiplying this sum to the last power of 2. In the exercise that follows, you are going to use this method to determine the next two perfect numbers. The first few rows in the table demonstrate the calculations being carried out to compute the first three perfect numbers. Apply this technique now, and let's see how fast you can find the fourth perfect number.
35th Mersenne Prime Discovered Mersenne prime. Not all Mersenne numbers between the 31st and 35th have been checked.There is a wellknown formula that generates a perfect number from a http://www.mersenne.org/1398269.htm
Extractions: 2^1398269-1 is the Largest Known Prime. ORLANDO, Fla., November 23, 1996 On November 13, Joel Armengaud discovered the largest known prime number using a program written by George Woltman. Joel Armengaud, a 29-year-old programmer for Apsylog, is from Paris, France. George Woltman is a 39-year-old programmer living in Orlando, Florida. Early this year, Woltman launched the Great Internet Mersenne Prime Search (GIMPS). This web site offers free software for ordinary personal computer owners to use in searching for big prime numbers. Large prime numbers were once the exclusive domain of supercomputer users. "By using a large number of small computers, we negate the supercomputer's speed advantage," said Woltman. Armengaud is one of more than 700 people searching for new primes. Even though Armengaud was the one lucky enough to find this new prime, credit must also go to all the other searchers . Without their efforts, this discovery would not have been possible. The new prime number, 2^1398269-1 is the 35th known Mersenne prime.
38th Mersenne Prime Discovered Second is to further research in prime numbers and computer algorithms. is a wellknownformula that generates a perfect number from a Mersenne prime. http://www.mersenne.org/6972593.htm
Extractions: -1 is now the Largest Known Prime. ORLANDO, Florida, June 30, 1999 Nayan Hajratwala, a participant in the Great Internet Mersenne Prime Search (GIMPS) , has discovered the first known million-digit prime number using software written by George Woltman and the distributed computing technology and services of Scott Kurowski's company, Entropia.com, Inc. The prime number, 2 -1, contains 2,098,960 digits qualifying for the $50,000 award offered by the Electronic Frontier Foundation (EFF) . An article is being submitted to an academic journal for consideration. The new prime number, discovered on June 1st, is one of a special class of prime numbers called Mersenne primes. This is only the 38th known Mersenne prime. Nayan used a 350 MHz Pentium II IBM Aptiva computer running part-time for 111 days to prove the number prime. Running uninterrupted it would take about three weeks to test the primality of this number. Richard Crandall, whose faster algorithms helped prove the number prime, has a poster that displays this huge number for sale at http://www.perfsci.com.
Least Primitive Root Of Prime Numbers Empirical and statistical results showing the smallest base required to prove a number is prime. Includes Category Science Math Number Theory prime numbers Primality Tests p. It is not difficult to verify that g(p) cannot be a perfect power. mentioned above,when the bases r_k used in this test are restricted to be prime numbers. http://www.ieeta.pt/~tos/p-roots.html
Extractions: Introduction Results References Links ... [Up] Let p be a prime number. Fermat's little theorem states that a^(p-1) mod p=1 (a hat (^) denotes exponentiation) for all integers a between and p-1 . A primitive root of p is a number r such that any integer a between and p-1 can be expressed by a=r^k mod p , with k a nonnegative integer smaller that p-1 . If p is an odd prime number then r is a primitive root of p if and only if for all prime divisors q of p-1 . If a number r can be found that satisfies these conditions, then p must be a prime number. In fact, it is possible to relax the above conditions in order to prove that p is prime ; it is sufficient to find numbers denotes the variable r with index k such that and (r_k)^(p-1) mod p=1 for all prime divisors of p-1 (these conditions guarantee the existence of a primitive root of p A famous conjecture of Emil Artin [3, problem F9] states that if a is an integer other than or a perfect square, then the number
Factoids > Mersenne Prime and n are natural numbers, with n greater than 1, and if m n 1 is prime, then mis 2 and n is prime. Each Mersenne prime corresponds to an even perfect number http://www-users.cs.york.ac.uk/~susan/cyc/m/mersenne.htm
Extractions: A prime number of the form where p is prime. ... it is the greatest that will ever be discovered for, as they are merely curious without being useful, it is not likely that any person will attempt to find one beyond it. Peter Barlow 1811, on M Mersenne numbers have a particulary simple test for primality, the Lucas-Lehmer test The number-theoretic interest in Mersenne primes comes from the following theorem: if m and n are natural numbers, with n greater than 1, and if m n -1 is prime, then m is 2 and n is prime. Each Mersenne prime corresponds to an even perfect number The GREAT Internet Mersenne Prime Search help find another Mersenne prime! Chris Caldwell's Mersenne Primes Page history, lists, theorems, conjectures, ... Luke Welsh's Marin Mersenne Page biographies, prime number lists, algorithms, bibliography, ...
Extractions: What do searching for extraterrestrials, curing cancer, and finding big prime numbers all have in common? These problems are all being attacked with grid computing, a a technique of breaking a large problem into small tasks that can be computed independently. While projects like Seti@home and The Greatest Internet Mersenne Prime Search have received plenty of press for using the Internet to distribute tasks to end users around the globe, grid computing also takes place in more controlled environments, such as research and financial settings. But it is by using the power of the Internet and the ability to discover and access idle processes on users' machines that grid computing (once called
