Mersenne Primer Mersenne primes and perfect numbers. That covers all the factors ofour alleged perfect number, provided that Mp is itself prime. http://www.sheeplechasers.org/prime/mersenne.htm
Extractions: A Mersenne prime is a prime number (a number divisible only by itself and 1) of the form 2^n-1; for instance, 7 = 2^3-1. They are named for Marin Mersenne , who investigated prime numbers, especially of this form. Not all numbers of the form 2^n-1 are prime; actually they are quite rare. For starters, for 2^n-1 to be prime, 'n' must be prime. For this reason you will usually see the form written as 2^p-1, where 'p' is a prime number. Even so, most numbers of the form 2^p-1 are still not prime. They are known as Mersenne numbers, and only called Mersenne primes if they are prime. Both are sometimes represented as 'Mp'. A perfect number is a number that is equal to the sum of its factors, provided you count '1' as a factor but not the number itself. Why is that? I don't know, you'd have to ask the ancient Greeks. Probably because if you don't use that definition then there are no perfect numbers. Now it just so happens that all known perfect numbers are of the form (2^p-1)*2^(p-1). The first two are 6 (=1+2+3) and 28 (=1+2+4+7+14). They are given by the formula for p = 2 and p = 3. You may have noticed that the first half of that formula is the form of a Mersenne prime. It also holds true that the number given by the formula will not be a perfect number unless 2^p-1 is prime. There's a good reason for that. Let's look again at that perfect number formula. It's always a Mersenne number times a power of 2. Let's let n = p-1, so that can be rewritten as "Mp * 2^n". The factors of 2^n are [1, 2, 4, 8,... 2^(n-1), 2^(n)], which add up to 2^(n+1)-1. So the factors of our alleged perfect number (Mp*2^n) will contain those powers of 2, plus Mp, plus Mp times powers of 2 up to 2^(n-1). (We can't count Mp times 2^n because that is the number itself.) Now the sum of Mp times powers of two from 1 to 2^(n-1) is Mp*(2^n-1).
American Scientist Online - Prime Time that has built up around special primes Numerologists have enjoyed the connectionwith perfect numbers, and the determination of Fermat primes (primes of the http://www.americanscientist.org/template/AssetDetail/assetid/14442
Extractions: Home Current Issue Archives Bookshelf ... Subscribe In This Section Reviewed in This Issue Book Reviews by Issue New Books Received Publishers' Directory ... Virtual Bookshelf Archive Site Search Advanced Search Visitor Login Username Password Help with login Forgot your password? Change your username see list of all reviews from this issue: January-February 2002 MATHEMATICS Stan Wagon Prime Numbers: A Computational Perspective . Richard Crandall and Carl Pomerance. xvi + 545 pp. Springer-Verlag, 2001. $49.95 Although everyone knows what a prime number is, the implications of this simple concept are amazingly deep and diverse. As Richard Crandall and Carl Pomerance note in Prime Numbers, "The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture." The advent of accessible and fast methods of computation has had a profound impact on how we view prime numbers, and this book summarizes the state of the art regarding algorithms for dealing with primes. The two central algorithmic problems in this area are the determination of primality for a given integer and the factoring of an integer into prime factors. The book covers these and lots more, touching on many important computational approaches to problems of number theory.
Colours Of Numbers known prime is probably a Mersenne prime and therefore green. Fermat numbers.All Fermat numbers 5, 17, 257, 65537, , except 3 are red. perfect numbers. http://www.hermetic.ch/misc/numcol.htm
Extractions: Colours of Numbers by Karl Palmen I discovered a way of colouring the natural numbers that I have found very fascinating. I use following eight colours: black, red, green, yellow, blue, magenta, cyan and white . (Before printing this page on a colour printer see the note at the bottom.) It started years ago when I realised that those numbers that can be expressed as the sum of just two squares (1, 2, 4, 5, 8, 9, 10, 13 etc.) contain all their multiplication products (e.g., 2x5=10). This arises as a consequence of De Moivre's theorem in complex numbers. I became quite fascinated by these numbers and worked out a large number of them. I soon suspected a relationship between such numbers and the remainders of their prime factors divided by 4. From the geometry of the complex plane I discovered a similar set of numbers. These are the numbers expressable as the sum of two squares and their geometric mean (1, 3, 4, 7, 9, 12, 13 etc.). These too contain their multiplication products (e.g., 3x4=12). I soon suspected a relationship between such numbers and the remainders of their prime factors divided by 3. These considerations eventually inspired me to find my way of colouring numbers. The numbers that are the sum of two squares are either
Halfbakery: Prime Melodies music. bristolz prime numbers and music. http//home22.inet .sic_and_numbers.htmprime interval (perfect fifth) mentioned. bristolz http://www.halfbakery.com/idea/Prime_20Melodies
Extractions: Prime numbers are those divisible only by themselves and one and are regarded as beautiful by pure mathematicians. There are no discovered patterns to the dispersal of prime numbers within number sets and indeed, for all our advances, there are no formulas to predict the next prime number. Within an octave of 8 notes, 5 of these notes are prime. Within the key of C, the notes C, D, E, G, A and B [notes 1, 2, 3, 5, 7 within the octave] are prime. I propose musical pieces composed only using prime notes : Prime Melodies. The chords and melodies available will have a mathematical beauty about them. Extra emphasis on chords such as CMaj7th should be used as it employs the 1, 3, 5, 7 of the key. It's a prime chord. jonthegeologist , Oct 30 2003
Extractions: Submit primes Over 2300 years ago Euclid proved that If 2 k -1 is a prime number (it would be a Mersenne prime ), then 2 k k -1) is a perfect number . A few hundred years ago Euler proved the converse (that ever even perfect number has this form). It is still unknown if there are any odd perfect numbers (but if there are, they are large and have many prime factors). Proof: Suppose first that p k -1 is a prime number, and set n k k -1). To show n is perfect we need only show sigma( n n . Since sigma is multiplicative and sigma( p p k , we know sigma( n ) = sigma(2 k sigma( p k k n This shows that n is a perfect number.
Perfect Number -- From MathWorld connected with a class of numbers known as Mersenne primes, which are prime numbersof the form . This can be demonstrated by considering a perfect number P of http://mathworld.wolfram.com/PerfectNumber.html
Extractions: etc. The first few perfect numbers are summarized in the following table together with their corresponding indices p (see below). n Perfect numbers were deemed to have important numerological properties by the ancients, and were extensively studied by the Greeks, including Euclid Perfect numbers are also intimately connected with a class of numbers known as Mersenne primes , which are prime numbers of the form This can be demonstrated by considering a perfect number P of the form where q is prime . By definition of a perfect number P
Odd Perfect Number -- From MathWorld 34, 10271032, 1980. Hagis, P. Jr.; and Cohen, G. L. Every Odd perfect NumberHas a prime Factor Which Exceeds10 6 . Math. Comput. 67, 1323-1330, 1998. http://mathworld.wolfram.com/OddPerfectNumber.html
Extractions: Odd Perfect Number In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 1957, p. 3), although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why a odd perfect number could not exist (Dickson 1957, p. 12). Descartes was therefore among the first to consider the existence of odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid's construction comprised all possible perfect numbers (Dickson 1957, pp. 6-12). In 1657, Frenicle repeated Descartes' belief that every even perfect number is of Euclid's form and that there was no reason odd perfect could not exist. Like Frenicle, Euler also considered odd perfect numbers. To this day, it is not known if any odd perfect numbers exist, although numbers up to 10
Math Forum - Ask Dr. Math More than two thousand years ago, the Greek mathematician Euclid explained a methodfor finding perfect numbers that is based on the concept of prime numbers. http://mathforum.org/library/drmath/view/57044.html
Extractions: Associated Topics Dr. Math Home Search Dr. Math Date: 11/3/96 at 9:28:54 From: Anonymous Subject: math question What is the next perfect number after 28? http://mathforum.org/dr.math/ Date: 11/04/96 at 19:44:54 From: Anonymous Subject: Re: math question Thank you for answering my question about the next perfect number after 28. I would be interested to learn more about how you could find a formula that finds perfect numbers. I am only in 6th grade so I cannot understand mathematical words, but that doesn't mean that I am a numbskull. Sincerely, DZINE http://www.utm.edu/research/primes/mersenne.shtml -Doctor Yiu, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Associated Topics
Mersenne Prime Numbers This means that the quest for perfect numbers is reduced to the quest for primesof the form 2^m 1 A Mersenne prime is such a number Mp, where p is prime. http://www.resort.com/~banshee/Info/mersenne.html
Extractions: Marin Mersenne (1588-1648) was a Franciscan friar who lived most of his life in Parisian cloisters. He was the author of Cognitata Physico-Mathematica which stated without proof that M p is prime for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 and for no other primes p for p p , Mersenne contributed to the development of number theory through his extensive correspondence with many mathematicians, including Fermat. Mersenne effectively served as a clearing house and a disseminator of new mathematical ideas in the 17th century. Kenneth Rosen, Elementary Number Theory; Addison Wesley The concept of a Mersenne Prime is evolved from that of a perfect number . A perfect number is an integer for which the sum of its divisors is twice the number. For example: (6) = 1 + 2 + 3 + 6 = 12 = 2*6 thus 6 is a perfect number.
Numbers: Glossary Whenever someone discovers a new Mersenne prime they also automaticallydiscover a new perfect number. (2 1) x 2 is the world s http://richardphillips.org.uk/number/gl/prime.htm
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 4,053946 digits and probably get through quite a few pencils. The record was broken in November 2001 by Michael Cameron using Prime 95 software by George Woltman.
Mathematics Enrichment Workshop: The Perfect Number Journey Mersenne. So the search for perfect numbers became the search for more Mersenneprimes, ie prime numbers of the form 2 n 1. But this turned out to be a very http://home1.pacific.net.sg/~novelway/MEW2/lesson2.html
Extractions: How are Mersenne primes related to perfect numbers? If a Mersenne number turns out to be a prime number, then it is called a Mersenne prime You have computed the first 5 Mersenne primes: 3, 7, 31, 127, 8191. Each of these numbers in turn gives a perfect number when multiplied by its previous power of 2. (b) Two perfect numbers were discovered in 1588, both by Cataldi. These two perfect numbers can be obtained from the Mersenne primes M - 1 and M - 1. Can you compute these two perfect numbers with the help of your calculator? (c) Do you think M is a Mersenne prime? By now, you should have realised why numbers of the form 2 n - 1 have so much appeal. Whenever a prime number of this form is found, a perfect number is immediately obtained, as was proven by Euclid.
Prime Patterns The 5 can be replaced with an 8. It is also Superperfect. All order 2 and 3 perfectprime squares contain palindromes and contain duplicate prime numbers. http://www.geocities.com/~harveyh/primes.htm
Extractions: Patterns in P rimes This palindromic prime number reads the same upside down or when viewed in a mirror. CONTENTS More Prime series Factorial n n n Unfortunately the next factorial results in a composite number. The above shows the number 1 as a prime, although it is normally considered neither prime nor composite. Assign the value 1 to A, 2 to B, 3 to C, . . . , 26 to Z. Then i.e. 16 + 18 + 9 + 13 + 5 = 61 More Consecutive Prime series Above is shown three of the five series that use 2, the only even prime number. Charles w. Trigg , JRM 18(4),1985-86, p.247-248 All primes!
The Prime-perfect Numbers The primeperfect numbers. A Problem Proposal The sequence a(n) of prime-perfectnumbers begins. 30, 60, 70, 84, 90, 105, 120, 140, . http://www.geocities.com/SoHo/Exhibit/8033/primeperfect/primeperfect.html
Extractions: A Problem Proposal Consider the numbers n with at least two prime factors, the sum of whose prime factors divides n. In obvious analogy to the perfect numbers, I call these the prime-perfect numbers . (Clearly, the sum of the prime factors of n is almost always less than n, so to require equality of n to the sum, as in the definition of perfect numbers, will be fruitless.) The sequence a(n) of prime-perfect numbers begins (Note: This is EIS Sequence A066031 .) The numbers k with just one prime factor have been excluded from the sequence since they trivially satisfy the requirement that the sum of the prime factors of k divide k. The exclusion thus highlights the interesting numbers satisfying the requirement. It is easy to see that if p is a prime factor of the prime-perfect number n, then p m n is also prime-perfect for any m. Hence, a is an infinite sequence. But what about the elementary (or primitive ) terms of a, that is, terms which are not multiples of any previous terms? For example, 84 is elementary, since it is not a multiple of the preceding terms, 30, 60, 70. But 90 is not elementary because 90 is a multiple of 30. Are there also infinitely many elementary terms? A related problem: Find an expression generating elementary prime-perfect numbers.
Perfect Numbers What is a perfect Number? My Talk on Aliquot Parts Mersenne primes The GREAT InternetMersenne prime Search GIMPS Plot of Mersenne primes Plot of Mersenne http://pw1.netcom.com/~hjsmith/Perfect.html
Mathematics Archives - Numbers museum. Includes information on various topics as perfect numbers, primenumbers, Pythagorean triples, pi, and Fermat s Last Theorem. http://archives.math.utk.edu/subjects/numbers.html
Extractions: Hear and see the prime numbers! A Common Book of p The number p has been the subject of a great deal of mathematical (and popular) folklore. It's been worshipped, maligned, and misunderstood. Overestimated, underestimated, and legislated. Of interest to scholars, crackpots, and everyday people. Continued Fractions A senior Honor's Project at Calvin College by Adam Van Tuyl which gives the history, theory, applications and bibliography on the thery of continued fractions. In the section on applications there are a number of interactive programs that convert rationals (or quadratic irrationals) into a simple continued fraction, as well as the converse. Data Powers of Ten A petabyte?
Mathematics Archives - Topics In Mathematics - Number Theory of Integers, Divisibility, prime numbers, The Greatest Common Divisor, Unique Factorization,Linear Diophantine Equations, perfect numbers, Mersenne numbers http://archives.math.utk.edu/topics/numberTheory.html
Article: Prime Numbers primes). Euclid also showed that if the number 2 n 1 is prime thenthe number 2 n-1 (2 n - 1) is a perfect number. The mathematician http://www1.physik.tu-muenchen.de/~gammel/matpack/html/Mathematics/Primes.html
Extractions: Ancient greek mathematicians started studying prime numbers and their properties. 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 also interested in perfect and amicable numbers. An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. For example, the prime divisors of 10 are 2 and 5, and the first six primes are 2, 3, 5, 7, 11 and 13. 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. 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 (except for the order of the factors).
Least Primitive Root Of Prime Numbers 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
What's A Number? if, for a prime p, p+1 = 2 k , then 2 k1 p is perfect. Leonhard Euler (1707-1783)in a paper published posthumously, showed that every even perfect number has http://www.cut-the-knot.org/do_you_know/numbers.shtml
Extractions: Philosophical Library, 1965 Indeed there are many different kinds of numbers. Let's talk a little about each of these in turn. A number r is rational if it can be written as a fraction r = p/q where both p and q are integers. In reality every number can be written in many different ways. To be rational a number ought to have at least one fractional representation. For example, the number may not at first look rational but it simplifies to 3 which is 3 = 3/1 a rational fraction. On the other hand, the number 5 by itself is not rational and is called irrational. This is by no means a definition of irrational numbers. In Mathematics, it's not quite true that what is not rational is irrational. Irrationality is a term reserved for a very special kind of numbers. However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational). Much of the scope of the theory of rational numbers is covered by Arithmetic. A major part belongs to Algebra. The theory of irrational numbers belongs to Calculus. Using only arithmetic methods it's easy to prove that the number
SS > Factoids > Perfect Number it is divisible by a prime component greater that 10 20. Exhaustive computersearch has shown that there are no odd perfect numbers less than 10 300 . http://www-users.cs.york.ac.uk/~susan/cyc/p/perfect.htm
