Least Primitive Root Of Prime Numbers If p is an odd prime number then r is a primitive root of p if and only if be the smallest prime number q for which g(q)=n, with n not a perfect power, http://www.ieeta.pt/~tos/p-roots.html
Extractions: Least base necessary to prove the primality of a number Introduction Results References Links ... [Up] Let p be a prime number. Fermat's little theorem states that a^(p-1) mod p=1 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 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 N(x;a)
SQL And The Search For Prime And Perfect Numbers in reference to writing SQL statements to find prime perfect numbers.After a bit of clarification, (primes perfect numbers less than 1 million) http://weblogs.sqlteam.com/davidm/archive/2003/10/30/412.aspx
Perfect Number: Definition And Much More From Answers.com perfect number n. A positive integer that is equal to the sum of its positiveintegral prime numbers of the form 2n 1 are known as Mersenne primes, http://www.answers.com/topic/perfect-number
Extractions: Wikipedia perfect number In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors , excluding itself. Six ( and (sequence in OEIS ). These first four perfect numbers were the only ones known to the ancient Greeks Euclid discovered that the first four perfect numbers are generated by the formula 2 n n Noticing that 2 n prime number in each instance, Euclid proved that the formula 2 n n n Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2
Ivars Peterson's MathTrek - Cubes Of Perfection Hence, every Mersenne prime automatically leads to a new perfect number. An introduction to Mersenne primes and perfect numbers can be found at http://www.maa.org/mathland/mathtrek_5_18_98.html
Extractions: Ivars Peterson's MathTrek May 18, 1998 Playing with integers can lead to all sorts of little surprises. A whole number that is equal to the sum of all its possible divisors including 1 but not the number itself is known as a perfect number (see A Perfect Collaboration ). For example, the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 equals 6. Six is the smallest perfect number. Twenty-eight comes next. Its proper divisors are 1, 2, 4, 7, and 14, and the sum of those divisors is 28. Incidentally, if the sum works out to be less than the number itself, the number is said to be defective (or deficient). If the sum is greater, the number is said to be abundant. There are far more defective and abundant numbers than perfect numbers. However, do abundant numbers actually outnumber defective numbers? I'm not sure. Steven Kahan, a mathematics instructor at Queens College in Flushing, N.Y., has a long-standing interest in number theory and recreational mathematics. "I often play with number patterns," he says. In the course of preparing a unit on number theory for one of his classes, he noticed a striking pattern involving the perfect number 28:
Perfect Numbers number is prime, and so the largest known prime is often a Mersenne prime.Here are the 37 known Mersenne primes and perfect numbers (from MathWorld) http://www.jimloy.com/number/perfect0.htm
Extractions: Go to my home page 6 and 28 are called Perfect Numbers. The proper divisors (the divisors of a number, not including the number itself) of 6 are 1, 2, and 3, and 6=1+2+3. Similarly, the proper divisors of 28 are 1, 2, 4, 7, and 14 and 28=1+2+4+7+14. Are there any other perfect numbers, numbers equal to the sum of their proper divisors? Euclid (in Book IX, Proposition 36) actually showed that if p Primes of the form (2^p)-1 are called Mersenne Primes. A number of the form (2^n)-1 cannot be a prime unless n is a prime. The nth Mersenne number is M n =M(n)=(2^n)-1. Many people call M(n) a Mersenne number only if n is prime. The first few Mersenne primes are 3, 7, 31, 127, 8191, etc., which are primes. So the first few perfect numbers are 6, 28, 496, 8128, 33550336, etc. It was quickly shown that M(11), M(23), M(83), M(131) and others were not prime, even though n is prime in those cases. Mersenne apparently conjectured that M(n) was prime for n=2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and that M(n) was not prime for any other n below 257. He was shown to be wrong for n=61, 67, 89, 107, and 257. It is relatively easy to test to see if a Mersenne number is prime, and so the largest known prime is often a Mersenne prime.
Landon Curt Noll's Prime Pages Landon Curt Noll s prime number and large prime pages. prime numbers, Mersenneprimes, perfect numbers, etc. Mersenne prime Digits and Names English http://www.isthe.com/chongo/tech/math/prime/
SAVEgateway Document Delivery perfect numbers and what later came to be called Mersenne primes date actuallyback to the legendary Greek mathematician, Euclid, said Carl Pomerance, http://www.isthe.com/chongo/tech/math/prime/mercnews.html
Extractions: Page: 1A When the British mountaineer George Leigh Mallory was asked why he wanted to scale Mount Everest, he replied: ''Because it's there.'' A related urge sparks computer scientists at Silicon Graphics Inc.'s Cray Research unit, who will announce today that they've discovered the world's largest-known prime number - and a special kind of prime number at that. This one is 378,632 digits long, roughly 120 single-spaced typewritten pages - and ''a rare jewel,'' said co-discoverer Paul Gage. But the way they found it, using sophisticated programming on high-powered supercomputers, goes well beyond mathematical mountain climbing. The techniques help create and test computer systems that in turn help solve real-world problems such as cryptography, improving weather forecasts and designing safer cars, said David Slowinski, the other co-discoverer of the latest record number. Using a Cray T94 supercomputer, Slowinski and Gage found what is currently the biggest example of a Mersenne prime number, named after a 17th-century French monk, Father Marin Mersenne, who had a thing for numbers. A prime number is an integer greater than zero whose divisors are only itself and 1. (The number 2 is prime because it can only be divided evenly by 1 and 2, for example). Mersenne numbers are primes that take the form 2 to some power, minus 1 - in other words, 2 multiplied by itself a certain number of times with 1 subtracted from the result.
Mathematics Archives - Numbers Includes information on various topics as perfect numbers, prime numbers,Pythagorean triples, pi, and Fermat s Last Theorem. The PI Page 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?
Straight Dope Staff Report: What's The Story On Perfect Numbers? Also, given a Mersenne prime, applying the formula will always give a perfectnumber. That is, Mersenne primes and even perfect numbers go hand in hand. http://www.straightdope.com/mailbag/mperfectnumbers.html
Extractions: Home Page Message Boards News Archive ... FAQs, etc. A Staff Report by the Straight Dope Science Advisory Board What's the story on perfect numbers? 05-Jul-2005 Dear Straight Dope: What exactly is the deal with perfect numbers and why are only 38 recognized? Is there some sort of math mafia that limits this sort of thing or what? Mac McCartney Guest contributor Mathochist replies: Perfect numbers are a holdover from the days of the Pythagoreans, when mathematicians were mystics as much as anything else and put a lot more stock in coincidence. Start with a number. Find all the numbers that divide it evenly. Add them all up (other than the number itself), and sometimes you'll get your original number back again. These are the perfect numbers. For example, 6 is evenly divisible by 1, 2, and 3. 1+2+3 = 6. Others are (see reference 1):
Resources An integer is perfect if it equals the sum of its divisors (excluding itself).Read about the connection between perfect numbers and Mersenne primes. http://www.teachers.ash.org.au/mikemath/resources/prime.html
Integers As Prime Or Composite perfect numbers and Mersenne Primes; Cardinal vs. Whenever another mersenneprime is found, another perfect number is generated. http://www2.andrews.edu/~calkins/math/webtexts/numb03.htm
Extractions: Back to the Table of Contents Numbers and Their Application - Lesson 3 The Naturals as Prime or Composite Factors, Prime, Composite, 1 is Unique The natural numbers have been studied intensely for millenia. Several fascinating properties relate to their factors. A factor is a natural number which divides another natural number evenly (as in without a remainder). The word factor will be used later in a less restricted sense as in x +1 is a factor of x Divisor is essentially a synonym of factor and is also commonly used interchangeably. A prime number only has factors of itself and one. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.... Twin primes are primes which differ by 2. Examples of twin primes are: 3 and 5, 5 and 7, 11 and 13, 17 and 19, .... A composite number has factors in addition to itself and one. One (1) is unique in that it is considered neither prime nor composite.
Mersenne Primer Mersenne primes and perfect numbers. A perfect number is a number that is equalto the sum of its factors, provided you count 1 as a factor but not the 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).
What's A Number? 5 is a prime number (it has no other divisors but itself and 1). Quite rigorously.QED. It is obvious that all Euclid s perfect numbers are triangular. http://www.cut-the-knot.com/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.
Gnist.no A fascinating look at the math and mystique of prime numbers prime numbers patents on prime numbers; Pepin s test for Fermat numbers; perfect numbers; http://www.gnist.no/vare.php?ean=9780471462347
Extractions: Prime Numbers Introduction A prime number is an integer that has exactly two factors. These numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, .... There are infinitely many of these, as can be proven in a variety of ways. Here is one fascinating way to do this. As shown on the theorems page, the harmonic series diverges to infinity. Now we use this to prove that there are infinitely many prime numbers. First we make the bold statement: . How can we make such a statement? Well, we note that every integer greater than 1 can be written as the product of primes, so it follows that . Conveniently, each of the parentheses encloses a geometric series that we can sum, so we get . Thus we reach the infinite product above. If there are only a finite number of primes, then the right side will not diverge, but we already know that the left side will, so we have a contradiction, so there must be an infinite number of primes. Obviously, there are simpler ways of proving this, but I believe that this method gives the most insight into the distribution of primes. More on the Distribution of the Primes Let us define a function to be the number of prime numbers not exceeding x. Thus
Identifying Perfect Numbers Prerequisite vocabulary factors, prime numbers, divisibility rules, and powers.On the overhead or whiteboard, write down the first perfect number (six). http://www.microsoft.com/Education/IdentifyingPerfectNumbers.aspx
Extractions: Extensions During this lesson, students will learn the definition of a perfect number and use various methods for determining what numbers are perfect. Top of page Student Directions -save this document to your classroom computers. Adjust the directions as needed for your lesson. When presenting your lesson to the students, have them use the student directions sheet as a jump point into the activity. The Perfect Number Journey Top of page Math: Numbers and operations, reasoning and proof ISTE NETS Standards for Students Students use technology tools to enhance learning, increase productivity, and promote creativity.
