81. 4-dim HyperDiamond Lattice Mersenne Primes; perfect numbers; Complex Gaussian and Eisenstein Primes;Quaternion and Octonion Primes; Primes over Algebraic Extensions http://www.valdostamuseum.org/hamsmith/PrimeFC.html

Tony Smith's Home Page Prime Numbers Mersenne Primes Perfect Numbers ... Prime? 127 = 2^7 - 1 is a Mersenne prime, called M7. 65,537 = 2^2^4 + 1 is the largest known Fermat prime. It is called F4, but is not likely to be confused with the exceptional Lie algebra F4. 2,147,483,647 = 2^31 - 1 is a Mersenne prime. It was shown to be prime by Euler. It is called M31, but is not likely to be confused with the Andromeda galaxy M31. ( see Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin, 1986 The Mersenne prime 2^859433-1 (258716 digits) found by Slowinski and Gage in 1994 For a number of the form 2^p + 1 to be prime, p must be of the form 2^k. The Fermat primes, of the form 2^2^k + 1, include: 2^2^3 + 1 = 2^8 + 1 = 257 is the only other known Fermat prime. 2^2^5 + 1 = 641 x 6,700,417 is not prime. 2^2^8 + 1 has factor 1,238,926,361,552,897. 2^2^2^2^2 + 1 = 2^65,536 + 1 = 2^2^16 + 1 has factor 825,753,601. The primalities of the Fermat Numbers 2^2^24 + 1 and 2^2^28 + 1 are not now known. The Mersenne Primes, of the form 2^k - 1 for prime k, include:

82. The Square Root Of Any Prime Number Is Irrational@Everything2.com Suppose that a prime number does have a rational square root that is, (If p is a perfect square then (vp c) is zero and the rest of the proof goes http://www.everything2.com/index.pl?node=The square root of any prime number is

83. ASPN : Python Cookbook : A Script To Generate And Print Perfect Numbers In this case the perfect numbers can be generated from Mersenne primes, and thereare only 42 or so of those which are known. Testing Mersenne candidates is http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/437072

sign in join Cookbooks Documentation ... All Cookbooks View by Category Algorithms CGI Databases and ... Debugging and Testing ... XML Programming Title: A script to generate and print perfect numbers Submitter: Aditya Gopalakrishnan ( other recipes Last Updated: Version no: Category: Rate 5 - Excellent 4 - Very good 3 - Good 2 - Needs work 1 - Poor Not Rated yet Description: A perfect number is a number whose factors, including the number itself, sum up to twice the number. For example,6 is a perfect number since its factors (1, 2, 3, 6) sum up to 12. Source: Text Source # Filename: perfect.py a = True n = while a == True: # To loop the following commands running = True while running == True: r = n = n+ 1 for x in range(1,n + 1): if n%x == 0: # To determine the factors of n r = r + x if x == n: if r == 2*n: print n running = False else : running = False Discussion: I got the idea for writing a program to generate perfect numbers from a website which had posted the problem of generating them as a challenge. I took up the challenge and decided to write a program in Python to accomplish that.Solutions for the problem in other languages are there on the page, but nobody had posted a solution in Python. Add comment Number of comments: Aditya Gopalakrishnan, 2005/07/17

84. Prime Number --  Britannica Concise Encyclopedia - Your Gateway To All Britann prime number body Any positive integer greater than 1 and exactly divisible only However, the Swiss mathematician Leonhard Euler perfect number http://concise.britannica.com/ebc/article-9375892

Home Browse Store Help Search Britannica Concise Again prime number Concise Encyclopedia Article Page 1 of 1 Any positive integer greater than 1 and exactly divisible only by 1 and itself. number theory . Primes have been recognized at least since Pythagoras . It has been known that there are infinitely many of them at least since Euclid . The prime-number factor s of an integer are the prime numbers whose product is that integer ( see fundamental theorem of arithmetic var mm = [["Jan.","January"],["Feb.","February"],["Mar.","March"],["Apr.","April"],["May","May"],["June","June"],["July","July"],["Aug.","August"],["Sept.","September"],["Oct.","October"],["Nov.","November"],["Dec.","December"]]; To cite this page: MLA style: "prime number." Britannica Concise Encyclopedia http://concise.britannica.com/ebc/article-9375892 APA style: prime number. ( Britannica Concise Encyclopedia . Retrieved http://concise.britannica.com/ebc/article-9375892 Britannica style: "prime number." Britannica Concise Encyclopedia http://concise.britannica.com/ebc/article-9375892

85. Review Of "Prime Obsession" with perfect accuracy, the number of prime numbers — numbers only divisible by These days, prime numbers are integral to digital audio quality — ie, http://olimu.com/Riemann/Reviews/NewYorkSun.htm

Review of Prime Obsession Navigate up Reviews The New York Sun June 18th, 2003 Prime Number Time By Eric Wolff In educated society, it is unacceptable to forget the date of the Declaration of Independence, or to not know that William Shakespeare was a playwright. But, for some reason, it’s acceptable to claim ignorance of grade school arithmetic. Consider, as an example, the tolerant chuckle given to a diner unwilling to work out the party’s tip because he was "never any good at math." Despite the general acceptance of, even pride in, our incompetence at this basic skill, we must be thankful that there are men and women in our midst who spend their days pondering the mysteries of numbers and working to create those most definite of all truths, mathematical proofs. Occasionally an individual appears, a visionary, who can see in the numbers broad patterns that are not proven, but still probably true. Pierre de Fermat was one of these, and his Last Theorem (there are no whole-number solutions to the equation x n +y n =z n for Riemann’s Hypothesis is both more difficult to express and vastly more useful (Carl Gauss, arguably the greatest mathematician of all time, felt that Fermat’s Last Theorem wasn’t worth his effort), and mathematicians have spent the last 150 years delving into its mysteries.

86. Sci.math FAQ: Unsolved Problems Euler proved that if N is an odd perfect number, then in the prime powerdecomposition of N , exactly one exponent is congruent to 1 mod 4 and all the other http://www.faqs.org/faqs/sci-math-faq/unsolvedproblems/

Usenet FAQs Search Web FAQs Documents ... RFC Index sci.math FAQ: Unsolved Problems There are reader questions on this topic! Help others by sharing your knowledge Newsgroups: sci.math alopez-o@neumann.uwaterloo.ca hv@cix.compulink.co.uk (Hugo van der Sanden): To the best of my knowledge, the House of Commons decided to adopt the US definition of billion quite a while ago - around 1970? - since which it has been official government policy. dik@cwi.nl (Dik T. Winter): The interesting thing about all this is that originally the French used billion to indicate 10^9, while much of the remainder of Europe used billion to indicate 10^12. I think the Americans have their usage from the French. And the French switched to common European usage in 1948. gonzo@ing.puc.cl alopez-o@barrow.uwaterloo.ca Rate this FAQ N/A Worst Weak OK Good Great Related questions and answers Usenet FAQs Search Web FAQs ... RFC Index Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update June 15 2004 @ 00:31 AM

87. Perfect Number -- From MathWorld Iannucci, DE The Second Largest prime Divisor of an Odd perfect Number ExceedsTen Thousand. Math. Comput. 68, 17491760, 1999. http://mathworld.wolfram.com/PerfectNumber.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Special Numbers Divisor-Related Numbers ... Unsolved Problems Perfect Number Perfect numbers are positive integers such that where is the restricted divisor function (i.e., the sum of proper divisors of ), or equivalently where is the divisor function (i.e., the sum of divisors of including itself). For example, the first few perfect numbers are 6, 28, 496, 8128, ... (Sloane's ), since etc. The first few perfect numbers are summarized in the following table together with their corresponding indices (see below). 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 of the form where is prime . By definition of a perfect number Now note that there are special forms for the divisor function for a prime, and

88. Prime Patterns Here are 45 prime number patterns. Included are; reversible primes, SPsquare.gif (1437 bytes) 1 of the 24 possible order3 perfect prime squares (not http://www.geocities.com/~harveyh/primes.htm

Patterns in P rimes This palindromic prime number reads the same upside down or when viewed in a mirror. CONTENTS Primes From Factorials ! PRIME is Prime More Prime series Near Repdigit Primes ... Pairs equal Prime Primes From Factorials ! 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. PRIME is Prime 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 Near Repdigit Primes All primes! The next prime number in this series, though, is 17 threes with a one at the end. Some other numbers in this series (with less then 1800 threes) are: 1 (that's 39 threes with a one at the end Near Repdigit Primes consist of a series of the same digit, then one different digit; or one digit and then a series of a different digit. The above series is particularly attractive because the number of threes in the first seven primes increase by one. There are only eleven other primes in this series with less then 1800 threes.

89. Ivars Peterson's MathTrek - Prime Listening 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? http://www.maa.org/mathland/mathtrek_7_6_98.html

Search MAA Online MAA Home Ivars Peterson's MathTrek July 6, 1998 Prime Talent 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.

90. 42nd Mersenne Prime Discovered The formula for the new prime number is 2 to the 25964951st power minus 1. There is a wellknown formula that generates a perfect number from a http://www.mersenne.org/25964951.htm

Mersenne.org Project Discovers New Largest Known Prime Number, 2 ORLANDO, Florida - February 27, 2005 Dr. Martin Nowak, an eye surgeon in Michelfeld, Germany, and a long-time volunteer in the Great Internet Mersenne Prime Search (GIMPS) distributed computing project, has discovered the largest known prime number. Dr. Nowak used one of his business PCs and free software by George Woltman and Scott Kurowski. His computer is a part of a world-wide array of tens of thousands of computers working together to make this discovery. The formula for the new prime number is 2 to the 25,964,951st power minus 1. The number belongs to a special class of rare prime numbers called Mersenne primes . This is only the 42nd Mersenne prime found since Marin Mersenne , a 17th century French monk, first studied these numbers over 350 years ago. Written out the number has 7,816,230 digits , over half a million digits larger than the previous largest known prime number. It was discovered February 18th after more than 50 days of calculations on a 2.4 GHz Pentium 4 computer. The new prime was independently verified in 5 days by Tony Reix of Grenoble, France using a 16 Itanium CPU Bull NovaScale 5000 HPC running the Glucas program by Guillermo Ballester Valor of Granada, Spain. The discovery is the eighth record prime found by the GIMPS project. In recognition of every GIMPS contributor's effort, credit for this new discovery will go to "Nowak, Woltman, Kurowski, et al".

91. 38th Mersenne Prime Discovered The new prime number, discovered on June 1st, is one of a special class of prime There is a wellknown formula that generates a perfect number from a http://www.mersenne.org/6972593.htm

GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award. -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.

92. Unsolved Problems To understand them you need to understand the concept of a prime number. A perfect number is one which equals the sum of its proper divisors. http://www.math.utah.edu/~pa/math/conjectures.html

Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah Some Simple Unsolved Problems One of the things that turned me on to math were some simple sounding but unsolved problems that were easy for a high school student to understand. This page lists some of them. Prime Number Problems To understand them you need to understand the concept of a prime number A prime number is a natural number greater than 1 that can be divided evenly only by 1 and itself. Thus the first few prime numbers are You can see a longer list of prime numbers if you like. The Goldbach Conjecture. Named after the number theorist Christian Goldbach (1690-1764). The problem: is it possible to write every even number greater than 2 as the sum of two primes? The conjecture says "yes", but nobody knows. You can explore the Goldbach conjecture interactively with the Prime Machine applet.

93. Amazing Number Facts No The new prime number, discovered on June 1st, is one of a special class of prime A perfect number is one whose factors add up to the number itself. http://www.madras.fife.sch.uk/maths/amazingnofacts/fact028.html

No 28 "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 2098960 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. There is a well-known formula that generates a "perfect" number from a Mersenne prime. A perfect number is one whose factors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3. The newly discovered perfect number is This number is 4,197,919 digits long!

94. The Music Of The Primes Following on from his article The prime number lottery in last issue of The graph of the sound wave of the tuning fork looks like a perfect sine wave. http://plus.maths.org/issue28/features/sautoy/

about Plus support Plus ... Plus search plus with google home latest issue explore the archive ... news Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 28 January 2004 Contents Features Pools of blood Making the grade: Part II The music of the primes Practice makes perfect Career interview Interview: Maths student Regulars Plus puzzle Pluschat Outer space Reviews 'Dicing with death' 'Strange curves, counting rabbits' 'A mathematician plays the market' News from January 2004 ... posters! January 2004 Features The music of the primes by Marcus du Sautoy Many people have commented over the ages on the similarities between mathematics and music. Leibniz once said that "music is the pleasure the human mind experiences from counting without being aware that is counting". But the similarity is more than mere numerical. The aesthetics of a musical composition have much in common with the best pieces of mathematics, where themes are established, then mutate and interweave until we find ourselves transformed at the end of the piece to a new place. Just as we listen to a piece of music over and over, finding resonances we missed on first listening, mathematicians often get the same pleasure in rereading proofs, noticing the subtle nuances that make the piece hang together so effortlessly. The one advantage that music has over mathematics is the physical connection that our body has with the sound of a composition. The hairs on the back of my neck never fail to stand on end when I hear Schubert's

95. Numbers: Glossary For example, 23 is a prime number because it cannot be made by multiplying togethersmaller The Mersenne primes are a special type of prime number. http://richardphillips.org.uk/number/gl/prime.htm

Prime Numbers The first twelve prime numbers are - A number is called prime if its only factors are one and itself. Many numbers can be made by multiplying smaller numbers together. For example - 3 x 7 = 21 - 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 - - and they can be expressed as a power of two minus one - 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.

96. MathSteps: Grade 5: Prime Factors: When Students Ask For example, a perfect number is one whose proper factors (factors less than the The greatest prime number discovered so far has 895, 932 digits, http://www.eduplace.com/math/mathsteps/5/b/5.primefact.ask.html

Prime Factors Why should I bother learning this? The prime factorization of a number is used in many algorithms such as finding the least common multiple and the greatest common divisor. These in turn are used in working with fractions. The least common multiple is used when finding the lowest common denominator and the greatest common factor is used in simplifying a fraction. Many patterns, formulas, and number concepts in number theory rely on the ability to express a number as a product of prime numbers. For example, a perfect number is one whose proper factors (factors less than the number) add up to the given number. The smallest perfect number is six, and its proper factors are 1, 2 and 3. After illustrating six as being perfect, you could ask students to find the next perfect number (28). What is the greatest prime number? There is no greatest prime number. The greatest prime number discovered so far has 895, 932 digits, but there are undoubtedly greater ones. A famous mathematician named Euclid was able to prove many years ago that there is no greatest prime number. Are there rules for divisibility for 6, 7, 8 and 11?

97. Types Of Number The Delannoy numbers are the number of lattice paths from to (b, a) in which onlyeast (1, 0), perfect numbers. mathworld.wolfram.com/perfectNumber.. http://www.virtuescience.com/numbertypes.html

site the web Number Data-Base Types of Number Fibonacci Numbers mathworld.wolfram.com/FibonacciNumber.. The first few Fibonacci numbers are Lucas Numbers mathworld.wolfram.com/LucasNumber.. The first few are Motzkin Numbers mathworld.wolfram.com/MotzkinNumber.. The first are Mersenne Numbers mathworld.wolfram.com/MersenneNumber.. The first are Abundant Numbers mathworld.wolfram.com/AbundantNumber.. The first are Deficient Numbers mathworld.wolfram.com/DeficientNumber.. The first are Delannoy Numbers mathworld.wolfram.com/DelannoyNumber.. The Delannoy numbers are the number of lattice paths from to (b, a) in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed. The first few are Schröder Numbers mathworld.wolfram.com/SchroederNumber.. The Schröder number is the number of lattice paths in the Cartesian plane that start at (0, 0), end at (n, n), contain no points above the line y = x, and are composed only of steps (0, 1), (1, 0), and (1, 1). The first few are Catalan Numbers mathworld.wolfram.com/CatalanNumber..

98. Big Number Calculator The tag for this number is b58 sp8 . (This prime is a perfect prime whichmeans that (p1)/2 is also a prime. You can easily check this on the calculator http://world.std.com/~reinhold/BigNumCalc.html

Big Number Calculator Applet by Arnold Reinhold, diceware.com , Rev 0.9 This Applet requires a Java enabled browser (Java version 1.1 or later). The calculator works just like an ordinary desk calculator. To compute 2+3 you click the button, then the + button, then the button and finally the button. There are three display windows. The top window is the X display and shows the final result. The second window is the Y display and shows the value that you are entering. You can also paste numbers into the Y display (but not the X display). In general the calculator uses infix notation for most operations. Enter a number in Y, press an operation (e.g. * or "gcd"), enter another number in Y, then press "=". To do work mod N, enter N in Y , press "=" and then press "setmod". Do a setmod of zero to clear the modulus and work in the group of integers. The bottom window shows information about the number in the X window, including: its length in bits whether or not it is a prime the number base it is displayed in the master modulus the current operation its value as a decimal floating point number its tag The tag is an six character hashcode tag that is displayed for very large numbers. This value is handy when exchanging large values or rechecking calculations. A tag is also show instead of the modulus when the modulus is longer than 9 digits. (The tag vaule is the high order 30 bits of the SHA1 hash of the number, shown in base-32.)

99. Number Theory Topics If Mpis also a prime number, than it s a Mersenne prime (number). A perfectGolomb ruler is one, in which every such difference from 1 to the length of http://www.lb.shuttle.de/apastron/numbers.htm

Stars among the numbers... (last update see Golomb ruler search on 22. Apr 2000) The Mersenne Numbers Database The numbers defined as M p p - 1 are the Mersenne Numbers, there p means a prime number. If M p is also a prime number, than it's a Mersenne prime (number). Early discovered in the 18 th century by Marin Mersenne, they have gained some fame especially in the last decades, because the highest so far known prime numbers are without exception such Mersenne primes (remark: if the exponent is no prime, than M p is always also composite). This has two reasons: first, only numbers of the type of Mersenne or Fermat allow an enormous efficient algorithm to work for checking if they are primes. And the second reason is, that (see below) only the Fermat numbers with the lowest possible exponents are primes, leaving all checked higher ones as composite numbers, and furthermore it's believed, that these are all Fermat primes. Therefore for higher exponents only the Mersenne numbers can be primes - at least today it seems so. The scheme of checking is called Lucas-Lehmer Test and goes like this: a sequence s of numbers is calculated with s(1) = 4, s(n+1) = (s(n)

100. Prime Numbers A prime number is one which cannot be expressed as a product of two whole For example, 6 equals 2 times 3, so 6 is not prime. 7 is a prime number http://argyll.epsb.ca/jreed/javaMath/prime/sgprime.htm

Prime Numbers Prime numbers have long been an interest of mine. The main reason is that with large numbers it takes a very long time to reliably decide whether that number is prime or not. This gives rise to more creative methods being devised. Interest in this area is heavily funded from the cryptography because large prime numbers play a key role in the development of encryption (secure, coded) systems. Here is a simple Java applet which allows you to type in a number and find out whether or not it is prime: Prime Number Test [ Sorry, you cannot see the applet because your browser does not support Java applets ] source: Prime Numbers A prime number is one which cannot be expressed as a product of two whole numbers other than itself and 1. For example, 6 equals 2 times 3, so 6 is not prime. 7 is a prime number because only 1 and 7 can be multiplied together to give 7. The applet above simply checks all whole numbers from 2 up to the square root of the number to see if any of them divide evenly into the number being tested. Both the interface and the core algorithm could do with some work. Some of the imaginative methods people have used to see if a number is prime or not include: just check whole numbers from 2 up to the square root of the number you are testing - if it has any factors, one of them must be less than its square root

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