61. MATHEWS: Zip Primes true for all k = 1,2, ,K we call the number a zip prime of order K. If K is equalthe number of digits of the starting number we call it a perfect zip prime. http://www.wschnei.de/digit-related-numbers/zip-primes.html

Zip Primes Walter Schneider 2001 (last updated 1/3/2003) In this article we extend the class of primes which remain prime when rearranging the digits in a certain way by so-called zip primes . As the name says rearranging the digits means to zip the number into k parts by alternately distributing the left-most digit to the parts. For the number for example the zipping goes as follows: k = 1: 27239 prime k = 2: 229 prime 73 prime k = 3: 23 prime 79 prime 2 prime k = 4: 29 prime 7 prime 2 prime 3 prime k = 5: 2 prime 7 prime 2 prime 3 prime 9 not prime If for given k all parts are prime we say that the number is a k-zippable prime . If this is true for all k = 1,2,...,K we call the number a zip prime of order K . If K is equal the number of digits of the starting number we call it a perfect zip prime The above example shows that 27239 is a zip prime of order 4.

62. Mersenne Prime Mersenne primes have a close connection to perfect numbers, whichare numbers that are equal to the sum of their proper divisors. http://www.fact-index.com/m/me/mersenne_prime.html

Main Page See live article Alphabetical index Mersenne prime A Mersenne prime is a prime number that is one less than a power of two Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist. It is currently unknown whether there is an infinite number of Mersenne primes. More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than an odd power of two; the notation M n n shows that M n can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true; M n may be composite even though n is prime. For example, 2

63. Perfect Numbers At the heart of every perfect number is a Mersenne prime. be a perfect number with(2c+1 1) being the embedded Mersenne prime. Then the divisors of n P are. http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/prfctno.htm

Proficiency Tests Mathematical Thinking in Physics Aeronauts 2000 CONTENTS Introduction Fermi's Piano Tuner Problem How Old is Old? If the Terrestrial Poles were to Melt... ... A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File On Expansion of the Universe - PDF File Perfect Numbers - A Case Study Perfect numbers are those numbers that equal the sum of all their divisors including 1 and excluding the number itself. Most numbers do not fit this description. At the heart of every perfect number is a Mersenne prime. All of the other divisors are either powers of 2 or powers of 2 times the Mersenne prime. Let's examine the number 496 - one of the known perfect numbers. In order to demonstrate that 496 is a perfect number, we must show that 496 = (the sum of all its divisors including 1 and excluding 496) We might just start by dividing and working out the divisors the long way. Or, we might begin by noting that, in the notation that includes a Mersenne prime, x 31.

64. Special Numbers numbers. Euclid proved that a number n of the form (2 n 1)*2 n-1is a perfect number if the factor 2 n -1 is prime. For example http://www.math.wichita.edu/history/topics/snumbers.html

Topics About Special Numbers Topic Tree Home Following are some items relating to special numbers discussed in the history of mathematics. Contents of this Page Perfect Numbers Square Numbers The Nature of Prime Numbers ... The History of Zero Perfect Numbers The Pythagoreans produced a theory of numbers comprised of numerology and scientific speculation. In their numerology, even numbers were feminine and odd numbers masculine. The numbers also represented abstract concepts such as 1 stood for reason, 2 stood for opinion, 3 stood for harmony, 4 stood for justice, and so on. Their arithmetica had a theory of special classes of numbers. There were “perfect” numbers of two kinds. The first kind included only 10, which was basic to the decimal system and the sum of the first four numbers 1 + 2 + 3 + 4 = 10. The second kind of “perfect” numbers were those equal to the sum of their proper divisors. A perfect number is a positive integer that is equal to the sum of it divisors. However, for the case of a perfect number, the number itself is not included in the sum. The Greeks called a number such as 6 or 28 a “perfect” number because the sum of the proper divisors in each case is equal to the number; the proper divisors of 6 are 1, 2, and 3, and their sum is 6. Although perfect numbers are regarded as arithmetical curiosities, their study has helped to develop the theory of numbers. Euclid proved that a number n of the form (2

65. Number Patterns, Curves & Topology other than itself). Investigates perfect numbers, their properties,and their connection to Mersenne primes. Title Amicable numbers. http://ccins.camosun.bc.ca/~jbritton/jbfunpatt.htm

Investigating Patterns Number Patterns Fun with Curves TOPIC LINKS TOPIC 1 (Prime Numbers / Magic Squares) Title: Sieve of Eratosthenes Comment: A natural number is prime if it has exactly two positive divisors, 1 and itself. Eratosthenes of Cyrene (276-194 BC) conceived a method of identifying prime numbers by sieving them from the natural numbers. Web page uses the sieve to find all primes less than 50. Includes a link to a Sieve of Eratosthenes Applet which also begins with a size or upper boundary of 50. Eratosthenes' Sieve contains a similar applet preset to find all primes less than 200. Both applets require a JAVA-capable browser. Title: Prime Number List Comment: Once you have entered the lower bound and upper bound, this javascript applet will display all prime numbers within the selected range. Another Prime Number List will generate prime numbers until you click Stop or until your computer runs out of memory. Title: Prime Factorization Machine Comment: A positive integer (natural number) is either prime or a product of primes. This applet decomposes any positive integer less than 1,000,000 into its prime factors. The bigger the number, the longer it will take. Requires a JAVA-capable browser. Title: Comment: Includes a link to Mini-Lessons demonstrating how to find the Common Divisor Factor (GCF) or Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of two or more natural numbers using prime factorization. Features an interactive applet with detailed explanations and solutions.

66. Ivars Peterson's MathTrek - Cubes Of Perfection even perfect number. primes of the form are now known as Mersenne primes, and thesenumbers figure prominently in the search for the largest known prime (see http://www.maa.org/mathland/mathtrek_5_18_98.html

Search MAA Online MAA Home Ivars Peterson's MathTrek May 18, 1998 Cubes of Perfection 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:

67. Ivars Peterson's MathTrek - Prime Listening 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. 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.

68. Number Theory --  Encyclopædia Britannica Elementary number theory. perfect numbers and Mersenne primes. , perfect numbersand Mersenne primes from number theory The true origin of number theory is http://www.britannica.com/eb/article?eu=117296&tocid=52279&query=number theory

69. The Music Of The Primes The harmonics are in some perfect balance, creating the endless ebb and flow of the tellus how far Gauss s guess is from the way the prime number dice really http://plus.maths.org/issue28/features/sautoy/

@import url(../../../newinclude/plus_copy.css); @import url(../../../newinclude/print.css); @import url(../../../newinclude/plus.css); 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 ... poster! 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.

70. In Perfect Harmony In perfect harmony. Since the remaining denominators are all prime, and the primenumbers are very thinly scattered, it is indeed surprising that the series of http://plus.maths.org/issue12/features/harmonic/

@import url(../../../newinclude/plus_copy.css); @import url(../../../newinclude/print.css); @import url(../../../newinclude/plus.css); 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 12 September 2000 Contents Features Death and statistics Fishy business In perfect harmony Take a break ... Career interview: Avalanche researcher Career interview Career interview: Avalanche researcher Regulars Plus puzzle Pluschat Mystery mix Letters Reviews 'The Maths Gene' 'Ingenious Pursuits' 'The Cogwheel Brain' News from September 2000 ... poster! September 2000 Features In perfect harmony by John Webb How would you like your maths displayed? If the character p doesn't look like the greek letter "pi", and the character isn't a square root sign

71. Prime Formulas is a Mersenne prime. Thus the investigation of perfect numbers is closelyrelated with Mersenne primes. The first 4 perfect numbers http://heja.szif.hu/ANM/ANM-000926-A/anm000926a/node3.html

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-000926-A Prime formulas The great mathematicians for centuries were trying to give formulas, which would always produce primes, or at least infinitely many primes. For the second part of this question a nice answer was given by the following theorem, which analyses the occurence of prime numbers in arithmetic sequences. T HEOREM 6. (B Y D IRICHLET) Let and be integer numbers, for which gcd . In this case the sequence produces infinitely many primes R EMARKS As special cases of Theorem 6., there are infinitely many primes in the form We can rephrase the results as follows: the polynomial with gcd produces infinitely many primes. In this context we can formulate some other questions, e.g. a) Is there a polynomial in the form , which produces infinitely many primes? b) Is there a polynomial which always produces prime numbers? In the first case it is easy to prove, that necessary conditions are the irreducibility of the polynomial and gcd , but the complete answer is still unknown. To question b), for

72. Sublime Numbers Assuming there are no odd perfect numbers, there can be no more even sublime numbersunless there are other (presently unknown) Mersenne prime exponents that http://www.mathpages.com/home/kmath202/kmath202.htm

Sublime Numbers For any positive integer n let t (n) denote the number of divisors of n, and let s (n) denote the sum of those divisors. The ancient Greeks classified each natural number n as "deficient", "abundant", or "perfect" according to whether s (n) was less than, greater than, or equal to 2n. Notice that the number 12 has 6 divisors, and the sum of those divisors is 28. Both 6 and 28 are perfect numbers. Let's refer to a natural number n as "sublime" if the sum and number of its divisors are both perfect. Do there exist any sublime numbers other than 12? To answer this question, recall that for any integer N with prime factorization we have Also, every even perfect number is of the form (2 s s-1 where 2 s - 1 is a prime. Thus an even perfect number has exactly one odd prime factor. Now suppose N is divisible by exactly k powers of 2. It follows that s (N) is divisible by 2 k+1 1, which is odd, so this must be a prime (else it would factor into two odd primes). Also, all the other factors of N must then contribute a combined factor of 2 k to s (N). But each odd prime power p

73. Finding Perfect Numbers The product will be a perfect number. For example, the sum of 16 is31, a prime number. Therefore, 31 x 16 = 496 (a perfect number). http://www.wpunj.edu/icip/itm/Lessonpl/calc/portos/perfect.html

Mark Porto and June Porto Hackensack School District TOPIC: The purpose of this lesson is to help students understand the differences among deficient, abundant, and perfect numbers and to help them determine a few perfect numbers. LEVEL: Grades five through eight TI-82 PROFICIENCY: A basic understanding of TI-82 is required. CLASS TIME: Two class periods TASK: Step 1: Students should find the proper factors for numbers one through fifty using either the divisibility laws or TI-82. Step 2: Students should determine the sum of the proper factors for each number. If the sum of the factors is less than the number itself, then this number is considered "deficient." If the sum of the factors is more than the number itself, then this number is considered "abundant." If the sum of the factors equals the number, then this number is considered a "perfect" number. Note: The first perfect number is 6 and the second is 28. These are the only two perfect numbers between one and fifty. Perfect numbers are few and far between. The next perfect number after 28 jumps to 496! The next few steps provide a quick method to determine a perfect number without having to find the sum of the factors. Students should use the TI-82 calculator when following these steps. Step 3: Starting with two as the first number, double the numbers up to 128. For example:

74. 4-dim HyperDiamond Lattice In fact computers have led to a revival of interest in the discoveryof Mersenne primes, and therefore of perfect numbers. At the http://www.innerx.net/personal/tsmith/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:

75. Perfect Numbers MD Sayers, An improved lower bound for the total number of prime factorsof an odd perfect number, M.App.Sc. Thesis, NSW Inst. Tech., 1986. http://www.math.swt.edu/~haz/prob_sets/notes/node13.html

Next: Exercises Up: The Factorization of Integers Previous: Exercises Perfect Numbers Let denote the sum of all positive divisors of n Proof. n has the form and every such number is a divisor of n . Therefore, The following theorem follows immediately from the above theorem. For example, 6 and 28 are perfect numbers because Proof. that Now let a be an even perfect number. Suppose that By Theorem which implies that Noting that u and are divisors of u , we have that u is a prime and because is the sum of all positive divisors. We still do not know if any odd perfect number exists, which is a famous difficult problem in number theory. Brent, Cohen and te Riele showed that the lower bound for an odd perfect number is M. S. Brandstein, New lower bound for a factor of an odd perfect number, No. 82T-10-240, Abstracts Amer. Math. Soc. R. P. Brent, G. L. Cohen and H. J. J. te Riele, Improved techniques for lower bounds for odd perfect numbers, Math. Comput. M. D. Sayers, An improved lower bound for the total number of prime factors of an odd perfect number, M.App.Sc. Thesis, NSW Inst. Tech., 1986. Donald Hazlewood and Carol Hazlewood Wed Jun 5 14:35:14 CDT 1996

76. Prime Numbers be the case so making n a perfect square square roots of these numbers are all numberswith two the interestingness of this class (now renamed `Primes by Lenat http://web.media.mit.edu/~haase/thesis/node59.html

Next: Properties of Primes Up: The AM Program Previous: Factorization Prime Numbers AM finds some simple patterns among these definitions: the set of numbers with zero divisors is empty; the set of numbers with one divisor is itself a singleton (this is found uninteresting); the set of numbers with two divisors (primes without one) is somewhat interesting because there are a reasonable number of examples. The excitment begins as AM notices that all numbers with three divisors are also perfect squares;To see this, consider an integer n with three divisors. One must be 1 and one must be n itself. The remaining divisor p must produce n by repeated multiplication so that . If , must also be a factor of n . Since n has only three divisors, this cannot be the case so making n a perfect square. this makes them interesting enough to consider what their square roots are (AM had earlier invented SQUARE-ROOT by inverting SQUARE). It turns out that the square roots of these numbers are all numbers with two divisors. This regularity boosts the interestingness of this class (now renamed `Primes' by Lenat) to a point where detailed examiniations of the properties of primes are undertaken. Ken Haase Sun Nov 3 16:17:57 EST 1996

77. 11A: Elementary Number Theory M^(3^n) ). Show there is a prime of the form What numbers are sums of two Egyptianfractions? solutions to the 4/n problem; perfect numbers recent literature; http://www.math.niu.edu/~rusin/known-math/index/11AXX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 11A: Elementary number theory Introduction History Applications and related fields For analogues in number fields, See 11R04 Subfields Multiplicative structure; Euclidean algorithm; greatest common divisors Congruences; primitive roots; residue systems Power residues, reciprocity Arithmetic functions; related numbers; inversion formulas Primes Factorization; primality Continued fractions, For approximation results, See 11J70; See also 11K50, 30B70, 40A15 Radix representation; digital problems, For metric results, See 11K16 Other representations None of the above but in this section Parent field: 11: Number Theory Browse all (old) classifications for this area at the AMS. Textbooks, reference works, and tutorials Well-known texts with an elementary focus include: LeVeque, William J.: "Fundamentals of number theory", Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977, 280 pp. ISBN 0-201-04287-8 Dudley, Underwood: "Elementary number theory", W. H. Freeman and Co., San Francisco, Calif., 1978. 249 pp. ISBN 0-7167-0076-*

78. Mersenne Primes Now Mersenne primes are especially interesting in another fascinating class of numbers with considerable psycho-mathematical significance ie perfect numbers. http://indigo.ie/~peter/prime.htm

New Largest Prime Numbers! Mersenne Primes Biologically, every male as an infant foetus initially enjoys a feminine identity. What may not be quite so obvious is that this parallels the very nature of the prime number system, which starts with an even (feminine) number. Indeed, this would suggest that every prime number can ultimately be derived from 2. A famous example of this approach is the set of Mersenne primes. (Another is the set of Fermat primes). Now Mersenne primes are especially interesting in that they also generate another fascinating class of numbers - with considerable psycho-mathematical significance i.e. perfect numbers. A Mersenne prime is always of the form 2 n - 1 (n is a positive integer). For example, - 1 = 31 is a Mersenne prime. Now, there are two points which I wish to point out which illustrate the transrational approach. 1) The power of 2 (i.e. the qualitative vertical number) must itself be prime, if the resulting number (i.e. the reduced quantitative horizontal number) is to be prime. In our example, the qualitative number 5 is prime, and the (reduced) quantitative number 31 is prime. 2) The resulting prime number is closely associated with a highly composite number.

79. Number Theory - Technology Services lookalike; perfect numbers; fermat test; a curious paradox; Pi r^2 /3; primeNumber Distribution Solved? How do we define division? prime numbers searching;Pi http://www.physicsforums.com/archive/f-80

Physics Help and Math Help - Physics Forums Mathematics View Forum : Number Theory Is the number of twin primes really infinite? Proof: Numbers with repeating blocks of digits are rational 7-million digit prime number discovered ... Car Body Parts Physics Forums is a network of science forums with an emphasis on physics help and star astronomy. Register to post your physics questions or just hang out and talk about general science topics!

80. Sci.math FAQ: Unsolved Problems Euler proved that if N is an odd perfect number, then in the prime power decompositionof 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 Current Top-Rated FAQs Are you an expert in this area? Share your knowledge and earn expert points by giving answers or rating people's questions and answers! This section of FAQS.ORG is not sanctioned in any way by FAQ authors or maintainers. Questions strongly related to this FAQ: what are some unsolved problems by Dylan McDonald (11/3/2003) Choose 4 prime numbers such that adding the first of them equals the fourth, for example:...

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