81. 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-*

82. What's A Number? perfect numbers. Every 2n. Euclid in his Elements, IX.36 proved that if,for a prime p, p+1 = 2 k , then 2 k1 p is perfect. Leonhard http://www.cut-the-knot.com/do_you_know/numbers.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site What is a number? When I considered what people generally want in calculating, I found that it always is a number. Mohammed ben Musa al-Khowarizmi. From The Treasury of Mathematics , p. 420 H. O. Midonick Philosophical Library, 1965 Indeed there are many different kinds of numbers. rational and irrational real and complex imaginary algebraic and transcendental ... square and triangular numbers Let's talk a little about each of these in turn. Rational and Irrational numbers 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

83. Biography Of Marin Mersenne It remains an open question as to whether there are any odd perfect numbers. Wheneveranother mersenne prime is found, another perfect number is generated. http://www.andrews.edu/~calkins/math/biograph/199899/biomerse.htm

Back to the Table of Contents Marin Mersenne His Life Marin Mersenne was a 17th century monk and mathematician, who mainly studied the numbers 2 p Marin Mersenne is best known for his role as a sort of clearing house for correspondence between eminent philosophers and scientists, and for his work in number theory. Mersenne's Accomplishments Many early writers felt that the numbers of the form 2 p - 1 were prime for all primes p, but in 1536 Hudalricus Regius showed that 2 - 1 = 2047 was not prime. By 1603 Pietro Cataldi had correctly verified that 2 - 1 and 2 - 1 were both prime, but then incorrectly stated 2 n -1 was also prime for 23, 29, 31 and 37. In 1640 Fermat showed Cataldi was wrong about 23 and 37; then Euler in 1738 showed Cataldi was also wrong about 29. Sometime later Euler showed Cataldi's assertion about 31 was correct. Enter Mersenne. Marin Mersenne investigated prime numbers and he tried to find a formula that would represent all prime numbers. Although he failed in this, his work on the numbers 2

84. SYNERGETICS: INDEX 420.041, 443.02, 464.08 See also Equanimity Exact Ideal perfect imperfect systems,430.06, 1074.0013 prime nucleus, 427.03 prime numbers, 202.03, 223.67 http://www.rwgrayprojects.com/synergetics/index/INDEXP.html

@import url("../menu/menu.css"); INDEX for 'P' Package: Packaged, A, ,Table See also: Discontinuity Discrete Energy package Finite ... VE packages Panic, See also Fire in a theater Paper: Sheet of paper, Parabola, See also Hyperbolic-parabola Parallax, Parallelogram of forces. See Force lines: Omnidirectional lines of force Parallels, footnote, See also Convergent vs parallel perception Parameters, Parity, See also Disparity Partial generalization, Partially tuned, See also: Threshold Twilight zone Particle: Particular, See also: Building blocks: No building blocks Quark Particle accelerator, Parting the grass, Parts: Partiality, See also: Components Overlapping: Partially overlapping Starting with parts Pass: And it came to pass, Passive. See Past: Historic past, See also: History Now Patents, Path. See Parting the grass Paths of least resistance, Patience, Pattern: Patternings, See also: Big patterns Bounce patterns of energy Circuit pattern tensegrity Holding patterns ... Wave pattern Pattern cognizance, Pattern conservation, Pattern divisibility

85. Ivars Peterson's MathTrek - Cubes Of Perfection From Ivars Peterson's MathTrek column in MAA Online. Curious relationships satisfied by perfect numbers.Category Science Math Number Theory Factoring perfect numbers...... theorem All even perfect numbers must have the form specified by Euclid's formula.Hence, every Mersenne prime automatically leads to a new perfect number. http://www.maa.org/mathland/mathtrek_5_18_98.html

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:

86. Prime & Perfect Numbers Test - C, C++, C__ prime perfect numbers Test. http://www.planet-source-code.com/vb/scripts/ShowCode.asp?txtCodeId=5161&lngWId=

87. 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/

PRIME NRICH PLUS Current Issue ... Subject index Issue 22: Jan 03 Issue 21: Sep 02 Issue 20: May 02 Issue 19: Mar 02 Issue 18: Jan 02 Issue 17: Nov 01 Issue 16: Sep 01 Issue 15: Jun 01 Issue 14: Mar 01 Issue 13: Jan 01 Issue 12: Sep 00 Issue 11: Jun 00 Issue 10: Jan 00 Issue 9: Sep 99 Issue 8: May 99 Issue 7: Jan 99 Issue 6: Sep 98 Issue 5: May 98 Issue 4: Jan 98 Issue 3: Sep 97 Issue 2: May 97 Issue 1: Jan 97 In perfect harmony by John Webb Do the symbols below on the left match the descriptions on the right? p The greek letter "pi" x x squared x Square root of x If not, an alternative version of this page that should work with most browsers is available here. Introduction Two elementary series are studied in school mathematics: arithmetic series, such as n and geometric series, such as n There is an equally elementary series, called the harmonic series n Though elementary in form, the harmonic series contains a good deal of fascinating mathematics, some challenging Olympiad problems, several surprising applications, and even a famous unsolved problem. There are a number of questions about the harmonic series which have answers that initially offend our intuition, and therefore have particular relevance to mathematics teaching and learning. Why is the series called "harmonic"?

88. Wayne McDaniel Publication Page that all Even perfect numbers are of Euclid's Type, Math. Mag., 48, No. 2, March,(1975), 107108. On the Largest prime Divisor of an Odd perfect Number, II http://www.cs.umsl.edu/~mcdaniel/publication.html

Publications The Non-Existence of Odd Perfect Numbers of a Certain Form, Archiv der Mathematik Vol. XXI, (1970), 52-53. On Odd Multiply-Perfect Numbers, Bollettino del( Unione Mathematica Italiana , N. 2 (1970), 185-190. On the Divisibility of an Odd Perfect Number by the Sixth Power of a Prime, Math of Comp. , vol. 25 (1971), 383-385. A New Result Concerning the Structure of Odd Perfect Numbers, (with Peter Hagis), Proceedings of the American Mathematical Society , vol. 32 (1972), 13-15. Some Results Concerning the Non-Existence of Odd Perfect Numbers of the form p(M2(, (with Peter Hagis), Fibonacci Quarterly , 13, February, (1975), 25-28. Perfect Gaussian Integers, Acta Arithmetica , XXV (1974), 137-144. On the Largest Prime Divisor of an Odd Perfect Number, (with Peter Hagis), Math. of Comp. , 27, October, (1973), 955-957. On Multiple Prime Divisors of Cyclotomic Polynomials, Math. of Comp. , 28, July, (1974), 847-850. On the Proof that all Even Perfect Numbers are of Euclid's Type, Math. Mag. , 48, No. 2, March, (1975), 107-108. On the Largest Prime Divisor of an Odd Perfect Number, II, (with Peter Hagis)

89.  All The Prime Factors Of The Reversed Smarandache Concatenated Numbers Upto Th The first one with an unknown prime factor is of factors of the normal SmarandacheConcatenated numbers by Patrick Easyspace your perfect partner for the web. http://www.worldofnumbers.com/revfact.htm

W orld O f N umbers HOME plate WON Reversed Smarandache Concatenated Numbers. Prime factors from n (n=2,3,...,64) downto 1 Normal Smarandache Concatenated Numbers Repunits Factorization In the table below you'll find all the prime factors of the reversed concatenation of numbers from n downto These numbers are called Reversed Smarandache Concatenated Numbers. The first one with an unknown prime factor is when n = If there is a breaktrough in completely factorising , please let me know, so that I can update the list. For the factorizations I closely follow the source from Micha Fleuren Reversed Smarandache factors Other subject related sources on the web : Smarandache Numbers by Dr. M. L. Perez Smarandache factors by Micha Fleuren Primes by Listing by Carlos Rivera Consecutive Number Sequences by Eric W. Weisstein Smarandache Sequences by Eric W. Weisstein List of factors of the normal Smarandache Concatenated Numbers by Patrick De Geest Book Sources : "Some Notions and Questions in Number Theory" , by C.Dumitrescu and V.Seleacu

90.  All The Prime Factors Of The Smarandache Concatenated Numbers Upto The First N The first one with an unknown prime factor is factors of the Reversed SmarandacheConcatenated numbers by Patrick Easyspace your perfect partner for the web. http://www.worldofnumbers.com/factorlist.htm

W orld O f N umbers HOME plate WON Normal Smarandache Concatenated Numbers Prime factors from 1 upto n (n=2,3,...,72) Reversed Smarandache Concatenated Numbers Repunits Factorization In the table below you'll find all the prime factors of the concatenation of numbers from upto n These numbers are called Smarandache Concatenated Numbers. The first one with an unknown prime factor is when n = If there is a breaktrough in completely factorising , please let me know, so that I can update the list. For the factorizations above I follow the sources from Micha Fleuren Smarandache factors Hans Havermann Other subject related sources on the web : Smarandache Numbers by Dr. M. L. Perez Smarandache factors by Micha Fleuren Primes by Listing by Carlos Rivera Consecutive Number Sequences by Eric W. Weisstein Smarandache Sequences by Eric W. Weisstein List of factors of the Reversed Smarandache Concatenated Numbers by Patrick De Geest Book sources : "Some Notions and Questions in Number Theory" , by C.Dumitrescu and V.Seleacu , Glendale, AZ:Erhus University Press, 1994. (communicated to me by Marin Petrescu from Bucharest) "CRC Concise Encyclopedia of Mathematics" , by Eric W. Weisstein

91. Mathematics Fermat's Last Theorem Fermat's Last Theorem (n=4) Goldbach's Conjecture Wanless'Theorem Twin prime Conjecture Odd perfect numbers Catalan's Conjecture http://www.bearnol.pwp.blueyonder.co.uk/Math/

Geometry Pythagoras' Theorem Pythagoras' Big Theorem Euclid's Elements Number Theory Wanless' Theorem - Fermat's Big Theorem Fermat's Last Theorem Fermat's Last Theorem (n=4) Goldbach's Conjecture ... The Panfur Project Diophantus' Arithmetica (work in progress...) Book I Book II Book III Book VIII ... Book X Please email comments/suggestions/additions to: james@grok.ltd.uk Sponsored by: The Answer Shop

92. Euclid's Elements, Book IX, Proposition 36 If as many numbers as we please beginning from a proportion until the sum of all becomesprime, and if the last makes some number, then the product is perfect. http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html

Proposition 36 If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect. Let as many numbers as we please, A, B, C, and D, beginning from a unit be set out in double proportion, until the sum of all becomes prime, let E equal the sum, and let E multiplied by D make FG. I say that FG is perfect. For, however many A, B, C, and D are in multitude, take so many E, HK, L, and M in double proportion beginning from E. Therefore, ex aequali A is to D as E is to M. Therefore the product of E and D equals the product of A and M. And the product of E and D is FG, therefore the product of A and M is also FG. VII.14 VII.19 Therefore A multiplied by M makes FG. Therefore M measures FG according to the units in A. And A is a dyad, therefore FG is double of M. But M, L, HK, and E are continuously double of each other, therefore E, HK, L, M, and FG are continuously proportional in double proportion. Subtract from the second HK and the last FG the numbers HN and FO

93. PLACE VALUE NUMBER SYSTEM Composite numbers perfect Squares. • prime numbers. - Discover prime numbersin Eratosthenes Sieve - Multiples of 2, 3, 5 7 - prime Number Song. http://www.cogtech.com/Algebra1/

Covers topics and concepts essential for advancing on to algebra. Concepts are developed through the use of highly visual interactive explorations which develop life long understanding. Google, your colorful guide, will guide you through each lesson, giving hints and instructions. The program is kid, parent, and teacher approved. Kids love it because it is fun and easy to learn. Parents and teachers like it because it ACTUALLY teaches, and does not blast or destroy anything. The following is an outline of the program and its sections. - Binary - Decimal Conversion - Building Binary Numbers - Mayan - Egyptian - Arabic - Sanskrit - Roman - Build Place Value Chart - Money and Place Value - Notation - Music Scale to Place Value - Powers of Ten - Scientific Notation - Observatory - Mt. Everest to Pluto - A Fly to DNA Approaches to estimating answers to math problems are presented with a basketball theme. Step by step methods for estimating with examples are presented in detail. - Learn to round in a basketball court - Compelling Visuals - Basketball Game Exercises - Learn Estimation in a Pre-Game Briefing - Front End Multiplication - Division - Practice Shots with Free Throws - Discover how Eratosthenes estimated the circumference of Earth 2000 years ago.

94. A02-PrimesAndFactorizaton.html Notice that squarefree numbers are always products of distint primes with an exponentof 1! If a prime has a then it contains at least a perfect square of that http://www.mapleapps.com/powertools/alg1/html/A02-PrimesAndFactorizaton.html

95. Free2Code.net - Code Has Moved Other code list Comments prime numbers. Author said Self explain code that will check if the entered integer is prime or not. http://www.free2code.net/code/other/code_79

Currently 18 users online. Pascal /n./ A programming language named after a man who would turn over in his grave if he knew about it. Home About Us Tutorials Code ... Login You are not logged in, Login We have members: is online now guests are visiting threads posts Code has moved. The Code you are attempting to view has moved, it can be accessed here Please view our

96. Searching For Primes Illustrated Hypography article on how prime numbers are found, with reviewed links to prime number Category Science Math Number Theory prime numbers......Did you know that there are prime numbers with billions of digits? This weekwe take a look at how number theorists search for these monsters. http://www.hypography.com/topics/searchingforprimes.cfm

document.write('<');document.write('! '); home hypographies links quizzes ... about Tuesday, March 18, 2003 Not logged in Searching for Primes Did you know that there are prime numbers with billions of digits? This week we take a look at how number theorists search for these monsters. Created by Tormod Guldvog Last updated September 14 2001 Viewed 7473 times. Do you believe that there is an infinite amount of primes? Yes No results For beginners, the world of prime numbers may seem to be a mixture of chaos and strict order. If you start at 1 and count all the integers until, say, 1 million, there is a finite number of primes. The problem is that it is very difficult to find out exactly how many primes there are, and how they are spaced out over the endless list of integers. But over the years, new theories have emerged which predict where and how often primes will emerge. Using these theories, modern prime hunters have located primes which are so large that they defy the imagination. And the only thing we can say for sure about prime numbers, is that there are many more to come. Some of the links below point to sites where you can actively take part in the search for primes. You can also test your skills to see how good you are at determining whether a given number is prime.

97. Encyclopædia Britannica branch of mathematics concerned with the integers, or whole numbers,and generalizations of the integers. Number theory grew out http://www.britannica.com/eb/article?eu=117296

Page 5     81-97 of 97    Back | 1  | 2  | 3  | 4  | 5