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
Extractions: POINTERS: Texts Software Web links Selected topics here For analogues in number fields, See 11R04 Parent field: 11: Number Theory Browse all (old) classifications for this area at the AMS. 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-*
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
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
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
Extractions: 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. 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
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
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
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:
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/
Extractions: 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 by John Webb 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"?
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
Extractions: 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)
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/
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
Extractions: 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
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/
Extractions: 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
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
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
Searching For Primes Illustrated Hypography article on how prime numbers are found, with reviewed links to prime number Category Science Math Number Theory prime numbersDid 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
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
