61. Jlpe's Number Recreations Page Features original number recreations by the author, such as generalized perfect numbers, digital diversio Category Science Math Recreations......jlpe's number recreations page. The concept of number is the obviousdistinction between the beast and man. Thanks to number, the http://www.geocities.com/windmill96/numrecreations.html

jlpe's number recreations page The concept of number is the obvious distinction between the beast and man. Thanks to number, the cry becomes song, noise acquires rhythm, the spring is transformed into a dance, force becomes dynamic, and outlines figures. Joseph Marie de Maistre I have hardly ever known a mathematician who was capable of reasoning. Plato On a Generalization of Perfect Numbers Ana's Golden Fractal The Picture-Perfect Numbers Fractal Dimension, Primes, and the Persistence of Memory ... The Justice of Numbers : The author will not be responsible for any frustration or sleepless nights suffered by the would-be solver. Number of Visits: J. L. Pe Last update: 16 February 2003.

62. Puzzle 111. Spoof Odd Perfect Numbers On the contrary, as you know, there are many even perfect numbers,at least as many as Mersenne prime numbers (38 at the moment). http://www.primepuzzles.net/puzzles/puzz_111.htm

Puzzles Puzzle 111. Spoof odd Perfect numbers It's almost believed that there is not any odd perfect number. But what about "almost" or "quasi" or "spoof" odd perfect numbers? Descartes found one o dd Spoof Perfect Number that is odd only if you suppose (incorrectly) that is prime. You can verify the above statement if you remember that: If n = p a .q b ..., then s (n)=[(p a+1 -1)/(p-1)].[(q b+1 -1)/(q-1)].... If n is perfect then s (n) = 2.n Questions: 1. Find the least and/or other o dd Spoof Perfect Numbers On the contrary, as you know, there are many even Perfect Numbers , at least as many as Mersenne prime numbers (38 at the moment). But, is there any even Spoof Perfect Number ? The answer is "yes", and in this case there are many of them. The following are the least 3 examples of even Spoof Perfect Number when you suppose incorrectly that one of its factor is prime is an even Spoof Perfect Number if you suppose incorrectly that 4 is a prime is an even Spoof Perfect Number if you suppose incorrectly that 9 is a prime is an even Spoof Perfect Number if you suppose incorrectly that 15 is a prime and many more...

63. Prime Numbers rules to knock out a large proportion of numbers which are not prime, eg no primenumber can end a number as the difference of two perfect squares because http://dev1.epsb.edmonton.ab.ca/math14_Jim/java/prime/sgprime.htm

64. Unsolved Problems It can be shown that p must be prime for (*) to be prime. As of December 2002,39 Mersenne primes are known. There are thus 39 known even perfect numbers. http://www.math.utah.edu/~alfeld/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.

65. Message 46 Containing Perfect, Out The fact that numbers of this form (last factor a mersenneprime) are perfect wasknown by the ancient greeks, but the other way round is not obvious to see. http://pw1.netcom.com/~hjsmith/Perfect/46Out.html

Message 46 Containing Perfect, Out Return to Perfect Numbers Return to Harry's Home Page This page accessed times since June 7, 1997. Changes last made on: Sun Mar 08 01:20:53 pm PST 1998

66. Perfect Number -- From MathWorld intimately connected with a class of numbers known as Mersenne primes. This can bedemonstrated by considering a perfect number P of the form where q is prime. http://mathworld.wolfram.com/PerfectNumber.html

Foundations of Mathematics Mathematical Problems Unsolved Problems Number Theory ... Divisor-Related Numbers Perfect Number Perfect numbers are integers n such that where s n ) is the restricted divisor function (i.e., the sum of proper divisors of n ), or equivalently where is the divisor function (i.e., the sum of divisors of n including n itself). The first few perfect numbers are 6, 28, 496, 8128, ... (Sloane's ). This follows from the fact that etc. Perfect numbers were deemed to have important numerological properties by the ancients, and were extensively studied by the Greeks, including Euclid Perfect numbers are intimately connected with a class of numbers known as Mersenne primes . This can be demonstrated by considering a perfect number P of the form where q is prime . Then and using for q prime, and gives Therefore, if is prime , then is a perfect number, as was stated in Proposition IX.36 of Euclid's Elements (Dickson 1957, p. 3; Dunham 1990). The first few perfect numbers are summarized in the following table. p P While many of Euclid's successors implicitly assumed that all perfect numbers were of the form (9) (Dickson 1952, pp. 3-33), the precise statement that all

67. Odd Perfect Number -- From MathWorld Odd perfect numbers. Math. Proc. Cambridge Philos. Soc. 115, 191196, 1994. Iannucci,D. E. The Second Largest prime Divisor of an Odd perfect Number Exceeds http://mathworld.wolfram.com/OddPerfectNumber.html

Number Theory Special Numbers Divisor-Related Numbers Odd Perfect Number In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 1957, p. 3), although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why a odd perfect number could not exist (Dickson 1957, p. 12). Descartes was therefore among the first to consider the existence off odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid's construction comprised all possible perfect numbers (Dickson 1957, pp. 6-12). In 1657, Frenicle repeated Descartes' belief that every even perfect number is of Euclid's form and that there was no reason odd perfect could not exist. Like Frenicle, Euler also considered odd perfect numbers. To this day, it is not known if any odd perfect numbers exist, although numbers up to 10 have been checked without success, making the existence of odd perfect numbers appear unlikely (Brent

68. Perfect Numbers Most numbers do not fit this description. At the heart of every perfectnumber is a Mersenne prime. All of the other divisors 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.

69. Perfect Numbers expression 2 n 1 are known as Mersenne numbers, named after to Euclid, you can usethat Mersenne prime to automatically find a perfect number. http://66.137.204.220/plethorama/perfect.htm

Perfect Numbers Part 1 by Paul Becker updated September, 1999 with newly discovered perfects Since the beginning mathematicians have assigned mystical, magical, and even sacred properties to numbers. In the days of the Greeks, one such set of numbers were the perfects. A "perfect" number is a whole number that equals the sum of its divisors (except itself). Six is the first perfect, because 1,2, and 3 are factors, and 1+2+3 = 6. The next perfect is 28 (1,2,4,7,14). Early Christian and Jewish observers noted the obvious perfection of 6 and 28, for the Earth was created in six days and the moon circles the Earth every 28. In fact, St. Augustine argued that "Six is a number perfect in itself, and not because God created all things in six days; rather the inverse is true; God created all things in six days because the number is perfect. And it would remain perfect even if the work of the six days did not exist." (The City of God, Book 11, Chapter 30). The "perfection" seemingly ends here, because the next two perfects are 496 and 8128. With these four perfects; 6, 28, 496, 8128; the ancients made observations that:

70. VACETS Technical Column - Tc48 They do know, however, that all even perfect numbers have a direct relationshipto Mersenne primes, P = 2^(p1)*(2^p-1) where 2^p-1 is a Mersenne prime. http://www.vacets.org/tc/tc48.html

VACETS Regular Technical Column The VACETS Technical Column is contributed by various members , especially those of the VACETS Technical Affairs Committe. Articles are posted regulary on vacets@peak.org forum. Please send questions, comments and suggestions to vacets-ta@vacets.org September 10, 1996 Largest Known Prime Number Discovered About 2 years ago, Andrew Wiles, a researcher at Princeton, claims to have proved the Fermat's Last Theorem (FLT) and later a large gap was found in the proof. (The gap was filled later at the end of 1994.) At that time, we, the VACETSERS, had debated on proving the FLT using numerical methods (i.e., using computer to crank out the solutions to the famous theorem). One of the first steps in numerical method is to find the prime numbers, and from that, a "fastest prime number generator" war was waged among us the VACETSERS. The result of that "war" was that we were able to reduce the time from tens of seconds to find all the primes below 1 million to less than 1 second to find all the primes below 10 million. It was an improvement of more than 100. It was a fun war. (Actually, for me, anything involved with numbers, especially prime numbers, is fun.) Shortly after that "fastest prime number generator" war, Thomas R. Nicely, Professor of Mathematics at Lynchburg College, Virginia, computed the sums of the reciprocals of the twin primes (such as 11 and 13), triplets (such as 11, 13, and 17), and quadruplets (such as 11, 13, 17, and 19) up to a very large upper bound (about 10 trillion). He discovered during the summer and fall of 1994 that one of the reciprocals had been calculated incorrectly by a Pentium computer, although a 486 system gave the correct answer; this led to the publicization of the hardware divide flaw in the Pentium floating point unit.

71. InterMath | Investigations | Number Concept Historical References *perfect numbers *prime numbers *Fermat's last theorem *Babylonianand Egyptian mathematics Mathematicians *Christian Goldbach *Peirre de http://www.intermath-uga.gatech.edu/topics/nmcncept/integers/links.htm

Search the Site Investigations Number Concept Integers External Resources Applications and Connections Protons and antiprotons Harper's Index Factoids Mayan adventure Lessons, Activities, and Units Applications Show 'til you drop Negative times a negative Primes Prime numbers Prime numbers and prime factors Multiples and Divisors Understanding factoring through geometry Understanding algebraic factoring Factoring large numbers Factor game ... Positive and negative dominoes Sums and Differences Integer subtraction Fun with calendars Bean toss Integer addition ... Baseball attendance figures Multiple or Other Operations Multiples, factors, and prime numbers Integer war game Trinumbers Quadrinumbers ... Assorted problems Puzzles Measuring up Literature Bennett Jr., Albert B., and L. Ted Nelson. A Conceptual Model for Solving Percent Problems, Mathematics Teaching in the Middle School. Apr 1994, 20-25 Bezuszka, Stanley J., and Margaret J. Kenney, Honest Numbers: A Mathematics and Languages Connection, Mathematics Teaching in the Middle School. Hanselman, Cheryl A. Stop Using Foul Language in the Mathematics Classroom

72. Perfect Numbers for odd perfect numbers, Math. Comput., 57(1991), 857868. MD Sayers, An improvedlower bound for the total number of prime factors of an odd perfect number, M 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

73. 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

74. Sci.math FAQ: Unsolved Problems Furthermore, the prime occurring to an odd power must itself be congruent to 1 mod4. A sketch It has been shown that there are no odd perfect numbers 10^(300 http://isc.faqs.org/faqs/sci-math-faq/unsolvedproblems/

sci.math FAQ: Unsolved Problems Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76LD.Fnt@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math 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 By Archive-name By Author ... Help Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update March 05 2003 @ 01:20 AM

75. Science News Online, Ivars Peterson's MathTrek (7/4/98): Prime Talent What's the smallest prime the sum of whose digits is perfect? The prime 1999 alsocomes up in another context 1999 How many convenience store numbers are there http://www.sciencenews.org/sn_arc98/7_4_98/mathland.htm

Recently on MathTrek: First Digits 6/27/98 Prime Listening 6/20/98 Coins, Art, and Math in North Bay 6/13/98 July Prime Talent W hole 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. Cubes of Perfection ). What's the smallest prime the sum of whose digits is perfect? The answer is 1999. The prime 1999 also comes up in another context: 1999 = 2 . Pomerance calls any positive integer

76. StudyWorks! Online : Perfect Numbers How many perfect numbers are there? The number, 2 n – 1, if it is prime, is calleda Mersenne prime after Marin Mersenne, a seventeenth century French monk http://www.studyworksonline.com/cda/content/article/0,,EXP1720_NAV2-95_SAR1727,0

Algebra Explorations Astronomy Biology Chemistry ... CONTENTS NEXT >> Perfect Numbers Mersenne Primes and Perfect Pairs If you have StudyWorks, open the attachment below and keep it open in a StudyWorks window to help you with your work. How many perfect numbers are there? The number, 2 n Mersenne prime after Marin Mersenne, a seventeenth century French monk who found the first few of them. So far, there are 39 known Mersenne primes and each one gives a perfect number. In November 2001, it was discovered that 2 There are many things we don't know about perfect numbers. You may have noticed that all the perfect numbers you found using the formula are even Just for fun, add up the divisors of 220 and 284 and see what you find. TELL ME SPECIAL NUMBERS < PREV CONTENTS NEXT >> Related Files Perfect Numbers - Mersenne Primes and Perfect Pairs mersenne.swk (17KB) Credits Loretta Kelley

77. Valentin Vornicu's MathLinks z . Prove that if p 1 , p 2 , , p n are the first n prime numbersthen p­ 1 p 2 p n ±1 cannot be a perfect square. For which http://www.mathlinks.go.ro/nthproblems.htm

Valentin Vornicu's MathLinks EveryOne You are were brought here by a search engine - you should have been redirected to the actual site: www.mathlinks.go.ro Since you are here, it seems that you weren't redirected, so just press the link above. Thanks for visiting my site!

78. Prime Numbers - Mental Arithmetic Under Pressure! prime numbers. Story. The perfect place to play with your friends (the socialites!), to evaluate your knowledge (the anguished!) , to cultivate your mind (the http://www.cosmoquiz.com/en/prime_numbers_game.html

prime numbers Online contest Trivia rules Game requirements prime number - enigma Homepage ... Email Interactive trivia game - 350 questions - Compete for this week's best score! prime numbers Story Other Cosmoquiz games Multiplication mathematical enigmas Others... Un Dos Stress Mental arithmetic under pressure! Click here to play immediate, no download Category Science Difficulty Questions Update Results A good training... Seek the intruder! A clue... The right answer... The perfect place to play with your friends (the socialites!) , to evaluate your knowledge (the anguished!) , to cultivate your mind (the intellectuals!) , to beat the other players (the competitive ones!) , or to prove that you are the most cultured person on the Web (the megalomaniacs!)

79. F. Conjectures (Math 413, Number Theory) A collection of easily stated conjectures which are still open. Each conjecture is stated along with Category Science Math Number Theory Open Problems...... Thm M n is prime implies that 2 n 1 (2 n -1) is perfect. (Euclid); All evenperfect numbers are derived from Mersenne primes in this way. (Euler). http://www.math.umbc.edu/~campbell/Math413Fall98/Conjectures.html

F. Conjectures Number Theory, Math 413, Fall 1998 A collection of easily stated number theory conjectures which are still open. Each conjecture is stated along with a collection of accessible references. The Riemann Hypothesis Fermat Numbers Goldbach's Conjecture Catalan's Conjecture ... The Collatz Problem The Riemann Hypothesis Def: Riemann's Zeta function, Z(s), is defined as the analytic extension of sum n infty n s Thm: Z( s )=prod i infty p i s , where p i is the i th prime. Conj: The only zeros of Z( s ) are at s s Thm: The Riemann Conjecture is equivalent to the conjecture that for some constant c x )-li( x c sqrt( x )ln( x where pi( x ) is the prime counting function. Refs: Unsolved Problems Elementary Number Theory http://www.utm.edu/research/primes/notes/rh.html (notes in Chris Caldwell's Prime Pages) http://match.stanford.edu/rh/ (Dan Bump's notes) Def: n is perfect if it is equal to the sum of its divisors (except itself). Examples are 6=1+2+3, 28, 496, 8128, ... Def: The n th Mersenne Number, M n , is defined by M n n Thm: M n is prime implies that n n is perfect. (Euclid)

80. Unlucky 13 By Shyam Sunder Gupta of many countries is designated as prime MINISTER containing In Japan, the numbers4 and 9 are considered represented as sum of these two perfect squares but http://www.shyamsundergupta.com/unlucky13.htm

UNLUCKY 13 Why 13 is famous as Unlucky? Why people tend to believe on such superstitions? What is the effect of such superstitions? What is so interesting about 13? Why 13 is famous as Unlucky? Alexander wanted to be a God and accordingly made a 13 th statue on the place of his capital. He died and people considered that his will to be a 13 th God (12 gods, one for each month were known) was the reason of his death. After that this number was considered as unlucky. According to historians, there were 13 people at Christ's Last Supper and Christ was crucified on Friday 13 th . So, Friday 13 th is considered as Unlucky. Why people tend to believe on such superstitions? Let us see some reported incidences given below: The Rail Gazette International carried the following report in its April 1979 issue: " As reported in last month, opening of Indian Railway's Hassan-Mangalore line marks completion of a construction task every bit as difficult as the Waltair-Kirandul mineral line which so severely taxed I.R.Engineers a decade ago. In the highly unstable terrain of this part of Karnataka, new solutions had to be found to a number of complex engineering problems. Not the least of the difficulties was tunnel No. 13

Page 4     61-80 of 97    Back | 1  | 2  | 3  | 4  | 5  | Next 20