21. ONJava.com: Web Services And The Search For Really Big Prime Numbers [Aug. 29, 2 Finally, again from the Euler theorem, perfect numbers are related to prime numbersin that an even perfect number can be expressed in the form n = 2 k1 (2 k http://www.onjava.com/pub/a/onjava/2002/08/28/wsdc.html?page=2

22. Prime Page References HS76 M. Hausmann and H. Shapiro, perfect ideals over the gaussian integers MR 263684Huxley72 MN Huxley, The distribution of prime numbers, Oxford University http://www.utm.edu/research/primes/references/refs.cgi/refs.cgi?range=h

23. Prime Factorizations, Perfect Squares And Irrational Numbers prime Factorizations, perfect Squares and Irrational numbers. About this document prime Factorizations, perfect Squares and Irrational numbers http://bolzano.sf.edu/math_120/primefactor/

24. Puzzle 35.- 1999 And The Perfect Numbers be written as P = (1/2)(M)(M+1), where M is a Mersenne prime (2^p hold, and consequently6 != 1 (mod 9). However, this must hold for all other perfect numbers. http://www.primepuzzles.net/puzzles/puzz_035.htm

Puzzles Puzzle 35.- 1999 and the perfect numbers Tom Moore (5/1/99) made me notice that 1999 is the least prime number such that the sum of its digits is a perfect number (28). A perfect numbers is equal to the sum of all of its divisors, excluding itself. Vgr. 6 =1 + 2 + 3. The first 5 perfect numbers are: 6, 28, 496, 8128 and 33550336. Of course it does not exists any prime whose sum of digits is 6 (because it will be divisible by 3, and then a composite). On the other hand it is already known that the least prime whose sum of digits is 28 is 1999. a) Can you find the following 10 primes (apart from 1999) containing only two different kind of digits and whose SOD is 28? b) Can you find the least primes (or pseudo prime) whose SOD is 496, and 8128? c) Last: Any idea why (CBRF observation)? Solution Robert T. McQuaid (18/1/99) has solved a) and b): a) 9199, 338383, 383833, 1181881, 1881181, 1881811, 2222929, 2922229, 8118181, 8188111

25. Prime Queen Attacking Problem largest one for which we have found a perfect solution. the Queen sits on one of thetwo central odd numbers. and if the Queen also attacked the prime 2, then http://users.aol.com/s6sj7gt/primeq.htm

The Prime Queen Attacking Problem Problem by G. L. Honaker, Jr. These notes by Mike Keith This interesting problem was posed by G. L. Honaker, Jr. in November of 1998. First, create any knight's tour on an n x n chessboard, in which the knight starts on any square of the board and by successive knight's moves visits every square on the board exactly once. Number the squares visited by the knight in order starting with 1 for the starting square. When you are done, place a Queen on any square and count the number of prime numbers attacked by the Queen (note that the Queen is not considered to be attacking the square it sits on). Now, the problem: What is the largest number of primes that can be attacked by the Queen, for any placement of the Queen and any knight's tour? First, note that there are 18 primes between 1 and 64. Amazingly, there is a perfect knight's tour in which all 18 primes can be attacked! Here is the first solution ever constructed (by M. Keith, in Nov. 1998): where the location of the Queen is in this color and the attacked primes are shown in red Knights tours are impossible on 1x1, 2x2, 3x3, 4x4 boards, but it is natural to ask the same question for any

26. Perfect, Amicable And Sociable Numbers multiplied by a squarefree number relatively prime to every member of the cycle,so there are an infinite number of exponential perfect numbers, of exponential http://xraysgi.ims.uconn.edu/amicable.html

HTTP 200 Document follows Date: Tue, 18 Mar 2003 10:45:07 GMT Server: NCSA/1.5.2 Last-modified: Sun, 23 Jun 2002 01:44:17 GMT Content-type: text/html Content-length: 29239 Perfect, amicable and sociable numbers (how to have fun summing up divisors) Contents Introduction Perfect numbers Amicable numbers Sociable numbers ... Technical appendix Introduction For a number n , we define s(n) to be the sum of the aliquot parts of n, i.e., the sum of the positive divisors of n, excluding n itself: so, for example, s(8)=1+2+4=7, and s(12)=1+2+3+4+6=16. If we start at some number and apply s repeatedly, we will form a sequence: s(15)=1+3+5=9, s(9)=1+3=4, s(4)=1+2=3, s(3)=1, s(1)=0. If we ever reach 0, we must stop, since all integers divide 0. There are three obvious possibilities for the behavior of this aliquot sequence It can terminate at like the example above. It can fall into an aliquot cycle , of length 1 (a fixed point of s) , or greater It can grow without bound and approach infinity Perfect numbers A perfect number is a cycle of length 1 of s , i.e., a number whose positive divisors (except for itself) sum to itself. For example, 6 is perfect (1+2+3=6), and in fact 6 is the smallest perfect number. The next two perfect numbers are 28 (1+2+4+7+14=28) and 496 (1+2+4+8+16+31+62+124+248=496).

27. Perfect Numbers Harry J. Smith's pages of the mathematics behind perfection, aliquot sequences and Mersenne numbers .Category Science Math Number Theory Factoring perfect numbers......perfect numbers. Select a Subtitle What is a perfect Number? My Talk on AliquotParts Mersenne primes The GREAT Internet Mersenne prime Search GIMPS Plot of http://pw1.netcom.com/~hjsmith/Perfect.html

Perfect Numbers Select a Subtitle: What is a Perfect Number? My Talk on Aliquot Parts Mersenne Primes The GREAT Internet Mersenne Prime Search GIMPS ... Return to Harry's Home Page This page accessed times since June 7, 1997.

28. Resources prime Curios collection of curiosities, wonders and trivia related to prime numbers.The perfect Number Journey (6, 28, ) An integer is perfect if it equals http://michaelshepperd.tripod.com/resources/prime.html

Mathematics Resources by Topic - Number Prime numbers Home Resources by Topic Calculators and Plotters Reference ... Eratosthenes' Sieve 1 Remove multiples of selected numbers from a 1-400 number square. Eratosthenes' Sieve 2 Remove multiples of selected numbers. Select a maximum number up to 200. lucky numbers - Ivars Peterson A different type of sieve gives the lucky numbers which share many properties with prime numbers. prime number calculators successive primes starting from of digits Why is 1 not considered prime? Formulas for Primes - Cut The Knot These formulas work until ... Prime Curios collection of curiosities, wonders and trivia related to prime numbers The Perfect Number Journey (6, 28, ...) An integer is perfect if it equals the sum of its divisors (excluding itself). Read about the connection between perfect numbers and Mersenne primes. Goldbach Calculator - WWW Interactive Server The famous Goldbach conjecture states that any even number larger than 2 is the sum of two prime numbers. Enter an even number into the calculator and receive a list of such sums. history of prime numbers - MacTutor History Archive GIMPS Home Page The Great Internet Mersenne Prime Search Pulchritudinous Primes - Adrian Leatherland visualising the distribution of prime numbers prime numbers - collection of links - Jill Britton

29. Number Theory How are the prime numbers distributed? What is the largest known prime number? Whatare perfect, abundant, deficient, Mersenne, Fermat, and sublime numbers? http://mail.colonial.net/~abeckwith/numthry.html

30. Glossary-P For example, 6 is a perfect number because the sum of its proper factors is 1 + 2+ 3 = 6. See also abundant The first five prime numbers are 2, 3, 5, 7, and 11 http://www.kent.k12.wa.us/curriculum/math/edmath/glossary/glossary_P.html

E veryday M athematics G lossary P parallels Lines of latitude. parallel lines (segments, rays) Lines (segments, rays) that are the same distance apart and never meet. parallelogram A quadrilateral that has two pairs of parallel sides. Pairs of opposite sides of a parallelogram are congruent. pattern A set of shapes or numbers that are repeated in a predictable way. Patterns may often be described by a rule, using words or variables. pentagon A polygon with five sides. per capita Per person. percent (%) Per hundred, or out of a hundred. For example, "48% of the students in the school are boys" means that out of every 100 students in the school, 48 are boys. Percent Circle A device on the Geometry Template used to measure or draw figures (such as circle graphs) involving percents. perfect number A number for which the sum of all the proper factors is equal to the number itself. For example, 6 is a perfect number because the sum of its proper factors is 1 + 2 + 3 = 6. See also abundant numbers deficient numbers perimeter The distance around a two-dimensional shape. A formula for the perimeter of a rectangle is P = 2 * (l+ w), where l represents the length and w the width of the rectangle.

31. Math Trek : A Perfect Collaboration, Science News Online, Jan. 18, 2003 and these numbers figure prominently in the search for the largest known prime.Every Mersenne number uncovered automatically leads to a new perfect number. http://www.sciencenews.org/20030118/mathtrek.asp

Home page. Math Trek A Perfect Collaboration Food for Thought C-Minus—The Fallout of Parents' Smoking Science Safari World Development News TimeLine 70 Years Ago in Science News Week of Jan. 18, 2003; Vol. 163, No. 3 A Perfect Collaboration Ivars Peterson It seems an unlikely pairing. One was the most prominent mathematician of antiquity, best known for his treatise on geometry, the Elements . The other was the most prolific mathematician in history, the man whom his eighteenth-century contemporaries called "analysis incarnate." Together, Euclid of Alexandria ( c c BC ) and Leonard Euler (1707–1783), born in Switzerland and at various times resident in St. Petersburg and Berlin, collaborated on proving an interesting result in number theory—without the benefit of e-mail or time travel. Mathematician William Dunham describes this remarkable effort, which spanned nearly 20 centuries, in his book Euler: The Master of Us All The story begins with the fascination that numbers held for the followers of Pythagoras in ancient Greece. Among those of special interest were the perfect numbers, which have the property that their proper divisors add up to the number itself. For example, the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. Six is the smallest perfect number. The next highest is 28. Its divisors are 1, 2, 4, 7, and 14, so 1 + 2 + 4 + 7 + 14 = 28. Euclid also knew the next two perfect numbers: 496 and 8,128. Notice that each of the four numbers can be written as the following products: 2 x 3, 4 x 7, 16 x 31, and 64 x 127.

32. StudyWorks! Online : Perfect Numbers If you look closely at the method we used for computing perfect numbers, you willnotice that So if n is an integer and 2 n 1 is prime then is a perfect number http://www.studyworksonline.com/cda/content/article/0,,EXP1720_NAV2-95_SAR1724,0

Algebra Explorations Astronomy Biology Chemistry ... NEXT >> Perfect Numbers Finding an nth Term Formula If you have StudyWorks, open the attachment below and keep it open in a StudyWorks window to help you with your work. There is a formula for perfect numbers. If you look closely at the method we used for computing perfect numbers, you will notice that the sum of the powers of two is equal to the next power of two minus 1. For example: The next power of two would be 8. Notice: So: n equals the sum of the first nth powers of 2. This gives us our formula, but it only works if the sum is prime. So if n is an integer and 2 n - 1 is prime then: P(n) = 2 n is a perfect number. The number, 2 TELL ME SPECIAL NUMBERS < PREV CONTENTS ... NEXT >> Related Files Perfect Numbers - Finding the nth Term Formula perfect numbers.swk (23KB) Credits Loretta Kelley

33. Mathlinks.info - Numbers And Constants The perfect Number Journey Tutorial (O Heng); perfect numbers - Article (MacTutor theFirst 1000 Integers - Pages (SOS Math); prime numbers - Article (MacTutor http://www.mathlinks.info/em026_numbers_and_constants.htm

About the Numbers in Today's Date Pages (R Phillips - University of Nottingham) Aesthetics of the Prime Sequence Pages (T Armand) Arabic Numerals Article (MacTutor History of Mathematics Archive - University of St. Andrews) Arabic Numerals Tutorial (E Erhayiem) Babylonian Numerals Article (MacTutor History of Mathematics Archive - University of St. Andrews) Chronology of Pi Article (MacTutor History of Mathematics Archive - University of St. Andrews) Complex Numbers Tutorial (D Joyce, Clark University) Contents of 'Numbers' Tutorials (C Blomqvist) Counting to Infinity Tutorial (Bellevue Community College) Do "Imaginary Numbers" Really Exist? Tutorial (Mathematics Network - University of Toronto) Does "Infinity" Exist? Tutorial (Mathematics Network - University of Toronto) Egyptian Fractions Pages (D Eppstein - University of California at Irvine) Egyptian Numerals Article (MacTutor History of Mathematics Archive - University of St. Andrews) The Fibonacci Numbers Pages (D Schweizer - College of Holy Cross) Fibonacci Numbers, the Golden Section and the Golden String Pages (R knott - Surrey University) Fibonacci Numbers Spelled Out Tutorial (I Galkin - University of Massachusetts) Fundamental Physical Constants Pages (National Institute of Standards and Technology) Fun With Numbers Pages (S Weil) Greek Numbers Article (MacTutor History of Mathematics Archive - University of St. Andrews)

34. MGMSI | Instructor Manual | Course Outlines | Number Sense special classes of prime and composite numbers (ie, Mersenne primes, abundantnumbers, deficient numbers, perfect numbers, and relatively prime numbers). http://www.mgmsi.usg.edu/instruct/manual/courses/num_sense/homepg.htm

Home Page Intro Course Catalog Enrollment ... Contact Us Number Sense 1. PROGRAM TITLE. Number Sense 2. GOALS ADDRESSED Participants will: Explore concepts in the number sense QCC objectives Discover ways to teach the objectives so that all students can achieve at a high level Investigate support materials via Internet that will enhance their classroom presentation and student achievement Plan units and lessons that will enable students to work up to their learning potential 3. IMPROVEMENT PRACTICES Participants will gain knowledge and practice in developing excellent models for teaching number sense in the middle school classroom in order to positively affect student achievement. The course will specifically address deficiencies of certificated personnel as identified by evaluations. 4. COMPETENCIES Participants will be able to: Express an understanding of the various strategies proven to be effective in teaching number sense. Select suitable supplementary materials for developing number sense units and lessons in the middle school classroom. Demonstrate appropriate knowledge and techniques for teaching number sense in the middle school classroom.

35. Www.math.niu.edu/~rusin/known-math/98/perfect there are fairly efficient ways to determine whether a number 2^p 1 is prime,so very large primen numbers, and thus very large perfect numbers, have been http://www.math.niu.edu/~rusin/known-math/98/perfect

36. Department Of Mathematics And Statistics - News and Events of the subject of distribution of primes, and then I'll discuss a new connectionof certain prime forms (especially quadratic primes) and odd perfect numbers. http://www.math.ucalgary.ca/events/index.php3?newstypeid=12&Which=380

37. Number Theory And Arithmetic In The Pentagon numbers, Robert W. Prielipp 37, 1, 1618, The Relationship of Pascal's Triangleand perfect numbers, Robert A. Antol 39, 2, 94-98, On prime Powers Which http://www.kme.eku.edu/numthe.htm

38. Welcome To The Sanford Circles including counting from 1 to 100 in 1's, 2's, 3's and 5's, the large colorful (2'x3')poster contains the Times Tables, prime numbers, perfect Squares, perfect http://www.ic-2000.com/sanford/

Welcome to 101 Church Street Windsor, NJ 08561-0010 24hr voice mail bsanford@quixnet.net http://www.ic-2000.com/sanford Every Classroom... Every Home A Math Reference Poster For All Ages A Step Beyond Manipulatives! Hands-On Development of Eye-Hand Coordination; Spatial Relationships; and Number/Symbol/Color Recognition. Taking nothing for granted and beginning simply with number and color fun, including counting from 1 to 100 in 1's, 2's, 3's and 5's, the large colorful (2'x3') poster contains the Times Tables, Prime Numbers, Perfect Squares, Perfect Cubes, all the Factors of the numbers from 1 - 100, The Powers of "10" up to One Trillion (10 ), Formulas, Vocabulary, Color Codes, Symbols, etc., etc... Also Available Is "THE SANFORD CIRCLES" of Numbers Patented Math BOARD GAME For All Ages! entitled M U L T I P L E   C H O I C E S by Bruce R. Sanford The Game Board is a laminated version of the large poster! The object of the game is to begin at ENTER "0", move clockwise around all eight circles and Win by being the first player to enter the ninth circle (The Inner Circle), valued at ONE TRILLION - 1,000,000,000,000 - 10

39. PBS TeacherSource - Mathline - Middle School Math Project Objective. Students will use a game setting to identify the properties of prime,composite, abundant, deficient and perfect numbers. Overview of the Lesson. http://www.pbs.org/teachersource/mathline/lessonplans/msmp/factor/factor_procedu

March 18, 2003 Factor Game (Number Theory) Middle School Math Project Download PDF Lesson Get Acrobat A Printable version of this lesson is available in PDF format. This requires a free plug-in, Adobe Acrobat Reader. You can find out if you need it by clicking on the PDF link. Procedure Resources Mathematically Speaking Ideas for Discussion ... More Lessons: Grades 6-8 Objective Students will use a game setting to identify the properties of prime, composite, abundant, deficient and perfect numbers. Overview of the Lesson Materials Transparency: Factor Game Board (or a copy of game board drawn on the board) Worksheet: Factor Game Board Worksheet: Analysis of First Moves Note: Transparency and Worksheets are located at the bottom of the .pdf file. Colored pencils Calculators Procedure Project a transparency of a Factor Game Board on the screen. (If an overhead projector is not available, draw a 5 by 6 grid containing the numbers from 1 to 30 on the board.). Use two different colors or two different symbols to distinguish the moves made by the two opposing teams. In the video, the teacher used circles and squares. Create a friendly opening for your students by informing them that they will be playing a game. The object of the game is to obtain more points than your opponent. In the first game, the teacher challenges the entire class. Inform the students that the rules of the game will be revealed as needed while the game is in play.

40. Math In The Media 1002 The proof is worth $10,000 (put up by fellow perfectgraph aficionado George Johnsongives us a meditation on prime numbers and their distribution ( From Here http://www.ams.org/new-in-math/10-2002-media.html

Math in the Media Math in the Media Archive October 2002 Perfect Graphs Algorithmic Architecture Home on the Fractal Range Ants vote for Wolfram ... Times Perfect Graphs and the "Strong Perfect Graph Conjecture" are the topic of a News Focus piece by Dana Mackenzie in the July 5 2002 Science. As Mackenzie explains it the definition involves two invariants of a graph. The first, omega, is the size of the biggest clique (set of nodes each of which is one step away from all the others). The second, chi, is the number of colors it takes to color the nodes so that no two adjacent nodes are the same color. The two essential imperfections: an odd hole and an odd anti-hole. So chi is always bigger than omega; if the numbers are equal, the graph is perfect. Mackenzie: "A perfect graph is like a perfect chocolate cake: It might be easy to describe, but it's hard to produce a recipe." A conjecture due to Claude Berge (CNRS, Paris) has been around since 1960: every imperfect graph contains either an "odd hole" or an "odd anti-hole." This is the Strong Perfect Graph Conjecture (SPGC). The odd hole is "a ring of an odd number (at least 5) of nodes, each linked to its two neighbors but not to any other node in the ring." The odd anti-hole is "the reverse: Each node is connected to every other node in the ring except its neighbors." The news is that a proof of the SPGC has been announced by Paul Seymour (Princeton), G. Neil Robertson (OSU) and Robin Thomas (Georgia Tech). The proof is worth $10,000 (put up by fellow "perfect-graph aficionado" Gerard Cornuejols) and "the early betting is that they will collect the prize."

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