81. LESSON PLANET - 30,000 Lessons And 2896 Lesson Plans For Prime Numbers 21. How to determine whether a number is prime or composite Math Lesson Plan Here sa lesson on prime vs. Composite numbers Grades 6-8. 22. perfect numbers? http://www.lessonplanet.com/search/Math/Prime_Numbers?startval=20

82. The Prime Glossary: Perfect Number It turns out that for 2 k 1 to be prime, k must also be primeso the searchfor perfect numbers is the same as the search for Mersenne primes. http://primes.utm.edu/glossary/page.php?sort=PerfectNumber

83. Prime Numbers These prime numbers form the foundation for the rest of the Bible. Many large numbersdivide by three (ie, 96/3=32 is thus proven to be the perfect language for http://biblecoderesearch.com/Prime_Numbers(1).htm

Center for Bible Code Research Your Exclusive Site for Comprehensive Bible Code Research. Prime Numbers Inside Section [ Prime Numbers ] Compound Numbers Exponent Numbers This section deals with the Prime Numbers of the Bible. These prime numbers form the foundation for the rest of the Bible. 1. ONE - Unity. The word "JAH" (3 letters) is found ONLY one time in all capital letters. Thus with 3 letters (Divinity) and one word (unity), it reveals the Trinity as being in "Divine Unity", although in three Persons. That is what the scripture says. Godhead - One (single) word, yet plural (3 IN 1) in meaning. ONLY the AV/KJV records it THREE times! The NEW KJV changes it once to 'Divine Nature' (NOT PLURAL-and WRONG!) God - One (single) word (with 3 letters-see item 3 below), yet plural (3 in 1) both theologically and numerically. "Man" is also 3 in one, body, soul, spirit. "Bishop" - with a CAPITAL B, and in the verse context referring to Jesus, is found only ONE time in KJV. Thus, numerically, Jesus, is God (1 Tim 3:16) and is the one and only true, and divine "Bishop". The Lord appeared to Abraham "in the plains of Mamre". "..lo, three men stood by him" (Genesis 18:1-2). The Godhead/God is three "Persons"!

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

85. : Perfect Number/prime Number Finder, Aaron At 4/20/2004 04:24 perfect number/prime number finder. Written by Aaron at 20 Apr 2004 042432 AnswersRe perfect number/prime number finder Bruno Schäfer 4/21/2004 1306 (2) http://f27.parsimony.net/forum67475/messages/139.htm

perfect number/prime number finder Written by Aaron at 20 Apr 2004 04:24:32: I wrote a few cool programs: prime number finder: rem a=possible prime # rem b=count up # rem c=# of divisors a=1 label zz a=a+2 label aa b=1 label ab c=0 label ac d=int(a/b) if a/b=d then c=c+1 if c=2 then goto zz endif endif if b>=sqrt(a) then print a, goto zz endif b=b+2 goto ac I wrote a program that finds perfect numbers: b=1 c=0 n=1 label aa n=n+1 a=(2^n)-1 label bb d=int(a/b) if a/b=d then c=c+1 if c=2 then c=0 b=1 goto aa endif endif if b>=sqrt(a) then print 2^(n-1)*((2^n)-1), b=1 c=0 goto aa endif b=b+1 goto bb do you have any idea of how to get rid if the small "E" after the first few numbers n the perfect number program?, and what does it mean? thanks Answers: Re: perfect number/prime number finder Bruno Schäfer Re: perfect number/prime number finder Aaron Re: perfect number/prime number finder Mike Huffman

86. : Re: Perfect Number/prime Number Finder, Aaron At 4/21/2004 16:56 Re perfect number/prime number finder. As an answer to Re perfect number/primenumber finder written by Bruno Schäfer at 21 Apr 2004 130614 http://f27.parsimony.net/forum67475/messages/141.htm

Re: perfect number/prime number finder Written by Aaron at 21 Apr 2004 16:56:16: As an answer to: Re: perfect number/prime number finder written by Bruno Schäfer at 21 Apr 2004 13:06:14: >>do you have any idea of how to get rid if the small "E" after the first few numbers n the perfect number program?, and what does it mean? >>thanks >The small "E" belongs to the number and is used in big numbers. It is exponential number to the base of ten. >Some examples: >Please have a look in a mathematical book. >Good Luck, >Bruno Thanks, do you have any idea of how to get rid of it? is it possable? Answers: Re: perfect number/prime number finder Mike Huffman

87. BletchleyPark.net Mersenne primes are related to the perfect number concept. Euclid demonstratedthat if MP is a Mersenne prime then MP(MP + 1) / 2 is a perfect number. http://www.bletchleypark.net/computation/primenumbers.html

Prime Numbers. Introduction The second classification of the natural numbers, after the classification of even and odd, is prime numbers. The Fundamental Theorem of Arithmetic, sometimes called the Unique Factorization Theorem as described below in more detail, states that any and every integer is either uniquely a product of prime numbers or is a prime number. One of the beauties of number theory is that, since prime numbers can play the role of generator, every integer has a unique signature, the collection of its prime divisors. Trial Division Prime Number Theorem The Prime Number Theorem is an approximation that is reasonably accurate, which can be used to estimate with the probability of 1/ln(N) that a randomly chosen integer N is prime. The Prime Number Theorem states that the primes near a prime are spaced on the average one every (ln(N))/2 integers. The 1/2 considers that even numbers don't count. For instance, a 512-bit prime number would require ln(2 )/2 or roughly 178 512-bit random numbers for primality testing.

88. 37th Mersenne Prime Discovered There is a wellknown formula that generates a perfect number from a Mersenneprime. A perfect number is one whose factors add up to the number itself. http://www.mersenne.org/3021377.htm

GIMPS Discovers 37th Known Mersenne Prime, 2^3021377-1 is now the Largest Known Prime. ORLANDO, Fla., February 2, 1998 Roland Clarkson has discovered the world's largest known prime number using a program written by George Woltman and networking software written by Scott Kurowski. The prime number, 2^3021377-1, is one of a special class of prime numbers called Mersenne primes. This is only the 37th known Mersenne prime. Roland Clarkson , a 19 year-old student at California State University Dominguez Hills, is from Norwalk, California. George Woltman is a retired programmer living in Orlando, Florida. Scott Kurowski is a software development manager and entrepreneur living in San Jose, California. The new prime number, discovered on January 27th, is 909,526 digits long! Roland used a 200 MHz Pentium computer part-time for 46 days to prove the number prime. Running uninterrupted it would take about a week to test the primality of this prime number. Clarkson is one of over 4000 volunteers world-wide participating in the Great Internet Mersenne Prime Search (GIMPS). This prime number is the third record prime found by the GIMPS project. Gordon Spence discovered the previous largest known prime number last August. Joel Armengaud discovered the

89. VACETS Technical Column - Tc48 Note that, with the discovery of the new prime number, a new perfect number canalso be generated. A perfect number is equal to the sum of its factors. 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.

90. Prime Queen Attacking Problem n x n board, with n 5. Denote by Q(n) the maximum number of primes that can Thoseshown in green are perfect solutions, with all primes in the grid being http://users.aol.com/s6sj7gt/primeq.htm

The Prime Queen Attacking Problem Problem by G. L. Honaker, Jr. These notes by Mike Keith, last updated May 2004 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 perfect solution ever constructed (by M. Keith, in Nov. 1998): where the location of the Queen is in blue 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

91. Maths Glossary Numerator The top number of a fraction,Numerator/ Denominator perfect Number A number that is the sum of its factors.EG 6=1+2+3 prime Number A number http://www.fortunecity.com/emachines/e11/86/mathglos.html

web hosting domain names email addresses Maths Glossary Algorithm : A fixed process which if caried out systematically produces a desired result.Thus the "Euclidean algorithm" is a set of rules which when applied to two integers produces their common divisor. Complex Number : A 2 dimensional number comprising a real and imaginary component of the form A+Bi,where i is the square root of -1.Such numbers are mapped on the Argand plane and form a matrix of numbers,the imaginary numbers being at right angles to the real ones.The conjugate takes the form a-bi. Continuum Hypothesis : In Cantorian set theory,the cardinal number of a set designates its "manyness". The cardinality of the set of integers 1,2,3,... is designated by .The cardinality of the set of real numbers is 2 .The continuum hyopothesis asserts that there is no set whose cardinal falls between and 2 Cube : 1. A number raised to the power 3,ie 2 x 2 x 2 = 2 2. A 3 dimensional figure with the height,width and depth having the same lengths,and all at right-angles to each other.

92. Simon Gregory's Prime Numbers Page Fermat s method of attempting to express a number as the difference of two perfectsquares because any such number cannot be prime (if X = A * A B * B, it http://www.domino.finesystem.co.uk/A556A4/HOME.NSF/Documents/SGPrimes

Prime Numbers Prime numbers have long been an interest of mine. The main reason is that with large numbers it takes a very long time to reliably decide whether that number is prime or not. This gives rise to more creative methods being devised. Interest in this area is heavily funded from the cryptography because large prime numbers play a key role in the development of encryption (secure, coded) systems. Here is a simple Java applet which allows you to type in a number and find out whether or not it is prime: Prime Number Test [ Sorry, you cannot see the applet because your browser does not support Java applets ] A prime number is one which cannot be expressed as a product of two whole numbers other than itself and 1. For example, 6 equals 2 times 3, so 6 is not prime. 7 is a prime number because only 1 and 7 can be multiplied together to give 7. The applet above simply checks all whole numbers from 2 up to the square root of the number to see if any of them divide evenly into the number being tested. Both the interface and the core algorithm could do with some work. Some of the imaginative methods people have used to see if a number is prime or not include: just check whole numbers from 2 up to the square root of the number you are testing - if it has any factors, one of them must be less than its square root

93. Prime Number These tests are not perfect. For a given test, there may be some composite numbersthat will be declared probably prime no matter what witness is chosen. http://www.fact-index.com/p/pr/prime_number.html

Main Page See live article Alphabetical index Prime number In mathematics , a prime number , or prime for short, is a natural number larger than that has as its only positive divisors (factors) 1 and itself. The first 25 prime numbers are This definition is used throughout the Wikipedia. See the Generalizations section, below, for another definition in common use. The property of being a prime is called primality . If a number greater than one is not a prime number, it is called a composite number Table of contents 1 Representing natural numbers as products of primes 2 How many prime numbers are there? 3 Finding prime numbers 4 Some properties of primes ... 16 Books Representing natural numbers as products of primes An important fact is the fundamental theorem of arithmetic , which states that every natural number can be written as a product of primes, and in essentially only one way. Primes are thus the "basic building blocks" of the natural numbers. For example, we can write See Prime factorization algorithm for details. How many prime numbers are there?

94. Mersenne Numbers - Encyclopedia Article About Mersenne Numbers. Free Access, No Mersenne primes have a close connection to perfect numbers In mathematics, a perfectnumber is an integer which is the sum of its proper positive divisors, not http://encyclopedia.thefreedictionary.com/Mersenne numbers

Dictionaries: General Computing Medical Legal Encyclopedia Mersenne numbers Word: Word Starts with Ends with Definition A Mersenne prime is a prime number In mathematics, a prime number , or prime for short, is a natural number greater than 1 whose only positive divisors are 1 and itself. The sequence of prime numbers (sequence A000040 in OEIS) begins See list of prime numbers for the first 500 primes. Click the link for more information. that is one less than a power of two In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. Note that one is a power (the zeroth power) of two. Written in binary, a power of two always has the form 10000...0, just like a power of ten in the decimal system. Because two is the base of the binary system, powers of two are important to computer science. Specifically, two to the power of Click the link for more information. More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than an odd power of two; hence, M n n Mersenne primes have a close connection to perfect numbers In mathematics, a

95. AMCA: Prime Gaps Modulo A Perfect Number By Rahul Athale prime Gaps Modulo a perfect Number by Rahul Athale Research Institutefor Symbolic Computation (RISC), Hagenberg, Austria. The difference http://at.yorku.ca/cgi-bin/amca/cakl-24

Atlas Mathematical Conference Abstracts Conferences Abstracts Organizers ... About AMCA Journées Arithmétiques XXIII July 612, 2003 University of Graz and University of Technology of Graz Graz, Styria, Austria Organizers S. Frisch, A. Geroldinger, P. Grabner, F. Halter-Koch, C. Heuberger, G. Lettl, R. Tichy View Abstracts Conference Homepage Prime Gaps Modulo a Perfect Number by Rahul Athale Research Institute for Symbolic Computation (RISC), Hagenberg, Austria The difference between any two consecutive prime numbers is called a prime gap. We consider the distribution of prime gaps modulo six, which is a perfect number with respect to the usual definition (A natural number is called a perfect number if the sum of all its divisors, excluding itself, is equal to the number.). We call six a perfect number due to the property of the resulting distribution of prime gaps modulo six: The number of prime gaps congruent to zero modulo six is approximately same as the number of prime gaps not congruent to zero modulo six. This also substantiates the claim made in a recent Science Update on the Nature web site; statistically the difference between consecutive prime gaps is rarely a multiple of six. We also give the estimate of the distribution of prime gaps modulo six using Hardy-Littlewood k-tuple conjecture.

96. Stephen Wolfram: A New Kind Of Science | Online currently known). It was shown by Euclid in 300 BC that 2^ n1 (2^ n-1) is a perfect number whenever 2^ n -1 is prime. Leonhard Euler http://www.wolframscience.com/nksonline/page-911d-text

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97. Large Numbers -- Notes At MROB M 13 = 8191, etc. For each Mersenne prime there is also a perfectnumber P p given by P p = 2 p1 (2 p -1). Here is a (reasonably http://home.earthlink.net/~mrob/pub/math/ln-notes1.html

This page goes into greater detail on the background of some of the large numbers and functions described on my large numbers page . The topics are presented in the same order as on that page. Contents of this page: Notes on Fermat numbers Perfect numbers Mersenne Numbers Mersenne Primes ... A Cantor-style Proof That the Set of Countable Ordinals is Uncountable Notes on Fermat numbers The Fermat Numbers, Sloane's , are all numbers of the form 2 N Their factorizations are: All of the factors of all of the Fermat numbers, arranged in ascending order, make Sloane's sequence . There is a project, similar to the Mersenne prime search, to find all terms in this sequence. Goldbach's Proof That There are an Infinite Number of Primes There are many proofs of the "infinitude" of primes. Eleven are listed in the book by Paulo Ribenboim, The New Book of Prime Number Records, 3rd edition (Springer-Verlag, New York, 1995, ISBN 0-387-94457-5). This is Goldbach's proof, which he gave in a letter written to Euler in 1730. I have expanded and rewritten it from this page (in the past this page has been sometimes unavailable).

98. Mathematics Of Computation 20. N. Robbins, The nonexistence of odd perfect numbers with less than seven distinctprime factors, Ph.D. Thesis, Polytechnic Institute of Brooklyn (1972). http://www.ams.org/mcom/1999-68-228/S0025-5718-99-01126-6/home.html

Journals Home Search Author Info Subscribe ... Help ISSN 1088-6842 (e) ISSN 0025-5718 (p) Previous issue Table of contents Next issue Articles in press ... All issues The second largest prime divisor of an odd perfect number exceeds ten thousand Author(s): Douglas E. Iannucci. Journal: Math. Comp. MSC (1991): Primary 11A25, 11Y70 Posted: May 17, 1999 Retrieve article in: PDF DVI TeX PostScript ... Additional information Abstract: Let denote the sum of positive divisors of the natural number . Such a number is said to be perfect if . It is well known that a number is even and perfect if and only if it has the form where is prime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions necessary for their existence. One class of necessary conditions would be lower bounds for the distinct prime divisors of an odd perfect number. For example, Cohen and Hagis have shown that the largest prime divisor of an odd perfect number must exceed

99. Halfbakery: NOT A Prime Number But saying that you can demonstrate the existence of an infinite series of nonprimenumbers isn t exactly revolutionary Those are the prime numbers. http://www.halfbakery.com/idea/NOT_20a_20prime_20number

h a l f b a k e r y Bite me. idea: new search annotate link , view, overview recent by name best ... random meta: news help about links ... report a problem account: Browse anonymously, or get an account and write. User: Pass: Login Create an account. 4 dimensional Venn Diagram Alphabet base Base-60 Alphanumerics count in gray code ... mathematics NOT a prime number interesting graph [vote for against Over the last few years I have been interested to see if there was a pattern in the prime number series. I have found a pattern in the numbers which are NOT prime, and would like to explain about this. Perhaps it will give someone another idea. Draw a graph with two axis. The vertical axis is 1,2,3,4..up to, for a start, 121. The horizontal axis is 1,2,3,4.. up to, say, 11. For position 1 on the horizontal axis, count up the vertical axis one at a time, and mark the position. For position 2, count vertically two at a time, and mark 2,4,6,8 etc. For position 3, count vertically three at a time, and mark 3,6,9,12 etc. Continue for position 4 up to 11. Now find a square on the vertical axis, which is ODD. For example 7 squared = 49.

100. Guinness Book Of World Records,Guiness World Records,Limca Indian Records, Limca 3. The highest known perfect number, is the 32nd so far discovered, is 2 756839 –1x 2 756838 . It is derived from the largest known Mersenne prime also the http://www.4to40.com/recordbook/default.asp?category=land&counter=30

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