41. Integer Bars: More About Multiplication Exercise Find all of the numbers that are perfect squares between 1 and 100 . Composites - A composite is the product of two or more prime numbers. http://www.arcytech.org/java/integers/multiplication2.html

More About Multiplication Perfect Squares - Mathematically, a perfect square is when you multiply a two numbers that are the same. The result of the multiplication is the perfect square. For example, 4 x 4 = 16 Therefore, 16 is a perfect square. To use the integer bars to find a perfect square, you can follow the methods described in Activity 1 and you will end up with an image that is a perfect square. Here is an example of using the integer bars to show the square of the number 4: The perfect square on the left is made of 4 bars of size 4. The one on the right is made by overlapping two size 4 bars then filling out the empty space to complete the perfect square. In bothe cases the answer is 16. Exercise - Find all of the numbers that are perfect squares between 1 and 100. You can use the Multiplication Table that you filled out in the exercise above. When you are done, check your answers Prime Numbers and Composites Prime Numbers - A prime number is a whole number that only has two factors which are itself and one. For example the number 7 has only two factors which are 1 x 7 = 7. There are no other factors that you can multiply to get the answer 7. Another example would be the number 13. The only two factors that you can multiply to get 13 are 1 x 13 = 13. Here is a list of all the prime numbers between 2 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97

42. Halfbakery: Prime Melodies prime numbers are those divisible only by themselves and one and are regarded as prime interval (perfect fifth) mentioned. bristolz, Oct 04 2004, http://www.halfbakery.com/idea/Prime_20Melodies

h a l f b a k e r y Keep out of reach of children. 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 account. Hark at the Moon Lip-Syncing for Change Melodies of Movement ... music: source Prime Melodies Mathematically perfect music [vote for against Prime numbers are those divisible only by themselves and one and are regarded as beautiful by pure mathematicians. There are no discovered patterns to the dispersal of prime numbers within number sets and indeed, for all our advances, there are no formulas to predict the next prime number. Within an octave of 8 notes, 5 of these notes are prime. Within the key of C, the notes C, D, E, G, A and B [notes 1, 2, 3, 5, 7 within the octave] are prime. I propose musical pieces composed only using prime notes : Prime Melodies. The chords and melodies available will have a mathematical beauty about them. Extra emphasis on chords such as CMaj7th should be used as it employs the 1, 3, 5, 7 of the key. It's a prime chord. jonthegeologist , Oct 30 2003 The Circle of Fifths http://www.mikemurp...essons/lesson18.htm

43. Large Numbers -- Notes At MROB For more perfect numbers, check the list of Mersenne primes below and use theformula For each Mersenne prime there is also a perfect number Pp given by 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).

44. [Prime] [Fwd: Prime Post From Sghodeif@hotmail.com Requires Approval] (2) “Between Fermat, Mersenne and the perfect numbers” Abstract; as well asFermat’s prime numbers F and the even perfect numbers E. We find any http://hogranch.com/pipermail/prime/2005-July/000825.html

[Prime] [Fwd: Prime post from sghodeif@hotmail.com requires approval] John R Pierce pierce at hogranch.com Thu Jul 28 18:19:08 PDT 2005 Previous message: [Prime] cpu affinity... Next message: [Prime] [Fwd: Prime post from sghodeif@hotmail.com requires approval] Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ah, the stuff that is sent blindly to this list. This bounced because the sender is not a subscriber, but I thought some of you may find it amusing. For that matter, someone may want to take him seriously, and reply. As he's not a subscriber, you'd have to email him directly. If any significant correspondence comes out of this, it might be interesting to post a summary to the list. Original Message Subject: primes news From: "Salim Ghodeif" < sghodeif at hotmail.com prime at hogranch.com Previous message: [Prime] cpu affinity... Next message: [Prime] [Fwd: Prime post from sghodeif@hotmail.com requires approval] Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... More information about the Prime mailing list

45. Math Forum: Ask Dr. Math FAQ: Perfect Numbers How can you calculate a perfect number from a Mersenne prime? After readingup on how known perfect numbers relate to Mersenne prime numbers, http://mathforum.org/dr.math/faq/faq.perfect.html

Ask Dr. Math: FAQ P erfect N umbers Dr. Math FAQ Classic Problems Formulas Search Dr. Math ... Dr. Math Home What is a perfect number? How do you find them? How many are there? A contribution from John Knoderer, the MazeMan: What is a perfect number? How many perfect numbers are there? How many perfect numbers are known? Are there any odd perfect numbers? ... Your computer can join the search! What is a perfect number? A perfect number is a whole number, an integer greater than zero; and when you add up all of the factors less than that number, you get that number. Examples: The factors of 6 are 1, 2, 3 and 6. The factors of 28 are 1, 2, 4, 7, 14 and 28. The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496. The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128. I'll let you add them up. According to The Merriam-Webster Dictionary, the term was first used in the fourteenth century. The Grolier Multimedia Encyclopedia says that perfect numbers are "another example of Greek progress in number theory," and credits the Pythagoreans for coining the term "perfect." If you are interested in learning more about "perfect" numbers, you should also read up about "Mersenne" prime numbers because they are closely related. The first four perfect numbers were known over 2,000 years ago. Some ancient cultures gave mystic interpretations to numbers that they thought were magic.

46. Math Forum - Ask Dr. Math What is the next perfect number after 28? The number 1 is not regarded as aprime number. So, the prime numbers begin with 2, 3, 5, 7, 11, http://mathforum.org/library/drmath/view/57044.html

Associated Topics Dr. Math Home Search Dr. Math Perfect Numbers - Basics, History Date: 11/3/96 at 9:28:54 From: Anonymous Subject: math question What is the next perfect number after 28? http://mathforum.org/dr.math/ Date: 11/04/96 at 19:44:54 From: Anonymous Subject: Re: math question Thank you for answering my question about the next perfect number after 28. I would be interested to learn more about how you could find a formula that finds perfect numbers. I am only in 6th grade so I cannot understand mathematical words, but that doesn't mean that I am a numbskull. Sincerely, DZINE http://www.utm.edu/research/primes/mersenne.shtml -Doctor Yiu, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Associated Topics Elementary Large Numbers Elementary Number Sense/About Numbers Elementary Prime Numbers Middle School Number Sense/About Numbers ... Middle School Prime Numbers Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age.

47. Prime Magic Squares All numbers are prime. Orders 4 5 perfect prime Squares Order3 Super-perfectprime Square prime Number Patterns http://www.geocities.com/~harveyh/primesqr.htm

P rime Magic Squares CONTENTS A Large order-3 This magic square consists of 9 consecutive, 93-digit prime numbers. Minimum consecutive primes -3 This order-3 uses consecutive primes not in arithmetic progression. Minimum consecutive primes -4 This order-4 has a magic sum of 258 Minimum consecutive primes -5 This order-5 has a magic sum of 1703 Minimum consecutive primes -6 An order-6 pandiagonal magic square with a surprisingly small sum. A Small order-3 This is the smallest possible with primes in arithmetic progression. Primes in arithmetic progression An order-4 pandiagonal magic square using 14 or 15 digit primes. 73 consecutive primes from 3 to 373 together form 2 magic squares. Orders 4, 5, 6 use consecutive primes Prime # 37 to 103, 107 to 239 and 241 to 457 make 3 magic squares. A Bordered prime magic square Orders 8, 6 and 4 using distinct 4-digit primes. Order-3 with smallest sum These primes are neither consecutive or in arithmetical progression. Two palprime magic squares All numbers in these order-3's are 11-digit palindromic primes. Order-13 constant difference Nested squares of orders 13, 11, 9, 7, 5, 3, 1.

48. Multiply Perfect Numbers and a further list of multiply perfect numbers sorted only by their Call thesize of a substitution the number of different primes involved. http://www.uni-bielefeld.de/~achim/mpn.html

The Multiply Perfect Numbers Page Introduction Let o(n) be the number theoretic function which denotes the sum of all divisors of a natural number n . If o(n) is an integral multiply of n , then n is denoted as a multiply perfect number or k-fold perfect number (also called multiperfect number or pluperfect number ). Call o(n)/n abundancy (also called index or multiplicity ) of n . A multiply perfect number is called proper 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120=o(120)=o(2^3*3*5)=o(2^3)*o(3)*o(5)=(1+2+4+8)*(1+3)*(1+5)=15*4*6=360=3*120 Hence 120 is a 3-fold perfect number. Status Abundancy Count When last number was discovered Which was last? Are all discovered? Rough total number yes and proved no, there are infinitely many oo yes yes yes yes almost surely yes probably not no no no In column "Which was last?" the identifier ln(ln(MPN)) is given for those which were verfied by me. I checked these numbers only for those 5147 MPNs claimed before 2005-01-01. known and claimed MPNs until 2005-03-02. Data Richard Schroeppel's archive of 2094 MPNs built 1995-12-13 The collection of 5147 MPNs from 2005-01-01 (gziped to 862 kB) sorted by abundancy and magnitude. It is grown out of Rich's database - thanks -, and transformed into a new format, such that each multiply perfect number allocates one line with all its additional informations in the form:

49. VACETS Technical Column - Tc48 Note that, with the discovery of the new prime number, a new perfect number can The new Mersenne prime has 378632 digits and the new even perfect number http://www.vacets.org/tc/tc48.html

50. Mersenne Prime - Wikipedia, The Free Encyclopedia Mersenne primes have a close connection to perfect numbers, which are numbersthat are equal to the sum of their proper divisors. Historically, the study of http://en.wikipedia.org/wiki/Mersenne_prime

Mersenne prime From Wikipedia, the free encyclopedia. In mathematics , a Mersenne prime is a prime number that is one less than a power of two . For example, 3 = 4 − 1 = 2 − 1 is a Mersenne prime; so is 7 = 8 − 1 = 2 − 1. On the other hand, 15 = 16 − 1 = 2 − 1, for example, is not a prime, because 15 is divisible by 3 and 5. More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than a power of two; hence, M n n Mersenne primes have a close connection to perfect numbers , which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exists. It is currently unknown whether there is an infinite number of Mersenne primes Contents Properties of Mersenne numbers Searching for Mersenne primes List of Mersenne primes See also ... edit Properties of Mersenne numbers Mersenne numbers share several properties: M n is a sum of binomial coefficients: If a is a divisor of M q (q prime) then a and: A theorem from Euler about numbers of the form shows that M q (q prime) is a prime if and only if there exists only one pair x y such that: M q x y with . More recently, Bas Jansen has studied

51. Mathematics Archives - Topics In Mathematics - Number Theory KEYWORDS Number theory, prime numbers, primality tests; Lehmer s Conjecture KEYWORDS perfect numbers, Mersenne numbers, Triangular numbers, http://archives.math.utk.edu/topics/numberTheory.html

Topics in Mathematics Number Theory Absolute Primes ADD. KEYWORDS: Article, Repunits SOURCE: Arkadii Slinko, University of Auckland and Mathematics Olympiad Learning Centre TECHNOLOGY: Adobe Acrobat Reader Aesthetics of the Prime Sequence Algorithmic Number Theory - Bibliography AMS - Fermat's Last Theorem ADD. KEYWORDS: Number theory, Fermat's Last Theorem, History, Reports, Articles AMS's Materials Organized by Mathematical Subject Classification ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases Arithmetic properties of Binomial Coefficients ADD. KEYWORDS: Elementary Number Theory, Lucas' Theorem, Pascal's triangle via cellular automata, Bernoulli numbers and polynomials, Theorems of Morley and Emma Lehmer and their generalizations, Some useful p-adic numbers Bernoulli (Beta) and Integer Part Sequences ADD. KEYWORDS: Article SOURCE: Rod Nillsen, Keith Tognetti and Graham Winley, University of Wollongong TECHNOLOGY: Adobe PDF, Postscript and DVI readers

52. SS > Factoids > Perfect Number No odd perfect numbers are known, but if one does exist, a lot is known about it.it is a perfect square multiplied by an odd power of a single prime http://www-users.cs.york.ac.uk/~susan/cyc/p/perfect.htm

home factoids perfect number A perfect number P is equal to the sum of its divisors (where the divisors include 1, but not P itself). Euclid : If 2 n -1 is prime then 2 n n -1) is perfect Euler all even perfect numbers are of the form 2 p p -1), where 2 p -1 is a Mersenne prime (and so p is prime Every even perfect number ends in a '6' or an '8'. All even perfect numbers are triangular numbers. general result n n n n Every even perfect number, other than 6, is the sum of consecutive odd cubes. (6, with p =2, does not fit the pattern) conjecture: n n n n No odd perfect numbers are known, but if one does exist, a lot is known about it: it is a perfect square multiplied by an odd power of a single prime it has at least 8 distinct prime factors it has at least 75 prime factors (not necessarily distinct) its largest prime factor greater that 10 its second largest prime factor is greater that 10 its third largest prime factor is greater that 10 it is divisible by a prime component greater that 10 Exhaustive computer search has shown that there are no odd perfect numbers less than 10 My thanks to Douglas Iannucci and Joshua Zelinsky for some of this information, some of which is from the work of Kevin Hare.

53. Kathryn Cramer: Teaching Perfect Squares With Legos Then I explained about prime numbers. Listed below are links to weblogs thatreference Teaching perfect Squares with Legos http://www.kathryncramer.com/kathryn_cramer/2005/07/teaching_perfec.html

Kathryn Cramer www.kathryncramer.com books IMPORTANT LINKS The NOLA clickable depth map New Orleans Levee Break(s) Before and After How to Find Out if Your New Orleans House Is Under Water WE HAVE A DEPTH MAP! ... FEMA Needs to Tell People What It Intends for Their Homes Katrina Help When submitting a request for help finding out about a specific address, or with other Katrina-related problems, please put KATRINA HELP in the subject line. My email address is kathryn.cramer at gmail.com . Thanks. My Katrina archive contains all blog posts related to Katrina. The archive page is updated each time I make a new Katrina post, so it would be the best place to bookmark. On the other hand, it contains many images, so on a dial-up connection it would be slow to load. Also, separately, I have an online album of Katrina map images, Katrina Floods New Orleans, 2005 Recent Comments Kathryn Cramer on "How could it be that people were not monitoring the walls?" Bridget Coila on "How could it be that people were not monitoring the walls?" zeno on "How could it be that people were not monitoring the walls?"

54. SMARANDACHE PRIMES, SQUARES, CUBES, M-POWERS, AND MORE GENERAL T number (for example perfect number, or Bell number, or almost prime number,etc.). Then A Smarandache T Number is a term of any of the Smarandache http://www.gallup.unm.edu/~smarandache/s-number.htm

55. A Prime Way To Find Perfect Numbers If 2n 1 is a prime number then 2n - 1(2n - 1) is a perfect number. The secondtheorem gives us a method of finding perfect numbers. http://users.tru.eastlink.ca/~brsears/math/primperf.htm

SINE WAVE A Prime Way to Find Perfect Numbers return Prime numbers are numbers that cannot be expressed as the product of 2 whole numbers, other than 1 and the number itself. They are called prime because they cannot be factored into smaller numbers. The number 1 is not defined as prime. Here is a list of all the prime numbers less than 1000: How can we determine whether a number is prime or not? One obvious way is to divide the number by each smaller number, except 1. If any divide evenly, then the number is not prime. Actually we only need to try the numbers less than or equal to the square root of the number we are testing. For very large numbers, this method can take a long time. Even using modern computers, there is a practical limit to how many prime numbers we can find. Numbers that are not prime, except for 1, are called composite. Any composite number can be expressed as the product of a unique set of prime numbers. It is possible to prove that there are infinitely many prime numbers. We can also make an observation about the relative frequency of prime numbers. The prime number theorem tells us: If P(N) is the number of primes less than or equal to N, then the ratio N/P(N) approaches the natural logarithm of N as N approaches infinity.

56. Sublime Numbers Also, every even perfect number is of the form (2s 1) 2s-1 where 2s - 1 is aprime. Thus an even perfect number has exactly one odd prime factor. 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

57. Computing Perfect(prime) Numbers A perfect number is a number whose divisors not including the original numberadd up to Fast? way to find primes, dividing by odd numbers } Function http://www.piclist.com/techref/language/delphi/swag/MATH0108.html

58. Grade 5: Prime Factorization: When Students Ask rely on the ability to express a number as a product of prime numbers.For example, a perfect number is one whose proper factors (factors less than the http://www.eduplace.com/math/mw/background/5/07/te_5_07_factors_ask.html

@import url(/math/mw/includes/styles.css); /*IE and NN6x styles*/ Prime Factorization: When Students Ask Why should I bother learning this? The prime factorization of a number is used in many algorithms such as finding the least common multiple and the greatest common divisor. These in turn are used in working with fractions. The least common multiple is used when finding the lowest common denominator, and the greatest common factor is used in simplifying a fraction. Many patterns, formulas, and number concepts in number theory rely on the ability to express a number as a product of prime numbers. For example, a perfect number is one whose proper factors (factors less than the number) add up to the given number. The smallest perfect number is six, and its proper factors are 1, 2 and 3. After showing that six is perfect, you could ask students to find the next perfect number (28). What is the greatest prime number? There is no greatest prime number. The greatest prime number discovered so far has 895,932 digits, but there are undoubtedly greater ones. A famous mathematician named Euclid was able to prove many years ago that there is no greatest prime number. Are there rules for divisibility for 6, 7, 8, and 11?

59. Prime Numbers prime numbers. AM finds some simple patterns among these definitions the set that all numbers with three divisors are also perfect squares;To see this, http://web.media.mit.edu/~haase/thesis/node59.html

Next: Properties of Primes Up: The AM Program Previous: Factorization Prime Numbers AM finds some simple patterns among these definitions: the set of numbers with zero divisors is empty; the set of numbers with one divisor is itself a singleton (this is found uninteresting); the set of numbers with two divisors (primes without one) is somewhat interesting because there are a reasonable number of examples. The excitment begins as AM notices that all numbers with three divisors are also perfect squares;To see this, consider an integer n with three divisors. One must be 1 and one must be n itself. The remaining divisor p must produce n by repeated multiplication so that . If , must also be a factor of n . Since n has only three divisors, this cannot be the case so making n a perfect square. this makes them interesting enough to consider what their square roots are (AM had earlier invented SQUARE-ROOT by inverting SQUARE). It turns out that the square roots of these numbers are all numbers with two divisors. This regularity boosts the interestingness of this class (now renamed `Primes' by Lenat) to a point where detailed examiniations of the properties of primes are undertaken. Ken Haase Sun Nov 3 16:17:57 EST 1996

60. Puzzle 111. Spoof Odd Perfect Numbers even perfect numbers, at least as many as Mersenne prime numbers (38 at is an even Spoof perfect Number if you suppose incorrectly that 4 is a prime 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...

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