21. Perfect Number - Wikipedia, The Free Encyclopedia Thus, every Mersenne prime will yield a distinct even perfect number—there isa concrete oneto-one association between even perfect numbers and Mersenne http://en.wikipedia.org/wiki/Perfect_number

Perfect number From Wikipedia, the free encyclopedia. In mathematics, a perfect number is an integer which is the sum of its proper positive divisors , excluding itself. Thus, and 8128 (sequence http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000396 in OEIS ). These first four perfect numbers were the only ones known to the ancient Greeks The Greek mathematician Euclid discovered that the first four perfect numbers are generated by the formula 2 n n for n for n for n for n Noticing that 2 n prime number in each instance, Euclid proved that the formula 2 n n n Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisily the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2 n The fifth perfect number would have five digits since the first four had 1, 2, 3, and 4 digits respectively. The perfect numbers would alternately end in 6 or 8. The fifth perfect number has 8 digits, thus debunking the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth also ends in a 6. (That the last digit of any even perfect number must be a 6 or an 8 is not difficult to show.)

22. 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

23. SQL And The Search For Prime And Perfect Numbers SQL and the search for prime and perfect numbers. I got an usual request todayin reference to writing SQL statements to find prime perfect numbers. http://weblogs.sqlteam.com/davidm/archive/2003/10/30/412.aspx

DavidM Garbage My Links Home Contact SQLTeam.com Weblogs SQLTeam.com ... Login Blog Stats Posts - 42 Stories - Comments - 201 Trackbacks - 39 Archives June, 2004 (1) May, 2004 (4) April, 2004 (4) March, 2004 (3) ... October, 2003 (6) Post Categories Code (rss) Garbage (rss) Database Critique DBDebunk Relational Model SQL and the search for Prime and Perfect numbers "How do we fill the Prime table?", was the obvious next question. This is surprisingly simple thanks to the Internet... Prime Site This sight has a list of Prime numbers... A simply parse and a BCP and we have a 100,000 row Prime table. But it did get me thinking.. If we had a Numbers table with numbers 1 to 1 million, could we write a SQL statement to find the Primes or Perfect Numbers? I am going to stick with numbers below a 1000 for the solution.. Below is the code for the numbers table (Only to 1000) CREATE TABLE Numbers(Number INT NOT NULL PRIMARY KEY CLUSTERED) GO DECLARE @i INT SET @i = 1 BEGIN     INSERT Numbers(Number) values (@i)     SET @i = @i + 1 END GO Now for the SQL Stuff Prime Numbers SELECT X.Number as Primes

24. Prime Numbers the socalled Bertelsen s number and mistakes in calculating pi(n) at Seattle,USA and also about Fermat primes Some information about perfect numbers etc. http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Prmnmbrs.htm

Prime numbers Previous topic Next topic History Topics Index Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians. The mathematicians of Pythagoras 's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers. (A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28. A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.) By the time Euclid 's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

25. Knight's Tour Art For additional analysis on primes, Mersenne primes, or perfect numbers, please seemy articles on perfect Number Solution Key and prime Consideration for http://www.borderschess.org/KTprimes.htm

From Knight Moves to Primes From looking at the moves of the knight on the chessboard, I wanted to find out what the slope angle in degrees was for the hypotenuse of the (2, 1) right triangle made by the knight. I remembered from my old trigonometry days that I could use the Pythagorean Theorem to solve this problem. I also decided to find the angles for other types of similar triangles. I ultimately ended up with a neat summation formula. Select the formula below to see my math analysis. After finding the angles in degrees from the formula, I plotted them on the following graph: Afterwords, I added x,y coordinates to the squares where the angled lines intersected whole integer pairs. I began to realize that the coordinates were the same numbers that represent the factors for each integer. I then made a mirror image of the chart and replaced all the x,y coordinates of each square with ones and zeros. Before showing the binary and prime chart, here is a chart created by writing down the first 11 x,y coordinates of each angle starting with (0,0) from the previous chart. This new chart can be used as a multiplication chart. Increasing the length of the angled slopes in the previous chart will reveal additional integers and their factors that can be used to increase the size of the multiplicaton chart. Since the factorial chart also looked like binary, I went ahead and created a new binary chart and overlaid the same angles (only inverted and rotated 90 degrees counter-clockwise) previously discovered. The chart ultimately reveals that all primes fit within a specific pattern made by the angles of right triangles found by my summation formula.

26. Integers As Prime Or Composite difficulty. perfect numbers and Mersenne primes. A numbers. Whenever anothermersenne prime is found, another perfect number is generated. The http://www.andrews.edu/~calkins/math/webtexts/numb03.htm

Back to the Table of Contents Numbers and Their Application - Lesson 3 The Naturals as Prime or Composite Lesson Overview Factors Prime Composite 1 is Unique ... Division Rules for 2,3,4,5,8,9,11,... Perfect Numbers and Mersenne Primes Cardinal vs. Ordinal Numbers ... Homework Factors, Prime, Composite, 1 is Unique The natural numbers have been studied intensely for millenia. Several fascinating properties relate to their factors. A factor is a natural number which divides another natural number evenly (as in without a remainder). The word factor will be used later in a less restricted sense as in x +1 is a factor of x Divisor is essentially a synonym of factor and is also commonly used interchangeably. A prime number only has factors of itself and one. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.... Twin primes are primes which differ by 2. Examples of twin primes are: 3 and 5, 5 and 7, 11 and 13, 17 and 19, .... A composite number has factors in addition to itself and one. One (1) is unique in that it is considered neither prime nor composite.

27. Resources if it equals the sum of its divisors (excluding itself). Read aboutthe connection between perfect numbers and Mersenne primes. http://www.teachers.ash.org.au/mikemath/resources/prime.html

Mathematics Resources by Topic - Number Prime numbers Home Resources by Topic Calculators and Plotters Reference ... More Stuff prime number calculators successive primes starting from of digits The Prime Pages 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. 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. The Prime Puzzles and Problems Connection solutions available 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

28. Prime Numbers thing we know about the distribution of prime numbers is called n 1, there is atleast one prime such that . perfect Squares, Modulo 4, and the Two Types of http://www.albanyconsort.com/primes/primes.html

Prime Numbers Introduction A prime number is an integer that has exactly two factors. These numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, .... There are infinitely many of these, as can be proven in a variety of ways. Here is one fascinating way to do this. As shown on the theorems page, the harmonic series diverges to infinity. Now we use this to prove that there are infinitely many prime numbers. First we make the bold statement: . How can we make such a statement? Well, we note that every integer greater than 1 can be written as the product of primes, so it follows that . Conveniently, each of the parentheses encloses a geometric series that we can sum, so we get . Thus we reach the infinite product above. If there are only a finite number of primes, then the right side will not diverge, but we already know that the left side will, so we have a contradiction, so there must be an infinite number of primes. Obviously, there are simpler ways of proving this, but I believe that this method gives the most insight into the distribution of primes. More on the Distribution of the Primes Let us define a function to be the number of prime numbers not exceeding x. Thus

29. Perfect Numbers any other perfect numbers, numbers equal to the sum of their proper divisors? Euclid(in Book IX, Proposition 36) actually showed that if p is prime, and if (2 http://www.jimloy.com/number/perfect0.htm

Return to my Mathematics pages Go to my home page Perfect Numbers 6 and 28 are called Perfect Numbers. The proper divisors (the divisors of a number, not including the number itself) of 6 are 1, 2, and 3, and 6=1+2+3. Similarly, the proper divisors of 28 are 1, 2, 4, 7, and 14 and 28=1+2+4+7+14. Are there any other perfect numbers, numbers equal to the sum of their proper divisors? Euclid (in Book IX, Proposition 36) actually showed that if p Primes of the form (2^p)-1 are called Mersenne Primes. A number of the form (2^n)-1 cannot be a prime unless n is a prime. The nth Mersenne number is M n =M(n)=(2^n)-1. Many people call M(n) a Mersenne number only if n is prime. The first few Mersenne primes are 3, 7, 31, 127, 8191, etc., which are primes. So the first few perfect numbers are 6, 28, 496, 8128, 33550336, etc. It was quickly shown that M(11), M(23), M(83), M(131) and others were not prime, even though n is prime in those cases. Mersenne apparently conjectured that M(n) was prime for n=2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and that M(n) was not prime for any other n below 257. He was shown to be wrong for n=61, 67, 89, 107, and 257. It is relatively easy to test to see if a Mersenne number is prime, and so the largest known prime is often a Mersenne prime.

30. Mathematical Recreations Some Padovan numbers, such as 9, 16 and 49, are perfect squaresare SPIRALING CUBOIDSalso form Padovan numbers. If n divides A (n), must n always be prime? http://www.fortunecity.com/emachines/e11/86/padovan.html

web hosting domain names email addresses Mathematical Recreations by Ian Stewart Tales of a Neglected Number Last month I described the mathematical sculptures of Alan St. George, who often makes use of the well known golden number ". The catalogue of his Lisbon exhibition mentions a less glamorous relative, referring to a series of articles in which "the architect Richard Padovan revealed the glories of the 'plastic number.' " The plastic number has little history, which is strange considering its great virtues as a design tool, but its provenance in mathematics is almost as respectable as that of its golden cousin. It doesn't seem to occur so much in nature, but, then, no one's been looking for it. A Pearly Nautilus and it's logarithmically spiral shell. [Image:"Life on Earth" D.Attenborough For purposes of comparison, let me start with the golden number: q =1 + 1/q = 1.618034, approximately. The golden number has close connections with the celebrated Fibonacci numbers . This series can be illustrated by a spiralling system of squares [see upper illustration on this page]. The initial square (in gray) has side 1, as does its neighbour to the left. A square of Side 2 is added above the first two, followed in turn by squares of side 3, 5, 8, 13, 21 and so on. These numbers, each of which is the sum of the previous two, form the

31. Integer Bars: More About Multiplication In bothe cases the answer is 16. Exercise Find all of the numbers thatare perfect squares between 1 and 100. prime numbers and Composites. 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

32. Grade 5: Prime Factorization: When Students Ask formulas, and number concepts in number theory rely on the ability to express anumber as a product of prime numbers. For example, a perfect number is one 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?

33. Perfect Numbers 859,433. These represented also the greatest known prime numbers, and(calculated from them) the greatest known perfect numbers. Then http://uzweb.uz.ac.zw/science/maths/zimaths/51/perfect.htm

Perfect Numbers A number n is said to be perfect if it is equal to the sum of its proper divisors (or, if the sum of all its divisors is equal to its double 2n). The first obvious example is 6 = 1+2+3. What is the next smallest perfect number? (Answers to questions in this article can be found - after trying to answer them by yourself first - on page ). Perfect numbers were first (to our knowledge) investigated by the Pythagoreans about 2500 years ago. They felt a deep reverence for such numbers, and in fact felt that all numbers had meaning and significance far beyond their purely mathematical properties. For example, even numbers were female , odd numbers male ; 3 was the number of marriage (sum of first male and first female numbers); 4 was the number of justice (we still talk about a square deal, meaning a fair one); 10 = 1+2+3+4 was revered as the tetractys , a very special number indeed. This ``number mysticism'' was a great inspiration to the Pythagoreans to study numbers diligently, and they are often regarded as the founders of Pure Mathematics, certainly of the purest branch of mathematics: Number Theory. They defined other interesting categories of numbers related to perfect numbers. First, let us introduce the notation

34. HAKMEM -- NUMBER THEORY, PRIMES, PROBABILITY -- DRAFT, NOT YET PROOFED ITEM 62 (Speciner) The first four perfect numbers are 6, 28, 496, 8128. Aprime decade is where N+1, N+3, N+7 and N+9 are all prime. http://www.inwap.com/pdp10/hbaker/hakmem/number.html

Beeler, M., Gosper, R.W., and Schroeppel, R. HAKMEM . MIT AI Memo 239, Feb. 29, 1972. Retyped and converted to html ('Web browser format) by Henry Baker, April, 1995. NUMBER THEORY, PRIMES, PROBABILITY Previous Up Next ITEM 28 (Schroeppel): After about 40 minutes of run time to verify the absence of any non-trivial factors less than 235, the 125th Mersenne number, was factored on Tuesday, January 5, 1971, in 371 seconds run time as follows: John Brillhart at the University of Arizona had already done this. M137 was factored on Friday, July 9, 1971 in about 50 hours of computer time: Current prime records H.B. ITEM 29 (Schroeppel): For a random number X, the probability of its largest prime factor being greater than sqrt(X)=X^(1/2) is ln 2. less than X^(1/3) is about 4.86%. This suggests that similar probabilities are independent of X; for instance, the probability that the largest prime factor of X is less than X^(1/20) may be a fraction independent of the size of X. RELEVANT DATA: ([] denote the expected value of adjacent entries.) RANGE COUNT CUMULATIVE SUM OF COUNT 10^12 to 10^6 7198 [6944] 10018 10^6 to 10^4 2466 2820 10^4 to 10^3 354 402 [487] 10^3 to 252 40 48 ;252 = 10^2.4 252 to 100 7 8 100 to 52 1 1 ;52 = 10^1.7 51 to 1

35. Defective Number - Encyclopedia Article About Defective Number. Free Access, No one is not a prime number, it is called a composite number. all prime powers and ofdeficient or perfect numbers In mathematics, a perfect number is an integer http://encyclopedia.thefreedictionary.com/Defective number

Dictionaries: General Computing Medical Legal Encyclopedia Defective number Word: Word Starts with Ends with Definition In mathematics Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. Click the link for more information. , a deficient number or defective number is a number n for which n n . Here n ) is the divisor function In mathematics the divisor function a n ) is defined as the sum of the a th powers of the divisors of n, or The notation d n n ), or the number of divisors of n . The sigma function n ) is Click the link for more information.

36. An Odd Perfect Number Has At Least 3 Distinct Prime Factors numbers. We know , since is odd. with 2.7 i) and and we get which is a contradiction.Therefore must have at least 3 distinct prime factors to be perfect. http://www-maths.swan.ac.uk/pgrads/bb/project/node12.html

Next: An odd perfect number Up: Perfect Numbers Previous: Restrictions for the prime Contents An odd perfect number has at least 3 distinct prime factors An odd perfect number must have at least three distinct prime factors. Proof. Let be an odd perfect number. Suppose where are distinct primes ( ) and are natural numbers. We know , since is odd. with 2.7 i) and and we get which is a contradiction. Therefore must have at least 3 distinct prime factors to be perfect. If you look closely to this proof, you might notice, the fact that is odd is only used to determine that . In the following proof we will use it also in an other way. Footnotes ...Proof. Based on a proof in [ B–ttcher 2002-08-07

37. Restrictions For The Prime Factors Of A Perfect Number Contents Restrictions for the prime factors of a perfect number. Letwhere the are natural numbers and the are distinct primes. If http://www-maths.swan.ac.uk/pgrads/bb/project/node11.html

Next: An odd perfect number Up: Perfect Numbers Previous: Euler's Theorem Contents Restrictions for the prime factors of a perfect number Let where the are natural numbers and the are distinct primes. If is a perfect number, then i) ii) for Proof. Let where the are natural numbers and the are distinct primes. When we divide by we get is perfect so: Put the two above together to get . This proves the statement i). We saw above that Therefore with So we proved ii). Footnotes since all $p_i$ are distinct since the only divisors of $p_i$ are its powers B–ttcher 2002-08-07

38. Notable Properties Of Specific Numbers At MROB Also, for the number to be perfect, 2 P 1 must be prime, and is called aMersenne prime. See here for a complete list of known perfect numbers. http://home.earthlink.net/~mrob/pub/math/numbers-8.html

Notable Properties of Specific Numbers Back to page 7 Forward to page 9 This is 8! (8 factorial See also and 45360 has 100 factors (including 1 and itself); no smaller number has as many. Its prime factorization is 2 A B C D = 351530410 and so on. Based on this, you can see that: Similarly, the exponents in the prime factorization of a record-setter get smaller but don't get larger. For example, 2 For more about factors record-setters, see and The highest value of k for which 2 k k is known to be prime, or is considered a "probable prime". The other lower values of k are 1, 3, 7, 237 and 1885. There ought to be about log( N ) such k values for k < N , but since 2 k grows so quickly, it is very difficult to find more. It also happens to be difficult to prove whether there is or is not an infinite number of such k 's. This is 2 , which is using the operator . Also, since is , 65536 is where is the operator. (or 8 ), the square of the product of its odd-numbered digits starting from the left. Only and These, and properties of other numbers (notably including

39. LESSON PLANET - 30,000 Lessons And 2896 Lesson Plans For Prime Numbers numbers Students will learn about proper factors, perfect numbers, abundant numbers,and defective numbers. Grades 6-8. 20. prime Factorization - Lesson http://www.lessonplanet.com/search/Math/Prime_Numbers?startval=10

40. I Love Philosophy :: View Topic - Prime Numbers A perfect number is a number which is equal to the sum of all of it s divisors excludingitself and takes the general form 2^n1*2(n-1) where 2^n-1 is prime, http://www.ilovephilosophy.com/phpbb/viewtopic.php?p=1586192

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