1. The Prime Glossary Perfect Number Perfect Numbers and Mersennes (definitions and theorems) Great Internet Mersenne Prime Search (which is the search for even perfect numbers) http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

2. Perfect Security - Prime Numbers Prime Numbers Definition. A number is called prime, if and only if its Design by Perfect Security TM, 2001 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

3. Mersenne Primes History, Theorems And Lists 2. Perfect Numbers and a Few Theorems http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

4. "Computing Perfect(prime) Numbers" By SAMIEL@FASTLANE.NET Perfect Numbers A Perfect number is a number whose divisors not including the original number add up to the original number. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Mathematics Enrichment Workshop The Perfect Number Journey What are perfect numbers? Mathematicians and nonmathematicians have been fascinated for centuries by the properties and patterns of numbers. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. Prime Numbers. Perfect Numbers. Prime numbers. Perfect numbers. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. C++ Programming Need Help Now Pls Check For Prime Numbers And Solution Title Need Help Now Pls Check for Prime Numbers and Perfect Numbers asked by liza_holland on 10/28/2002 0825PM PST http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. 36th Mersenne Prime Discovered 1, is one of a special class of prime numbers called Mersenne primes. that generates a "perfect" number from a Mersenne prime. A perfect http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. SQL And The Search For Prime And Perfect Numbers I got an usual request today in reference to writing SQL statements to find Prime Perfect Numbers. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Aliquot Sequences f(4)=3, f(3)=1, f(1)=0, f(0)=0, and one then loops on 0. One can also loop on perfect numbers, i.e. numbers such that f(n)=n, for example n http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. Techangel Forums - Perfect And Prime Numbers This is a discussion forum powered by vBulletin. To find out about vBulletin, goto http//www.vbulletin.com/ . http://www.techangel.co.uk/forums/archive/index.php/t-734.html

techangel forums technology software / troubleshooting programming ... PDA View Full Version : Perfect and prime numbers mrplow This doesnt seem to work rite atm... further down ;) » Edited on 29-11-2002 by mrplow pretty much complete c code listings lower down... just realised this is being linked to by a maths site :o » Edited on 6-7-2003 by mrplow mrplow perfect numbers sure as hell dont come out right :S dash does now.... ;) mrplow // Program 1 : Perfect numbers void main(void) int isprime; // is it a prime or not? use this to store or 1 int i; // counter int j; // second counter int theprimes[10000]; // have an array for the prime numbers, size 10000 int total=0; // keep track of number of items in array int amount=10000; // used to allow fewer numbers in testing, set at 10000 for normal use float k; int perf,l,p; // for perfects isprime=1; // check for numbers that will divide into the prime with no remainder.. starting from 2, cause 1 will divide anything isprime=0;

12. Techangel Forums - Perfect And Prime Numbers This is a discussion forum powered by vBulletin. To find out about vBulletin, goto http//www.vbulletin.com/ . http://www.techangel.co.uk/forums/showthread.php?t=734

13. Mersenne Primes: History, Theorems And Lists Mersenne primes (and therefore even perfect numbers) are found using the We know that all even perfect numbers are a Mersenne prime times a power of two http://primes.utm.edu/mersenne/

Mersenne Primes: History, Theorems and Lists A forty second Mersenne found Feb 2005: Contents: Early History Perfect Numbers and a Few Theorems Table of Known Mersenne Primes The Lucas-Lehmer Test and Recent History ... Conjectures and Unsolved Problems See also Where is the next larger Mersenne prime? and Mersenne heuristics For remote pages on Mersennes see the Prime Links' Mersenne directory Primes: Home Largest Proving How Many? ... Mailing List 1. Early History Many early writers felt that the numbers of the form 2 n -1 were prime for all primes n , but in 1536 Hudalricus Regius showed that 2 -1 = 2047 was not prime (it is 23 89). By 1603 Pietro Cataldi had correctly verified that 2 -1 and 2 -1 were both prime, but then incorrectly stated 2 n -1 was also prime for 23, 29, 31 and 37. In 1640 Fermat showed Cataldi was wrong about 23 and 37; then Euler in 1738 showed Cataldi was also wrong about 29. Sometime later Euler showed Cataldi's assertion about 31 was correct. Enter French monk Marin Mersenne (1588-1648). Mersenne stated in the preface to his

14. The Prime Glossary: Perfect Number Welcome to the prime Glossary a collection of definitions, information and factsall related to prime numbers. This pages contains the entry titled http://primes.utm.edu/glossary/page.php?sort=PerfectNumber

15. Prime Numbers They understood the idea of primality and were interested in perfect and amicable It is not known to this day whether there are any odd perfect numbers. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html

Prime numbers Number theory index History Topics Index Version for printing 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. You can see more about these numbers in the History topics article Perfect numbers 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.

16. Perfect Numbers no odd perfect numbers, unless they are composed of a single prime number, When the exponent is a prime number, I say that its radical less one is http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html

Perfect numbers Number theory index History Topics Index Version for printing It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity. It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked, see for example [17] where detailed justification for this idea is given. Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties. Before we begin to look at the history of the study of perfect numbers, we define the concepts which are involved. Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number. An aliquot part of a number is a proper quotient of the number. So for example the aliquot parts of 10 are 1, 2 and 5. These occur since 1 = , and 5 = . Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i.e. a quotient different from the number itself. A perfect number is defined to be one which is equal to the sum of its aliquot parts.

17. Mersenne Prime -- From MathWorld COMMENT On this Page. A Mersenne prime is a Mersenne number, ie, a number of theform Kraitchik, M. Mersenne numbers and perfect numbers. http://mathworld.wolfram.com/MersennePrime.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Special Numbers Prime-Related Numbers ... Mathematical Records Mersenne Prime A Mersenne prime is a Mersenne number , i.e., a number of the form that is prime . In order for to be prime must itself be prime . This is true since for composite with factors and . Therefore, can be written as , which is a binomial number that always has a factor The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (Sloane's ) corresponding to indices , 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (Sloane's Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number . L. Welsh maintains an extensive bibliography and history of Mersenne numbers. It has been conjectured that there exist an infinite number of Mersenne primes. Fitting a line fixed to pass through the origin to the asymptotic number of Mersenne primes with gives a best-fit line with . If the line is not restricted to pass through the origin, the best fit is

18. Mathematics Enrichment Workshop: The Perfect Number Journey The first three perfect numbers 6, 28 and 496 were known to the ancient of 2 (ie doubling the numbers), until you get a sum which is a prime number. http://home1.pacific.net.sg/~novelway/MEW2/lesson1.html

The Perfect Number Journey Lessons on number patterns and properties of numbers by Heng O.K. What are perfect numbers? Mathematicians and nonmathematicians have been fascinated for centuries by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). The smallest such example is , since = 1 + 2 + 3. Such numbers are called perfect numbers The search for perfect numbers began in ancient times. The first three perfect numbers: and were known to the ancient mathematicians since the time of Pythagoras (circa 500 BC). Exercise 1 Verify for yourself that the numbers 28 and 496 are in fact perfect numbers, by completing the table below. You may use a calculator to work out the answers. Table 1: The first three perfect numbers. How to find perfect numbers? Euclid (circa 300 BC), the famous Greek mathematician, devised a simple method for computing perfect numbers. Beginning with the number 1, and keep adding the powers of 2 (i.e. doubling the numbers), until you get a sum which is a prime number . A perfect number is then obtained by multiplying this sum to the last power of 2. In the exercise that follows, you are going to use this method to determine the next two perfect numbers. The first few rows in the table demonstrate the calculations being carried out to compute the first three perfect numbers. Apply this technique now, and let's see how fast you can find the fourth perfect number.

19. Mathematics Enrichment Workshop: The Perfect Number Journey These two perfect numbers can be obtained from the Mersenne primes M17 = 217 1 and Whenever a prime number of this form is found, a perfect number is http://home1.pacific.net.sg/~novelway/MEW2/lesson2.html

How are Mersenne primes related to perfect numbers? If a Mersenne number turns out to be a prime number, then it is called a Mersenne prime You have computed the first 5 Mersenne primes: 3, 7, 31, 127, 8191. Each of these numbers in turn gives a perfect number when multiplied by its previous power of 2. Just to summarise what we have done so far, let's examine Table 2 again. This time, we will express the numbers in powers of two, and delete those rows that do not carry perfect numbers. Exercise 4 (a) Complete the following table, expressing the first five Mersenne primes and perfect numbers in powers of two. Table 4: The first five Mersenne primes and the corresponding perfect numbers. (b) Two perfect numbers were discovered in 1588, both by Cataldi. These two perfect numbers can be obtained from the Mersenne primes M - 1 and M - 1. Can you compute these two perfect numbers with the help of your calculator? (c) Do you think M is a Mersenne prime? By now, you should have realised why numbers of the form 2 n - 1 have so much appeal. Whenever a prime number of this form is found, a perfect number is immediately obtained, as was proven by Euclid.

20. Ancient Greeks: Prime Numbers And Number Theory Ancient Greeks discovered the prime numbers or pt a?µ (protoi They defined as perfect numbers those equal to the sum of their parts (or proper http://www.mlahanas.de/Greeks/Primes.htm

Ancient Greeks: Prime Numbers and Number Theory Michael Lahanas Griechische Mathematik: Zahlentheorie und Primzahlen Pythagoras of Samos ( Πυθαγόρας ο Σάμιος) discovered the relation between harmony and numbers. The Pythagoreans saw the number one as the primordial unity from which all else is created. Two was the symbol for the female, three for the male and therefore five (two + three) symbolized marriage. The number four was symbolic of harmony, because two is even, so four (two times two) is "evenly even". Four symbolized the four elements out of which everything in the universe was made (earth, air, fire, and water). Ten that was the sum from one to four was a very special number. The ancient Greeks believed that all numbers had to be rational numbers. 2500 years ago Greeks discovered that if all the common prime numbers were removed from the top and bottom of the ratio then one of the two numbers had to be odd. This we can term reduced form . Obviously, if top and bottom were both even, then both could be divide by the number two and this could be eliminated from both.

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