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1. Perfect-Key
Prime numbers (blue background). Perfect numbers (yellow background). Continue thesequences indefinitely in the chart to reveal all perfect and prime numbers.
http://www.borderschess.org/Perfect-Key.htm
 Perfect Number Solution Key n:=1 n:=2 n:=3 n:=4 n:=5 n:=6 n:=7 n:=8 n:=9 a(n) b(n) c(n) f(n) g(n) Prime numbers (blue background). Perfect numbers (yellow background). The product sequence algorithm of c(n) is the solution key to unlocking all Prime numbers (shaded blue) and Perfect numbers (shaded yellow) for numbers greater than 6 in our number system. The idea for the algorithm comes from my article: "Prime Consideration for the Perfect Number" shown below. The ÂnÂ sequence (n:=1, n:=2, .. , n:=9) can be easily increased and iterated in a software program. Results for the ÂnÂ sequence, shown in the chart above, were derived from MathCad Professional and Mathematica computer programs. If f(n) is prime, then b(n) is perfect. Also, if f(n) is prime, it will be a Mersenne prime. There are various ways to check a number for primeness. Factoring f(n) can be very slow on extremely large numbers. Using the IsPrime function (Mathcad) or PrimeQ function (Mathematica) takes much less time but gives a probability that a number is Prime. Use of IsPrime or PrimeQ may need to be verified on very large numbers by an independent source. Check out GIMPS (The Great Internet Mersenne Prime Search) at http://www.mersenne.org/prime.htm

2. Large Prime Numbers
Mersenne primes. The new perfect number generated with the new Mersenneprime is the 34th known perfect number and has 757,263 digits.
http://www.isthe.com/chongo/tech/math/prime/prime_press.html
##### Large Prime Number Found by SGI/Cray Supercomputer
Now serving
text by: Landon Curt Noll
##### Note: This is no longer the largest known prime
The largest known prime number may be found in chongo's table of
Mersenne Prime Digits and Names
<== try me EAGAN, Minn., September 3, 1996 Computer scientists at SGI 's former Cray Research unit, have discovered a large prime number while conducting tests on a CRAY T90 series supercomputer. The prime number has 378,632 digits. Printed in newspaper-sized type, the number would fill approximately 12 newspaper pages. In mathematical notation, the new prime number is expressed as , which denotes two, multiplied by itself 1,257,787 times, minus one. Numbers expressed in this form are called Mersenne prime numbers after Marin Mersenne, a 17th century French monk who spent years searching for prime numbers of this type. See Chris Callwell's prime page for more information on prime numbers. Prime numbers can be divided evenly only by themselves and one. Examples include 2, 3, 5, 7, 11 and so on. The Greek mathematician

3. A Prime Way To Find Perfect Numbers
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. is 28 (1+2+4+7+14=28). These are called perfect numbers and they are very rare is an interesting relationship between even perfect numbers and certain prime numbers
http://users.andara.com/~brsears/primperf.htm
##### 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.

4. Perfect, Amicable And Sociable Numbers
perfect, amicable and sociable numbers. ( how to have fun perfect numbers, and, apart from 1 and 2, no augmented modified exponential perfect numbers. Also, any prime factor of
http://xraysgi.ims.uconn.edu:8080/amicable.html
HTTP 200 Document follows Date: Sun, 06 Jun 2004 07:20:00 GMT Server: NCSA/1.5.2 Last-modified: Tue, 30 Dec 2003 02:14:15 GMT Content-type: text/html Content-length: 29165
##### 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).
• 5. Mersenne Primes: History, Theorems And Lists
We know that all even perfect numbers are a Mersenne prime times a powerof two (theorem one above), but what about odd perfect numbers?
http://www.utm.edu/research/primes/mersenne/
##### Mersenne Primes: History, Theorems and Lists
A forty first Mersenne found May 2004: 2
##### 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 Cogitata Physica-Mathematica (1644) that the numbers 2 n -1 were prime for n 31, 67, 127 and 257
• 6. Mersenne Primes: History, Theorems And Lists
The definitive pages on the Mersenne primes and the related mathematics! We know that all even perfect numbers are a Mersenne prime times a power of two (theorem one above), but what about odd perfect numbers? If there is one
http://www.utm.edu/research/primes/mersenne.shtml
##### Mersenne Primes: History, Theorems and Lists
A forty first Mersenne found May 2004: 2
##### 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 Cogitata Physica-Mathematica (1644) that the numbers 2 n -1 were prime for n 31, 67, 127 and 257
• 7. Prime Numbers
Euclid also showed that if the number 2 n 1 is prime then the number2 n-1 (2 n - 1) is a perfect number. The mathematician Euler
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html
##### Prime numbers
Number theory index 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.
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.

8. The Prime Glossary: Perfect Number
Welcome to the prime Glossary a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'perfect number.' Come explore a new prime term
http://www.utm.edu/research/primes/glossary/PerfectNumber.html
 perfect number (another Prime Pages ' Glossary entries) Glossary: Prime Pages: Many ancient cultures endowed certain integers with special religious and magical significance. One example is the perfect numbers, those integers which are the sum of their positive proper divisors . The first three perfect numbers are 28 = 1 + 2 + 4 + 7 + 14, and The ancient Christian scholar Augustine explained that God could have created the world in an instant but chose to do it in a perfect number of days, 6. Early Jewish commentators felt that the perfection of the universe was shown by the moonÂs period of 28 days. Whatever significance ascribed to them, these three perfect numbers above, and 8128, were known to be "perfect" by the ancient Greeks, and the search for perfect numbers was behind some of the greatest discoveries in number theory. For example, in Book IX of Euclid 's elements we find the first part of the following theorem (completed by Euler some 2000 years later).

9. Perfect Numbers
11 12 13 1 3 7 15 31 63 127 255 511 1023 2047 4095 8191 let them be called the radicalsof perfect numbers, since whenever they are prime, they produce them.
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html
##### Perfect numbers
Number theory index History Topics Index
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  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. The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.

10. "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 where p is prime and 2^p1 is prime, so 2^2
http://www.bsdg.org/swag/MATH/0108.PAS.html
 Back to MATH SWAG index Back to Main SWAG index Original PROGRAM Perfect VAR tmp num longint j k byte Function IsPrime num longint boolean Var tmp boolean j longint Begin tmp true if num mod then tmp false for j to round sqrt num do if odd j then if num mod j then tmp false if num then tmp true IsPrime tmp End BEGIN tmp writeln 'Perfect numbers...' for j to do begin tmp tmp if IsPrime j then begin num tmp if IsPrime num then begin num num tmp div writeln num ' is perfect' end end end END Samiel samiel fastlane net http //www.fastlane.net/~samiel Considering that is prime and is the largest known prime there are obviously better methods Use the Lucas Lehmer test B n B n mod p where B if B n then p is prime The division is a special case and is done easily because of the all ONE 's when 2^p-1 is written in binary. All that is necessary is a fast implementation of multiplication and this can be done with FFT 's. See http //www.utm.edu/research/primes/mersenne.shtml or for software http //ourworld.compuserve.com/homepages/justforfun/prime.htm Well considering we are looking at numbers under bits your code would probably not get done as fast as mine though in the long run say over bits or so it would All the FFT 's and rather long multiplication would slow it down considerably Even if you could get a really fast implementation of FFT 's and multiplications, we'

11. ONJava.com: Web Services And The Search For Really Big Prime Numbers [Aug. 29, 2
Web services should open up new avenues of computing. Such as? This article shows how Web services are an ideal model for computing Mersenne prime numbers, some of the largest primes yet prime numbers, Mersenne numbers, Mersenne prime numbers, and perfect numbers. A prime number Also, a relationship between prime numbers and perfect numbers has been known
http://www.onjava.com/pub/a/onjava/2002/08/28/wsdc.html
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##### Web Services and the Search for Really Big Prime Numbers
by Eoin Lane What do searching for extraterrestrials, curing cancer, and finding big prime numbers all have in common? These problems are all being attacked with grid computing, a a technique of breaking a large problem into small tasks that can be computed independently. While projects like Seti@home and The Greatest Internet Mersenne Prime Search have received plenty of press for using the Internet to distribute tasks to end users around the globe, grid computing also takes place in more controlled environments, such as research and financial settings. But it is by using the power of the Internet and the ability to discover and access idle processes on users' machines that grid computing (once called

12. SS > Factoids > Perfect Number
is prime then 2 perfect. Euler all even perfect numbers are of the form 2(p1)(2p-1), where 2p-1 is a Mersenne prime ( and so p is prime
http://public.logica.com/~stepneys/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. 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 47 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 for some of this information Conway and Guy.

 13. The Prime Glossary: Multiply Perfect Welcome to the prime Glossary a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'multiply perfect.' Come explore a new primehttp://primes.utm.edu/glossary/page.php?sort=MultiplyPerfect

14. Landon Curt Noll's Prime Pages
Landon Curt Noll Landon Curt Noll s picture. prime numbers, Mersenneprimes, perfect numbers, etc. Mersenne prime Digits and Names
http://www.isthe.com/chongo/tech/math/prime/
##### Prime numbers, Mersenne Primes, Perfect Numbers, etc.
Mersenne Prime Digits and Names English Name of a Number Mathematics / Cryptology / Cryptography pages Technology pages ... chongo's home page
##### As a member of the Amdahl 6, I Co-Discovered these primes:

15. Math Forum - Ask Dr. Math Archives: Elementary Prime Numbers
and why are they prime numbers? prime numbers vs. prime Factors perfect numbers Basics, History 11/3/1996 What is the next perfect number after 28? prime and Composite numbers
http://mathforum.com/library/drmath/sets/elem_prime_numbers.html
 Ask Dr. Math Elementary Archive Dr. Math Home Elementary Middle School High School ... Dr. Math FAQ TOPICS This page: prime numbers Search Dr. Math See also the Dr. Math FAQ prime numbers Internet Library prime numbers ELEMENTARY Arithmetic addition subtraction multiplication ... Word Problems Browse Elementary Prime Numbers Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions: Prime factoring. Prime Numbers What are the prime numbers and why are they prime numbers? Prime Numbers vs. Prime Factors What is the difference between a prime number and a prime factor? 1 and 0 Is 1 or a prime number? 1 and 0: Prime or Composite? Is the number one a prime or a composite number? Why? What about zero? Abundant and Deficient Numbers What are abundant and deficient numbers, and what are they used for? Classifying the Number One My students have learned about prime and composite numbers. What classification fits the number 1? Composite Numbers What are Composite Numbers?

16. Math Forum - Ask Dr. Math Archives: Elementary Prime Numbers
perfect numbers Basics, History 11/3/1996 What is the next perfect number after28? prime and Composite numbers, Sieve of Eratosthenes 01/28/1997 I need
http://mathforum.org/library/drmath/sets/elem_prime_numbers.html
 Ask Dr. Math Elementary Archive Dr. Math Home Elementary Middle School High School ... Dr. Math FAQ TOPICS This page: prime numbers Search Dr. Math See also the Dr. Math FAQ prime numbers Internet Library prime numbers ELEMENTARY Arithmetic addition subtraction multiplication ... Word Problems Browse Elementary Prime Numbers Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions: Prime factoring. Prime Numbers What are the prime numbers and why are they prime numbers? Prime Numbers vs. Prime Factors What is the difference between a prime number and a prime factor? 1 and 0 Is 1 or a prime number? 1 and 0: Prime or Composite? Is the number one a prime or a composite number? Why? What about zero? Abundant and Deficient Numbers What are abundant and deficient numbers, and what are they used for? Classifying the Number One My students have learned about prime and composite numbers. What classification fits the number 1? Composite Numbers What are Composite Numbers?

 17. Computing Perfect(prime) Numbers 28=14*7*4*2*1=14+7+4+2+1. The definition of a perfect number can also be definedas, 2^(p1)*(2^p-1) where p is prime and 2^p-1 is a Mersenne primehttp://atlas.csd.net/~cgadd/knowbase/MATH0108.HTM

18. Trailpost 2: Properties Of Prime Numbers
Mersenne primes and perfect numbers. You can find perfect numbers by using theformula (2 n 1) * 2 n-1 , where n in 2 n - 1 makes a Mersenne prime.
##### What You Will Learn
• What prime numbers are. Mersenne Primes and perfect numbers. Sieve of Eratosthenes(a.k.a. Tex the Saskatchewan Ranger).
##### Learn It
Natural numbers are either prime or composite numbers. A prime number is a natural number that can only be divided by one and itself. In other words, it has exactly two factors. For example the number can only be divided by and , so is a prime number. has the factors and so is a composite number. Numbers that have more than two factors are composite numbers. Marin Mersenne 1588 - 1648
A special type of prime is called a Mersenne prime. Mersenne primes are calculated using the formula n . Any prime number calculated using the formula is a Mersenne prime. For example, the number is a Mersenne prime since and is a prime number. Note that the formula does not always produce prime numbers. For example, , which is not a prime number. Mersenne primes are very rare. In fact there are only known Mersenne primes as of January 1998. Mersenne primes can be used to calculate a special type of number called Perfect numbers. These are numbers whose factors when added together equal the number. For example the number

19. Ancient Greek Number Theory And Prime Numbers
was able to find that each of these numbers is of the form 2 n1 (2 n -1), where2 n -1 is prime. Euclid proved that all numbers of this form were perfect.
http://www.mlahanas.de/Greeks/Primes.htm
 Greeks and Prime Numbers T Aristotle Metaphysica A 5. 985 b 23 Pythagoras 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. The Greeks then went on to show that for a right triangle with sides [1:1:square root of two] that the hypotenuse of the triangle, the square root of two, in reduced form could not have either top or bottom number odd. Consequently, it cannot be a rational number.

20. Mathematics Enrichment Workshop: The Perfect Number Journey
Beginning with the number 1, and keep adding the powers of 2 (ie doubling the numbers),until you get a sum which is a prime number. A perfect number is then
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.

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