41. Background On 2002 Fields And Nevanlinna Awardees An important question that often arises in number theory is whether, upondividing two prime numbers, the remainder is a perfect square. http://www.ams.org/ams/fields2002-background.html

Background on 2002 Fields and Nevanlinna Awardees Laurent Lafforgue Laurent Lafforgue has made an enormous advance in the so-called Langlands Program by proving the global Langlands correspondence for function fields. His work is characterized by formidable technical power, deep insight, and a tenacious, systematic approach. The Langlands Program, formulated by Robert P. Langlands for the first time in a famous letter to Andre Weil in 1967, is a set of far-reaching conjectures that make precise predictions about how certain disparate areas of mathematics might be connected. The influence of the Langlands Program has grown over the years, with each new advance hailed as an important achievement. One of the most spectacular confirmations of the Langlands Program came in the 1990s, when Andrew Wiles's proof of Fermat's Last Theorem, together with work by others, led to the solution of the Taniyama-Shimura-Weil Conjecture. This conjecture states that elliptic curves, which are geometric objects with deep arithmetic properties, have a close relationship to modular forms, which are highly periodic functions that originally emerged in a completely different context in mathematical analysis. The Langlands Program proposes a web of such relationships connecting Galois representations, which arise in number theory, and automorphic forms, which arise in analysis. The global Langlands correspondence proved by Lafforgue provides this complete understanding in the setting not of the ordinary numbers but of more abstract objects called function fields. One can think of a function field as consisting of quotients of polynomials; these quotients can be added, subtracted, multiplied, and divided just like the rational numbers. Lafforgue established, for any given function field, a precise link between the representations of its Galois groups and the automorphic forms associated with the field. He built on work of 1990 Fields Medalist Vladimir Drinfeld, who proved a special case of the Langlands correspondence in the 1970s. Lafforgue was the first to see how Drinfeld's work could be expanded to provide a complete picture of the Langlands correspondence in the function field case.

42. MathSteps: Grade 5: Prime Factors: 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/mathsteps/5/b/5.primefact.ask.html

Prime Factors 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 illustrating six as being 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?

43. Ivars Peterson's MathTrek -Appealing Numbers numberthe sum of its three proper divisors 1, 2, and 3. The next perfect numberis 2 n + 1 1, and 3 2 x 2 2n + 1 - 1 are all prime numbers (divisible only http://www.maa.org/mathland/mathtrek_2_26_01.html

Ivars Peterson's MathTrek February 26, 2001 Appealing Numbers The ancient Greeks, especially the Pythagoreans, were fascinated by whole numbers. They defined as "perfect" numbers those equal to the sum of their parts (or proper divisors, including 1). For example, 6 is the smallest perfect numberthe sum of its three proper divisors: 1, 2, and 3. The next perfect number is 28, which is the sum of 1, 2, 4, 7, and 14. The Pythagoreans were also interested in what we now call amicable numberspairs in which each number is the sum of the proper divisors of the other. The smallest such pair is 220 and 284. The number 220 is evenly divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which add up to 284; and 284 is evenly divisible by 1, 2, 4, 71, and 142, which add up to 220.The Pythagorean brotherhood regarded 220 and 284 as numerical symbols of friendship. About 1,500 years later, in the ninth century A.D., Arab scholar Thabit ibn Qurra (826-901) discovered a remarkable formula for generating amicable numbers: If n is a positive integer such that the numbers 3 x 2 n - 1, 3 x 2

44. Solution For /arithmetic/consecutive.product 1. Then n(n^2 1) = k^2. But n and (n^2 - 1) are relatively prime. Therefore n^2- 1 is a perfect square, which is a contradiction. Four consecutive numbers http://rec-puzzles.org/sol.pl/arithmetic/consecutive.product

Solution to the /arithmetic/consecutive.product problem Three consecutive numbers: If a and b are relatively prime, and ab is a square, then a and b are squares. (This is left as an exercise.) Four consecutive numbers: n(n + 1)(n + 2)(n + 3) = (n^2 + 3n + 1)^2 - 1 Five consecutive numbers: Assume the product is a integer square, call it m. The prime factorization of m must have even numbers of each prime factor. Each of the consecutive naturals is one of: 1) a perfect square 2) 2 times a perfect square 3) 3 times a perfect square 4) 6 times a perfect square. By the shoe box principle, two of the five consecutive numbers must fall into the same category. If there are two perfect squares, then their difference being less than five limits their values to be 1 and 4. (0 is not a natural number, so and 1 and and 4 cannot be the perfect squares.) But 1*2*3*4*5=120!=x*x where x is an integer. If there are two numbers that are 2 times a perfect square, then their difference being less than five implies that the perfect squares (which are multiplied by 2) are less than 3 apart, and no two natural squares differ by only 1 or 2. A similar argument holds for two numbers which are 3 times a perfect square.

45. Consecutive.product product of three or more consecutive positive integers cannot be a perfect square.Solution Three consecutive numbers If a and b are relatively prime, and ab http://rec-puzzles.org/new/sol.pl/arithmetic/consecutive.product

Consecutive.product Prove that the product of three or more consecutive positive integers cannot be a perfect square. Solution: E-mail Puzzle index Arlet's home page Linux ... Apache

46. Spermatikos Logos #3 The four corners of the board are (in no particular order) a prime number, a perfectnumber, a perfect square, and one of the numbers mentioned in clue 1. http://www.mathnews.uwaterloo.ca/Issues/mn7805/logos3.php

In this issue: Front Page Information mast HEAD ISSN 0705-0410 look ... Careers in Math and CS MathSoc Space Prez Sez Distinguished Teacher Awards Damn, just take Tylenol or something Fed Speak ... ActSci Club News and Stuff Opinions and Comments Devastation in Nicaragua Cynic's Corner The National Post-Mortem Special Section - Square One Phone Call Man on the Street Spade Parade Backstage With Blackstone ... MathNet Humour prof QUOTES 2nd Annual Ross Willard Appreciation Day! Games and Puzzles math NEWSquiz #4 grid COMMENTS ... Spermatikos Logos #3 Spermatikos Logos #3 Hey everyone, I'm back again. Hope you've all survived as well. I only received two submissions this week, so I guess I'm not the only one who's been attacked by midterm stress. Thanks to both Lisa Harpur and Greg "Hologrami" Taylor for submitting solutions to Logos #2. Greg got the correct solution the way I figured it. Lisa actually found a solution that was different from mine, but still correct (forgive me, I was tired and didn't check it properly when I made it). By random draw, Greg gets the prize. Go pick it up in the MathSoc office. Submissions for Logos #3 are due November 16 th , at 6:30pm in the BLACK BOX ... Where's the BLACK BOX ? Good question. It's hiding in the depths of the

47. Number Theory - Wikipedia the Euclidean algorithm to compute greatest common divisors, factorization of integersinto prime numbers, investigation of perfect numbers and congruences http://www.wikipedia.org/wiki/Number_theory

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Other languages: Esperanto Nederlands Polski Number theory From Wikipedia, the free encyclopedia. Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers . More generally, it has come to be concerned with a wider class of problems that are "easily understood by laymen" - this expansion has occurred as the techniques are used to attack wider varieties of problems. Number theory may be subdivided into several fields according to the methods used and the questions investigated. In elementary number theory , the integers are studied without use of techniques from other mathematical fields. Questions of divisibility, the Euclidean algorithm to compute greatest common divisors , factorization of integers into prime numbers , investigation of perfect numbers and congruences belong here. Typical statements are

48. Numerology - Finding Your Perfect Soulmate Or Business Partner - Online Press Ki Book Title Finding Your perfect SOULMATE or Business Partner, Fun Facts About Oddnumbers and prime numbers turn up regularly in many different religious texts http://www.talion.com/soulmates.htm

Contact: David E. Smith Tel: E-mail: destech@pcrealm.net Online Press Kit Fun Fact Sheet Talk Show Topics Author Bio Who Says Arithmetic Isn't Fun? Simple Addition Reveals Personality Secrets NEWS RELEASE Numerology Fun Finding Your Perfect Soulmate Well, he's not timid. "How often do you come across books that offer no opinions, just facts ? When you're looking for a relationship, do you have time for fiction?" says multi-published author David Smith. A refreshing change from ponderous therapists and smug self-help gurus, Smith claims that all you need to know is a birthday to reveal another person's innermost secrets. HERE ARE TOPICS THAT YOUR READERS AND LISTENERS REALLY TUNE INTO: Book Title: Finding Your Perfect SOULMATE or Business Partner Buy the book Profiling your EX Smith explains why some chemistry doesn't work, and shows how to add up numbers to profile your ex-husband, ex-wife, ex-girlfriend. A fun topic that everyone can relate to! All about celebrity troublemakers and high-profile marriages You provide the celebrity birthday, Smith tells you why they got in trouble, or predicts possible results of the marriage. Examples for news reporters supplied below see

49. PlanetMath: Quadratic Sieve and, the zero vector in signals a perfect square a factor base such that and for eachodd prime in , is If can be completely factored by numbers in , then it is http://planetmath.org/encyclopedia/QuadraticSieve.html

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports information Docs Classification News Legalese ... TODO List quadratic sieve (Algorithm) Algorithm To factor a number using the quadratic sieve, one seeks two numbers and which are not congruent modulo with not congruent to modulo but have . If two such numbers are found, one can then say that . Then, and must have non-trivial factors in common with The quadratic sieve method of factoring depends upon being able to create a set of numbers whose factorization can be expressed as a product of pre-chosen primes . These factorizations are recorded as vectors of the exponents . Once enough vectors are collected to form a set which contains a linear dependence, this linear dependence is exploited to find two squares which are equivalent modulo To accomplish this, the quadratic sieve method uses a set of prime numbers called a factor

50. Mersenne Primes: History, Theorems And Lists Contents include some historical notes, discussions about perfect numbers and different theorems, and a table of known Mersenne primes. http://www.utm.edu/research/primes/mersenne.shtml

/export/home/users/staff/math2/mathMersenne Primes: History, Theorems and Lists 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

51. Math Forum - Ask Dr. Math perfect numbers can be formed every time a prime of a certain typeis found. Just last November a new prime of this type was found. http://mathforum.org/library/drmath/view/57043.html

Associated Topics Dr. Math Home Search Dr. Math Perfect Numbers Date: 08/14/97 at 18:52:01 From: Insa Thiele Subject: Perfect Numbers What is the highest perfect number that has been found? How many perfect numbers are there? What are they? I already know that 6 and 28 are perfect numbers, and I would like to know the other ones. http://www.utm.edu/research/primes/index.html That last part tells us that whenever a new Mersenne prime is found, a new perfect number also is found. We use the formula 2^(p-1) * (2^ p - 1) to do this. Check out that website to find out more about primes and perfect numbers. Have fun! -Doctor Terrel, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Associated Topics Elementary Definitions Elementary Number Sense/About Numbers Elementary Prime Numbers Middle School Definitions ... 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.

52. All Even Perfect Numbers Are A Power Of Two Times A Mersenne Prime FILL THIS IN! http://www.utm.edu/research/primes/notes/proofs/EvenPerfect.html

All even perfect numbers are a power of two times a Mersenne prime (from the Prime Pages ' list of proofs Home Search Site Largest ... Submit primes Over 2300 years ago Euclid proved that If 2 k -1 is a prime number (it would be a Mersenne prime ), then 2 k k -1) is a perfect number . A few hundred years ago Euler proved the converse (that ever even perfect number has this form). It is still unknown if there are any odd perfect numbers (but if there are, they are large and have many prime factors). Theorem: If 2 k -1 is a prime number, then 2 k k -1) is a perfect number and every even perfect number has this form. Proof: Suppose first that p k -1 is a prime number, and set n k k -1). To show n is perfect we need only show sigma( n n . Since sigma is multiplicative and sigma( p p k , we know sigma( n ) = sigma(2 k sigma( p k k n This shows that n is a perfect number. On the other hand, suppose n is any even perfect number and write n as 2 k m where m is an odd integer and k Again sigma is multiplicative so sigma(2 k m ) = sigma(2 k sigma( m k sigma( m Since n is perfect we also know that sigma( n n k m.

53. Large Prime Numbers do know, however, that all perfect numbers have a direct relationship to Mersenneprimes. The new perfect number generated with the new Mersenne prime is the 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 A Large Prime Number A Large Perfect Number The discovery team: David Slowinski ... : right 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

54. Prime Curios!: 9 contains 3021 digits. Williams. The sum of the first 9 consecutiveprime numbers = 10 2 , a perfect square. If odd perfect numbers http://primes.utm.edu/curios/page.php?short=9

55. Mersenne Prime Numbers This means that the quest for perfect numbers is reduced to the quest for primesof the form 2^m 1 A Mersenne prime is such a number Mp, where p is prime. http://www.resort.com/~banshee/Info/mersenne.html

Mersenne Primes Marin Mersenne (1588-1648) was a Franciscan friar who lived most of his life in Parisian cloisters. He was the author of Cognitata Physico-Mathematica which stated without proof that M p is prime for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 and for no other primes p for p < 257. It took over 300 years to totally settle this claim made by Mersenne. The final work done in 1947 showed that Mersenne made five errors in his work (namely that M is prime, M is composite, M is prime, M is prime, and M is composite). Besides his famous statement about primes of the form M p , Mersenne contributed to the development of number theory through his extensive correspondence with many mathematicians, including Fermat. Mersenne effectively served as a clearing house and a disseminator of new mathematical ideas in the 17th century. Kenneth Rosen, Elementary Number Theory; Addison Wesley The concept of a Mersenne Prime is evolved from that of a perfect number . A perfect number is an integer for which the sum of its divisors is twice the number. For example: (6) = 1 + 2 + 3 + 6 = 12 = 2*6 thus 6 is a perfect number.

56. Mathematics Enrichment Workshop: The Perfect Number Journey Mersenne. So the search for perfect numbers became the search for more Mersenneprimes, ie prime numbers of the form 2 n 1. But this turned out to be a very http://home.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.

57. Factoids > Perfect Number it is divisible by a prime component greater that 10 20. Exhaustive computersearch has shown that there are no odd perfect numbers less than 10 300 . http://www-users.cs.york.ac.uk/~susan/cyc/p/perfect.htm

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 29 prime factors (not necessarily distinct) it has a 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.

58. Mathematics Archives - Numbers museum. Includes information on various topics as perfect numbers, primenumbers, Pythagorean triples, pi, and Fermat's Last Theorem. http://archives.math.utk.edu/subjects/numbers.html

Numbers Facts about the number 17: 17 in history, computing, astronomy, etc. The 47 Society The 47 Society is an international interest-group that follows the occurence and recurrence of the quintessential random number: 47. Many suspect that the coinciential nature of 47 carries some mystical, metaphysical and/or scientific significance. What is special about the number 73939133? Aesthetics of the Prime Sequence Hear and see the prime numbers! A Common Book of p The number p has been the subject of a great deal of mathematical (and popular) folklore. It's been worshipped, maligned, and misunderstood. Overestimated, underestimated, and legislated. Of interest to scholars, crackpots, and everyday people. Continued Fractions A senior Honor's Project at Calvin College by Adam Van Tuyl which gives the history, theory, applications and bibliography on the thery of continued fractions. In the section on applications there are a number of interactive programs that convert rationals (or quadratic irrationals) into a simple continued fraction, as well as the converse. Data Powers of Ten A petabyte?

59. A Prime Of Record Size! 2^1257787-1 Slowinski noted that with the discovery of the new prime number, a new perfect addedtogether, equal 6. Mathematicians don't know how many perfect numbers exist http://www.utm.edu/research/primes/notes/1257787.html

A Prime of Record Size! 2 (Another of the Prime Pages ' resources Home Search Site Largest Finding ... Submit primes This is no longer the largest known prime. Click here for information on and new records. On 3 September 1996 Cray Research announced that once again Slowinski and Gage have set a new record by finding the prime which has 378,632 digits. This is the largest known prime by farthe next largest has "only" 258,716 digits. It is also the 34th Mersenne prime to be discovered (though it might not be the 34th in order of size as the entire region below it has not been checked). Looking at the graph of the largest known prime by year, we see this prime is roughly the size record we'd expect to find this year. The proof of this 378,632 digit number's primality (using the traditional Lucas-Lehmer test ) took about 6 hours on one CPU of a CRAY T94 super computer. Richard Crandall and others independently verified the primality. The first and the most interesting of these was George Woltman who was 90% of the way through that very number when asked to check the result on April 15th. According to the San Jose Mercury News

60. The Prime-perfect Numbers The primeperfect numbers. A Problem Proposal. The sequence a(n) of prime-perfectnumbers begins. 30, 60, 70, 84, 90, 105, 120, 140, . http://www.geocities.com/SoHo/Exhibit/8033/primeperfect/primeperfect.html

The Prime-perfect Numbers A Problem Proposal Consider the numbers n with at least two prime factors, the sum of whose prime factors divides n. In obvious analogy to the perfect numbers, I call these the prime-perfect numbers . (Clearly, the sum of the prime factors of n is almost always less than n, so to require equality of n to the sum, as in the definition of perfect numbers, will be fruitless.) The sequence a(n) of prime-perfect numbers begins (Note: This is EIS Sequence A066031 .) The numbers k with just one prime factor have been excluded from the sequence since they trivially satisfy the requirement that the sum of the prime factors of k divide k. The exclusion thus highlights the interesting numbers satisfying the requirement. It is easy to see that if p is a prime factor of the prime-perfect number n, then p m n is also prime-perfect for any m. Hence, a is an infinite sequence. But what about the elementary (or primitive ) terms of a, that is, terms which are not multiples of any previous terms? For example, 84 is elementary, since it is not a multiple of the preceding terms, 30, 60, 70. But 90 is not elementary because 90 is a multiple of 30. Are there also infinitely many elementary terms? A related problem: Find an expression generating elementary prime-perfect numbers.

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