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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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