Perfect Number. A Study of Perfect Numbers and Related Topics, With Special Emphasis on the Search for an Odd Perfect Number By J.W.
Gaberdiel Euclidub@hotmail.com Table of Contents 1 Abstract 2 A Quick Introduction To Terminology 3 How I Became Involved With Perfect Numbers 4 My Attention Turns To Odd Perfect Numbers 5 The Sigma and Tau Functions 6 Applying The Sigma Function To The Study Of Even Perfect Numbers 7 The Search For Mersenne Primes 8 Return To The Odd Perfect Number Problem 9 Curiousities Relating to Perfect Numbers and Related Topics 10 A Link To A List Of The Perfect Numbers Found To Date 1 Abstract The following questions will be investigated throughout the course of this project: 1) What is already known about perfect numbers?
2) All known perfect numbers are even. 3) Define the sigma function to yield the sum of the proper divisors of a given number. 2 A Quick Introduction To Terminology A "perfect" number is a number whose divisors sum to twice itself. Prime Numbers - Advanced. Prime Numbers A Prime Number can be divided evenly only by 1 or itself.
And it must be a whole number greater than 1. Example: 2, 3, 5, 7, 11, etc. Twin Primes A pair of prime numbers that differ by 2 (successive odd numbers that are both Prime numbers). Examples: (3,5), (5,7), (11,13), ... It is not known whether the set of twin prime numbers ends or not. Project Euler. Problem 70. Semiprime. In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers.
The semiprimes less than 100 are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, and 95. (sequence A001358 in OEIS). By definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its only factors are 1, 2, 13, and 26. Properties[edit] The total number of prime factors Ω(n) for a semiprime n is two, by definition. Non-divisible by primes is semiprime. For a semiprime n = pq the value of Euler's totient function (the number of positive integers less than or equal to n that are relatively prime to n) is particularly simple when p and q are distinct: φ(n) = (p − 1) (q − 1) = p q − (p + q) + 1 = n − (p + q) + 1.
If otherwise p and q are the same, (sequence A117543 in OEIS) See also[edit] Euler's Totient Function and Euler's Theorem. Coprime. In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime (also spelled co-prime)[1] if the only positive integer that evenly divides both of them is 1.
That is, the only common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1.[2] The numerator and denominator of a reduced fraction are coprime. In addition to and the notation is sometimes used to indicate that a and b are relatively prime.[3] For example, 14 and 15 are coprime, being commonly divisible by only 1, but 14 and 21 are not, because they are both divisible by 7. A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to a positive integer n, between 1 and n, is given by Euler's totient function (or Euler's phi function) φ(n).
Properties[edit] Figure 1. There are a number of conditions which are equivalent to a and b being coprime: Coprimality in sets[edit] Probabilities[edit]