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Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1.
A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. ... 20 p − 1 ≡ 1 (mod p 2): 281, 46457, ...
A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially, let p equal 2, the smallest prime number.
Fermat also investigated the primality of the Fermat numbers + , [20] and Marin Mersenne studied the Mersenne primes, prime numbers of the form with itself a prime. [21] Christian Goldbach formulated Goldbach's conjecture , that every even number is the sum of two primes, in a 1742 letter to Euler. [ 22 ]
Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers. A composite number has Ω(n) > 1.
A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted g n or g(p n) is the difference between the (n + 1)-st and the n-th prime numbers, i.e. = +. We have g 1 = 1, g 2 = g 3 = 2, and g 4 = 4. The sequence (g n) of prime gaps has been extensively studied; however, many questions and conjectures remain ...
The PrimePages is a website about prime numbers originally created by Chris Caldwell at the University of Tennessee at Martin [2] who maintained it from 1994 to 2023.. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms.
A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.
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