Prime number

In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. Or for short: A prime number is a natural number with exactly two natural divisors. A natural number that is greater than one and is not a prime is called a composite number. The numbers zero and one are neither prime nor composite. The property of being a prime is called primality. Prime numbers are of fundamental importance in number theory.

Some special types of primes

A prime p is called primorial or prime-factorial if it has the form p = Π(n) ± 1 for some number n, where Π(n) stands for the product 2 · 3 · 5 · 7 · 11 · ... of all the primes ≤ n. A prime is called factorial if it is of the form n! ± 1. The first factorial primes are:

Related Topics:
Primorial - Π(''n'') - Factorial - ''n''!

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:n! − 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,... {{OEIS|id=A002982}}

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:n! + 1 is prime for n = 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154... {{OEIS|id=A002981}}

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The largest known primorial prime is Π(24029) + 1, found by Caldwell in 1993. The largest known factorial prime is 3610! − 1 . It is not known if there are infinitely many primorial or factorial primes.

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Primes of the form 2n − 1 are known as Mersenne primes, while primes of the form 2^{2^n} + 1 are known as Fermat primes. Prime numbers p where 2p + 1 is also prime are known as Sophie Germain primes. Other special types of prime numbers include Wieferich primes, Wilson primes, Wall-Sun-Sun primes, Wolstenholme primes, unique primes, Newman-Shanks-Williams primes (NSW primes), Smarandache-Wellin primes, Wagstaff primes and supersingular primes.

Related Topics:
Mersenne prime - Fermat prime - Sophie Germain prime - Wieferich prime - Wilson prime - Wall-Sun-Sun prime - Wolstenholme prime - Unique prime - Newman-Shanks-Williams prime - Smarandache-Wellin prime - Wagstaff prime - Supersingular prime

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The base-ten digit sequence of a prime can be a palindrome, as in the prime 1031512 + 9700079 · 1015753 + 1.

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