Search results
Results From The WOW.Com Content Network
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
But it does have to be in the same order; so, for example, 991 is still a minimal prime even though a subset of the digits can form the shorter prime 19 by changing the order. Similarly, there are exactly 32 composite numbers which have no shorter composite subsequence:
To change this template's initial visibility, the |state= parameter may be used: {{Prime number classes | state = collapsed}} will show the template collapsed, i.e. hidden apart from its title bar. {{Prime number classes | state = expanded}} will show the template expanded, i.e. fully visible.
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. [1] [2] Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit.
In recreational number theory, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all of its digits (in base 10) is larger than the number of primes obtainable in the same way for any smaller natural number. Primeval numbers were first described by Mike Keith.
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.
Not all Euclid numbers are prime. E 6 = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number. Every Euclid number is congruent to 3 modulo 4 since the primorial of which it is composed is twice the product of only odd primes and thus congruent to 2 modulo 4. This property implies that no Euclid number can be a square.
p = 2 k ± 1 or p = 4 k ± 3 for some natural number k. (OEIS: A122834) 2 p − 1 is prime (a Mersenne prime). (OEIS: A000043) (2 p + 1)/3 is prime (a Wagstaff prime). (OEIS: A000978) If p is an odd composite number, then 2 p − 1 and (2 p + 1)/3 are both composite. Therefore it is only necessary to test primes to verify the truth of the ...