In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101.[1]
An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC.[2] The number 1 is by definition not a prime number. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any nonzero natural number n can be factored into primes, written as a product of primes or powers of primes (including the empty product of factors for 1). Moreover, this factorization is unique except for a possible reordering of the factors.
The property of being prime is called primality. Verifying the primality of a given number n can be done by trial division, that is to say dividing n by all smaller numbers m, thereby checking whether n is a multiple of m, and therefore not prime but a composite. For big primes, increasingly sophisticated algorithms which are faster than that technique have been devised.
There is no known formula yielding all primes and no composites. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large can be modeled. The first result in that direction is the prime number theorem which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or the logarithm of n. This statement has been proved at the end of the 19th century. The unproven Riemann hypothesis dating from 1859 implies a refined statement concerning the distribution of primes.
Despite being intensely studied, many fundamental questions around prime numbers remain open. For example, Goldbach's conjecture which asserts that any even natural number bigger than two is the sum of two primes, or the twin prime conjecture which says that there are infinitely many twin primes (pairs of primes whose difference is two), have been unresolved for more than a century, notwithstanding the simplicity of their statements.
Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, notably the notion of prime ideals.
Primes are applied in several routines in information technology, such as public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Searching for big primes, often using distributed computing, has stimulated studying special types of primes, chiefly Mersenne primes whose primality is comparably quick to decide. As of 2009, the largest known prime has about 13 million decimal digits.[3]
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Prime Numbers Poster 11 x 17 $8 order online
Dustin Kirkland
Sun, 21 Jun 2009 15:27:00 GM
The network status item is still the most expensive notification, but even it has improved. With the network item turned off, though, there should be no noticeable stutter in your Byobu sessions. . Prime numbers. for the win! :-Dustin ...
