# Prime Number

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because only 1 and 5 divide it, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because it is the multiplicative identity.

The property of being prime is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and . Algorithms that are much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for primes of special forms, such as Mersenne primes. As of 2011, the largest known prime number has nearly 13 million decimal digits.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known useful formula that yields all of the prime numbers 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, proven at the end of the 19th century, 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.

Many questions around prime numbers remain open, such as Goldbach's conjecture, which asserts that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, which says that there are infinitely many pairs of primes whose difference is 2. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals.

### Other articles related to "prime number, number, numbers, prime numbers, primes":

Cohn's Irreducibility Criterion
... The criterion is often stated as follows If a prime number is expressed in base 10 as (where ) then the polynomial is irreducible in ... to other bases as follows Assume that is a natural number and is a polynomial such that ... If is a prime number then is irreducible in ...
Full Cycle - Example 2 (in C++)
... true if (displayOnce) { printf("Predicable Random Numbersn") displayOnce = false } printf("%d ", generated_number) } for(unsigned int iterator = 0 ...
Prime Number - In The Arts and Literature
... Prime numbers have influenced many artists and writers ... The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena" ... he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms the primes 41, 43, 47 and 53 appear in the third étude, "Neumes ...
Mutually Unbiased Bases - Existence Problem
... Let denote the maximal number of mutually unbiased bases in the d-dimensional Hilbert space Cd ... In general, if is the prime number decomposition of d, where then the maximal number of mutually unbiased bases which can be constructed satisfies It follows that if the dimension ... This can be seen in the previous equation, as the prime number decomposition of d simply is ...
Timeline Of Number Theory - 20th Century
... - Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem ... Aaiyangar Ramanujan develops over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions ... modular forms, divergent series, hypergeometric series and prime number theory ...

### Famous quotes containing the words number and/or prime:

A considerable percentage of the people we meet on the street are people who are empty inside, that is, they are actually already dead. It is fortunate for us that we do not see and do not know it. If we knew what a number of people are actually dead and what a number of these dead people govern our lives, we should go mad with horror.
George Gurdjieff (c. 1877–1949)

And this must be the prime of life . . . I blink,
As if at pain; for it is pain, to think
This pantomime
Of compensating act and counter-act,
Defeat and counterfeit, makes up, in fact,
My ablest time.
Philip Larkin (1922–1986)