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 evenly 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 one can include arbitrarily-many instances of 1
in any factorization, e.g., 3, 1 × 3, 1 × 1 × 3, etc. are all valid
factorizations of 3.
The property of being prime (or not) 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 \sqrt{n}. Algorithms much more efficient
than trial division have been devised to test the primality of large numbers.
Particularly fast methods are available for numbers of special forms, such as
Mersenne numbers. As of February 2013, the largest known prime number has
17,425,170 decimal digits.
There are infinitely many primes, as demonstrated by
Euclid around 300 BC. There is no known useful formula that sets apart all of
the prime numbers from composites. However, the distribution of primes, that is
to say, the statistical behaviour of primes in the large, can be modelled. 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
to the logarithm of n.
Many questions around prime numbers remain open, such
as Goldbach's conjecture (that every even integer greater than 2 can be
expressed as the sum of two primes), and the twin prime conjecture (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.
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