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A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. The smallest twentyfive prime numbers (all the prime numbers under 100) are:
An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC,^{[2]} although the density of prime numbers within natural numbers is 0. 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 different 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. The simplest trial division method tests whether n is a multiple of an integer m between 2 and . If n is a multiple of any of these integers then it is a composite number, and so not prime; if it is not a multiple of any of these integers then it is prime. As this method requires up to trial divisions, it is only suitable for relatively small values of n. More sophisticated algorithms, which are much more efficient than trial division, have been devised to test the primality of large numbers.
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 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 proven since 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, there remain some open questions around prime numbers which can be stated simply. For example, Goldbach's conjecture, which asserts that any even natural number bigger than two is the sum of two primes, and 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 publickey 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 2010, the largest known prime number has about 13 million decimal digits.^{[3]}
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