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In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. There exists an infinitude of prime numbers, as demonstrated by Euclid, in about 300 B.C.. The first 30 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, 103, 107, 109, and 113 (sequence A000040 in OEIS); see the list of prime numbers for a longer list.
The property of being a prime is called primality, and the word prime is also used as an adjective. Since 2 is the only even prime number, the term odd prime refers to all prime numbers greater than 2.
The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers. Prime numbers have been the subject of intense research, yet some fundamental questions, such as the Riemann hypothesis and the Goldbach conjecture, have been unresolved for more than a century. The problem of modelling the distribution of prime numbers is a popular subject of investigation for number theorists: when looking at individual numbers, the primes seem to be randomly distributed, but the "global" distribution of primes follows well-defined laws.
The notion of prime number has been generalized in many different branches of mathematics.
In ring theory, a branch of abstract algebra, the term "prime element" has a specific meaning. Here, a non-zero, non-unit ring element a is defined to be prime if whenever a divides b c for ring elements b and c, then a divides at least one of b or c. With this meaning, the additive inverse of any prime number is also prime. In other words, when considering the set of integers ( ) as a ring, - 7 is a prime element. Without further specification, however, "prime number" always means a positive integer prime. Among rings of complex algebraic integers, Eisenstein primes, and Gaussian primes may also be of interest.
In knot theory, a prime knot is a knot which can not be disaggregated into a smaller prime knot.
In both the two above examples, the fundamental theorem of arithmetic (Every natural number can be 'uniquely' decomposed into a product of primes) does not apply.
The complete, up-to-date and editable article about Prime Number can be found at Wikipedia: Prime Number