In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. Any natural number greater than 1 has the two divisors 1 and itself, so a prime number has no other divisors.
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.
Another means of understanding primes is to represent numbers geometrically in terms of their prime factorization. The number 6 would be built with a side-by-side pair of 3's (with each 3 being built as a row of three unit squares). The number 9 would be a square built with three 3's placed side-by-side. The number 8 would be a cube built with two 2's stacked on top of another pair of 2's (with each 2 being built as a row of two unit cubes). The number 27 would be a cube similar to 8, except built with three layers of 9 instead of two layers of 4. Building a sequence of such numbers provides a clear illustration that primes are all one dimensional, having length but no width. Composites are two dimensional (having length and width) or higher. The number 16 is the first four-dimensional representation, being a 2-by-2-by-2-by-2 hypercube.