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In mathematics, a prime number, or a prime, is a positive integer that is greater than 1 and divisible only by itself and 1. There are infinitely many prime numbers, although their frequency of occurrence decreases as they become larger. All prime numbers that are smaller than 100 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 (sequence A000040 in the OEIS).

The property of being a prime is called primality (http://en.wiktionary.org/wiki/primality), and the word prime is also used as an adjective (http://en.wiktionary.org/wiki/prime). Since two is the only even prime number, any prime number greater than two is called an odd prime (http://mathworld.wolfram.com/OddPrime.html).

The theory of prime numbers is an integral part of elementary number theory, and appears also in number theory in general. There are many open questions regarding prime numbers. Some of these have been formulated more than a century ago, but still remain unanswered. The idea of primality has been generalized or adapted by other fields of mathematics such as ring theory, knot theory, order theory and other fields of number theory besides elementary number theory.

Generalizations:

prime ideals in ring theory: http://en.wikipedia.org/wiki/Prime_ideal
prime ideals in order theory: http://en.wikipedia.org/wiki/Ideal_(order_theory)
prime elements: http://en.wikipedia.org/wiki/Prime_element#Divisibility.2C_prime_and_irreducible_elements
prime knots: http://en.wikipedia.org/wiki/Knot_theory
eisenstein prime: http://en.wikipedia.org/wiki/Eisenstein_prime
prime ring: http://en.wikipedia.org/wiki/Prime_ring ?

Side topics:

Ulam_spiral http://en.wikipedia.org/wiki/Ulam_spiral
Sacks_spiral http://en.wikipedia.org/wiki/Sacks_spiral

For research:

http://mathworld.wolfram.com/PrimeNumber.html