This definition is used throughout the Wikipedia. See the Generalizations section, below, for another definition in common use. The property of being a prime is called primality. If a number greater than one is not a prime number it is called a composite number.

Representing natural numbers as products of primes
An important result is the fundamental theorem of arithmetic, which states that every natural number can be written as a product of primes, and in essentially only one way. Primes are thus the "basic building blocks" of the natural numbers. For example, we can write
How many prime numbers are there?
There are infinitely many prime numbers. The oldest known proof for this statement is a reductio ad absurdum dating to the Greek mathematician Euclid. The argument goes as follows:
Other mathematicians have given their own proofs; for example Kummer[?]'s is particularly elegant and Furstenberg provides one using topological terms.
Even though the total number of primes is infinite, one could still ask "how many primes are there below 100,000" or "How likely is a random 100digit number to be prime?" Questions like these are answered by the prime number theorem.
A simple but inefficient way to compute a list of all the prime numbers up to a given limit is the algorithm called the "Sieve of Eratosthenes".
In practice though, to check whether a given large number (say, up to a few thousand digits) is prime, one uses primality tests which are probabilistic tests. These typically pick a random number called a "witness" and check some formula involving the witness and the potential prime N. After several iterations, they declare N to be "definitely composite" or "probably prime". These tests are not perfect. For a given test, there may be some composite numbers that will be declared "probably prime" no matter what witness is chosen. Such numbers are called pseudoprimes for that test. Here's a description of the Fermat primality test.
A new algorithm which determines whether a given number N is prime and which uses time polynomial in the number of digits of N has recently been described.
There are many open questions about prime numbers. For example:
The largest known prime is 2^{13466917}1 (this number is 4,053,946 digits long). It is the 39th Mersenne prime M_{13466917} found by a collaborative effort known as GIMPS on November 14, 2001 and announced in early December 2001 after double checking. The next largest known is 2^{6972593}1, (this number is 2,098,960 digits long), also a Mersenne prime, found by GIMPS on June 1, 1999. All largest known primes are Mersenne primes, because there exists a particularly fast primality test for numbers of this form, the LucasLehmer test[?].
Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semirandom binary data, converting it to a number n, multiplying it by 256^{k} for some positive integer k, and searching for possible primes within the interval [256^{k}n + 1, 256^{k}(n + 1)  1]. In fact, as a publicity stunt against the Digital Millennium Copyright Act and other WIPO Copyright Treaty implementations, some people have applied this to various forms of DeCSS code, creating the set of illegal prime numbers. Such numbers, when converted to binary and executed as a computer program, perform acts encumbered by applicable law in one or more jurisdictions.
Extremely large prime numbers (that is, greater than 10^{100}) make public key cryptography possible. Primes are also used for hash tables and pseudorandom number generators.
A prime p is called primorial or primefactorial if it has the form p = Π(n) ± 1 for some number n, where Π(n) stands for the product 2 · 3 · 5 · 7 · 11 · ... of all the primes ≤ n. A prime is called factorial if it is of the form n! ± 1. The first factorial primes are:
The largest known primorial prime is Π(24029) + 1, found by Caldwell in 1993. The largest known factorial prime is 3610!  1 [Caldwell, 1993]. It is not known if there are infinitely many primorial or factorial primes.
Primes of the form 2^{2n}+1 are known as Fermat primes.
The baseten digit sequence of a prime can be a palindrome, as in the prime 10^{31512} + 9700079 · 10^{15753} + 1. Or 11. Or 2. This is a trivial statement.
The gap between the nth prime p_{n} and the n+1st prime p_{n+1} is defined to be the number of composite numbers between them, i.e. g_{n} = p_{n+1}  p_{n}  1 (slightly different definitions are sometimes used). We have g_{1} = 0 and g_{2} = 1. The sequence (g_{n}) of prime gaps has been extensively studied. One can show that gaps get arbitrarily large, i.e. for any natural number N, there is an index n with g_{n} > N. On the other hand, for any positive real number ε, there exists a start index n_{0} such that g_{n} < ε · p_{n} for all n > n_{0}.
We say that g_{n} is a maximal gap if g_{m} < g_{n} for all m < n. The largest known maximal gap is 1131, found by T. Nicely and B. Nyman in 1999. It is the 64th smallest maximal gap, and it occurs after the prime 1693182318746371.
Formulas yielding prime numbers
The curious polynomial f(n) = n^{2}  n + 41 yields primes for n = 0,..., 40, but f(41) is composite. There is no polynomial which only yields prime numbers in this fashion.
There exists a polynomial in 26 variables with integer coefficients such that, if you plug in integers for the variables, the set of positive values is equal to the set of prime numbers. However, for some values of the variables, the result is negative and can then be composite.
The following function yields all the primes, and only primes, for natural numbers n:
Using the floor function [x] (defined to be the largest integer less than or equal to the real number x), one can construct several formulas for the nth prime. These formulas are also based on Wilson's theorem and have little practical value: the methods mentioned above under "Finding prime numbers" are much more efficient.
Define
or, alternatively,
These definitions are equivalent; π(m) is the number of primes less than or equal to m. The nth prime number p_{n} can then be written as
Another approach does not use factorials and Wilson's theorem, but also heavily employs the floor function (S. M. Ruiz 2000): first define
and then
The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. The set {2,3,5,7,11,...} is the primes over the natural numbers. The set {...11,7,5,3,2,2,3,5,7,11,...} is the primes over the integers. When the word prime or prime number is used without qualification in the Wikipedia, it means a prime natural number. This is a common definition, but some mathematics dictionaries define it instead to mean a prime integer.
In number theory itself, one talks of pseudoprimes, integers which, by virtue of having passed a certain test, are considered probable primes but are in fact composite (such as Carmichael numbers). To model some of the behavior of prime numbers, one defines prime and irreducible polynomials. More generally, one can define prime and irreducible elements in every integral domain. Prime ideals are an important tool and object of study in commutative algebra and algebraic geometry.
_{n}  p_{n}  Binary  1/p  Len 
1  2  10  0·5  1 
2  3  11  0·3...  1 
3  5  101  0·2  1 
4  7  111  0·142857...  6 
5  11  1011  0·09...  2 
6  13  1101  0·076923...  6 
7  17  10001  0·0588235294117647...  16 
8  19  10011  0·052631578947368421...  18 
9  23  10111  0·0434782608695652173913...  22 
10  29  11101  0·0344827586206896551724137 931...  28 
11  31  11111  0·032258064516129...  15 
12  37  100101  0·027...  3 
13  41  101001  0·02439...  5 
14  43  101011  0·023255813953488372093...  21 
15  47  101111  0·0212765957446808510638297 872340425531914893617...  46 
16  53  110101  0·0188679245283...  13 
17  59  111011  0·0169491525423728813559322 0338983050847457627118644 06779661...  58 
18  61  111101  0·0163934426229508196721311 4754098360655737704918032 7868852459...  60 
19  67  1000011  0·0149253731343283582089552 23880597...  33 
20  71  1000111  0·0140845070422535211267605 6338028169...  35 
21  73  1001001  0·01369863...  8 
22  79  1001111  0·0126582278481...  13 
23  83  1010011  0·0120481927710843373493975 9036144578313253...  41 
24  89  1011001  0·0112359550561797752808988 7640449438202247191...  44 
25  97  1100001  0·0103092783505154639175257 7319587628865979381443298 9690721649484536082474226 804123711340206185567...  96 
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