In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The first thirty-four 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, 113, 127, 131, 137, 139. (sequence A000040 in OEIS)

CLICK ON THE LINKS BELOW TO SEE THE FIRST 50 MILLION PRIMES:

List of the first Fifty Million Prime Numbers

List	First Prime on list	Last Prime on list
1,000,000	2	15,485,863
2,000,000	15,485,867	32,452,843
3,000,000	32,452,867	49,979,687
4,000,000	49,979,693	67,867,967
5,000,000	67,867,979	86,028,121
6,000,000	86,028,157	104,395,301
7,000,000	104,395,303	122,949,823
8,000,000	122,949,829	141,650,939
9,000,000	141,650,963	160,481,183
10,000,000	160,481,219	179,424,673
11,000,000	179,424,691	198,491,317
12,000,000	198,491,329	217,645,177
13,000,000	217,645,199	236,887,691
14,000,000	236,887,699	256,203,161
15,000,000	256,203,221	275,604,541
16,000,000	275,604,547	295,075,147
17,000,000	295,075,153	314,606,869
18,000,000	314,606,891	334,214,459
19,000,000	334,214,467	353,868,013
20,000,000	353,868,019	373,587,883
21,000,000	373,587,911	393,342,739
22,000,000	393,342,743	413,158,511
23,000,000	413,158,523	433,024,223
24,000,000	433,024,253	452,930,459
25,000,000	452,930,477	472,882,027
26,000,000	472,882,049	492,876,847
27,000,000	492,876,863	512,927,357
28,000,000	512,927,377	533,000,389
29,000,000	533,000,401	553,105,243
30,000,000	553,105,253	573,259,391
31,000,000	573,259,433	593,441,843
32,000,000	593,441,861	613,651,349
33,000,000	613,651,369	633,910,099
34,000,000	633,910,111	654,188,383
35,000,000	654,188,429	674,506,081
36,000,000	674,506,111	694,847,533
37,000,000	694,847,539	715,225,739
38,000,000	715,225,741	735,632,791
39,000,000	735,632,797	756,065,159
40,000,000	756,065,179	776,531,401
41,000,000	776,531,419	797,003,413
42,000,000	797,003,437	817,504,243
43,000,000	817,504,253	838,041,641
44,000,000	838,041,647	858,599,503
45,000,000	858,599,509	879,190,747
46,000,000	879,190,841	899,809,343
47,000,000	899,809,363	920,419,813
48,000,000	920,419,823	941,083,981
49,000,000	941,083,987	961,748,927
50,000,000	961,748,941	982,451,653

The number one is by definition not a prime number; see the discussion below under Primality of one.

The property of being a prime is called primality, and the word prime is also used as an adjective. Since two is the only even prime number, the term odd prime refers to any prime number greater than two.

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 bc 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 written as the knot sum of two lesser nontrivial knots.