A prime number is a number which has precisely two positive factors: one and itself. Prime numbers are generally restricted to the domain of natural numbers. Note that the set of prime numbers excludes one, as one only has one positive factor. In any other algebraic structure with a multiplication operation, and a concept of primes, particularly rings with unity, the multiplicative identity is excluded as prime.

One very important theorem relating to prime numbers is that there are infinitely many primes. This can be proven by contradiction, called Euclid's proof:

If we assume that there are only a finite number of primes, then we can list them, ${\displaystyle p_1, p_2,...,p_k}$. If we multiply all primes together and add one, we get ${\displaystyle P = p_1 * p_2 * ... * p_k + 1}$. Because 1 is only divisible by itself, this new number P is not divisible by any of the primes and is therefore itself a prime number. Therefore, the assumption is false and there must be an infinite number of primes.

Currently, the largest known prime number is ${\displaystyle 2^{136,279,841}-1}$. It has 41,024,320 digits and was discovered in October 12, 2024.^[1]

Definition

A natural number (1, 2, 3, 4, 5, 6 and so on) is called prime if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called composite numbers.^[2] In other words, ${\displaystyle n}$ is prime if ⁠⁠${\displaystyle n}$ items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange ⁠⁠${\displaystyle n}$ dots into a rectangular grid that is more than one dot wide and more than one dot high. For example, out of the numbers from 1 to 10, the prime numbers are 2, 3, 5 and 7 because there are no numbers that divide them evenly without a remainder. 1 is not prime because it is excluded from the definition of prime numbers. 4=2×2, 6=2×3, 8=2³, 9=3×3 and 10=2×5 are composite.

First prime numbers up to 401

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, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401

See also

• Table of prime numbers

References