---
tag: 0SN0MAEKYX
kind: definition
name: Prime number
aliases: [prime]
msc: [11A41]
defines: 01826KMG2H
uses: [096PW7Q014]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_number
    locator: Definition and examples
---

## Definition

A prime number is a natural number $p$ such that $p > 1$ and the only positive divisors of $p$ are $1$ and $p$.

## Remarks

The first prime numbers are $2, 3, 5, 7, 11, 13$. The number $2$ is the only even prime.
The number $1$ is not prime by this definition. This choice makes the [[prime factorization|0AEKDFCYFA]] of each integer unique.
A natural number $n > 1$ that is not prime is a [[composite number|0N3ADXHNTQ]].
An equivalent property for $p > 1$ is [[Euclid's lemma|051VYKA1EX]]: if $p \mid a b$, then $p \mid a$ or $p \mid b$.

## Explanations

### L1 human

A prime number is a number greater than $1$ that you cannot split as a product of two smaller natural numbers.
For example, $7$ is prime, but $6 = 2 \cdot 3$ is not.
The primes are the building blocks of the natural numbers under multiplication.