# 13

From Number

This article is about a particular natural number.|View all articles on particular natural numbers

## Contents

## Summary

### Factorization

The number 13 is a prime number.

### Properties and families

Property or family | Parameter values | First few numbers | Proof of satisfaction/membership/containment |
---|---|---|---|

prime number | it is the 6th prime number | 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, [SHOW MORE]View list on OEIS |
A natural number is prime if and only if is not divisible by any prime less than or equal to . Since is between 3 and 4, we only need to check divisibility by primes less than or equal to 3, i.e., we need to verify that 13 is not divisible by the primes 2 and 3. |

Proth prime: prime of the form with | 3, 5, 13, 17, 41, 97, 113, [SHOW MORE]View list on OEIS |
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regular prime | fifth regular prime (2 is neither regular nor irregular) | 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, [SHOW MORE]View list on OEIS |

## Polynomials

### Prime-generating polynomials

Below are some polynomials that give prime numbers for small input values, which give the value 13 for suitable input choice.

Polynomial | Degree | Some values for which it generates primes | Input value at which it generates 13 |
---|---|---|---|

2 | all numbers 1-10, because 11 is one of the lucky numbers of Euler. | 2 |

### Irreducible polynomials by Cohn's irreducibility criterion

By Cohn's irreducibility criterion, we know that if we write 13 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here all the irreducible polynomials:

Base | 13 in base | Corresponding irreducible polynomial |
---|---|---|

2 | 1101 | |

3 | 111 | |

4 | 31 | |

5 | 23 | |

6 | 21 | |

7 | 16 | |

8 | 15 | |

9 | 14 | |

10 | 13 | |

11 | 12 | |

12 | 11 |

## Multiples

### Interesting multiples

Number | Prime factorization | What's interesting about it |
---|---|---|

1105 | 5 times 13 times 17 |
second Carmichael number, i.e., absolute pseudoprime |

1729 | 7 times 13 times 19 |
third Carmichael number, i.e., absolute pseudoprime |

2821 | 7 times 13 times 31 |
fifth Carmichael number, i.e., absolute pseudoprime |