# Carmichael number

From Number

Template:Pseudoprimality property

## Definition

A composite number is termed an **Carmichael number** or **absolute pseudoprime** if it satisfies the following equivalent conditions:

- The universal exponent (also called the Carmichael function) of divides .
- For any natural number relatively prime to , divides .
- is a Fermat pseudoprime to any base relatively prime to it.
- is a square-free odd number greater than 1 and divides for every prime divisor of .

## Occurrence

### Initial examples

561, 1105, 1729, 2465, 2821, 6601, [SHOW MORE]View list on OEISNote that Carmichael number is square-free and Carmichael number is odd, so each of these is the product of distinct odd primes. Further, because Carmichael number is not semiprime, there are at least three prime factors of each number. For the first few examples, we indicate the prime factors:

Carmichael number | Prime factors as list | 3? | 5? | 7? | 11? | 13? | 17? | 19? | 23? | 29? | 31? | Universal exponent (must divide number minus one) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

561 | 3, 11, 17 | Yes | No | No | Yes | No | Yes | No | No | No | No | 80 |

1105 | 5, 13, 17 | No | Yes | No | No | Yes | Yes | No | No | No | No | 48 |

1729 | 7, 13, 19 | No | No | Yes | No | Yes | No | Yes | No | No | No | 36 |

2465 | 5, 17, 29 | No | Yes | No | No | No | Yes | No | No | Yes | No | 112 |

2821 | 7, 13, 31 | No | No | Yes | No | Yes | No | No | No | No | Yes | 60 |