# Least primitive root

From Number

## Definition

Suppose is a natural number such that the multiplicative group modulo is cyclic (i.e., is either a prime number, or , or a power of an odd prime, or twice the power of an odd prime). The **least primitive root** or **smallest primitive root** modulo is the smallest natural number such that is a primitive root modulo .

## Particular cases

Here, we only list equal to a prime or the square of a prime. For something that is a higher power of an odd prime, the smallest primitive root modulo the square of the prime works.

Smallest primitive root modulo | |
---|---|

2 | 1 |

3 | 2 |

4 | 3 |

5 | 2 |

7 | 3 |

11 | 2 |

13 | 2 |

17 | 3 |