# Difference between revisions of "7"

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− | If 10 is a primitive root modulo a prime <math>p</math>, then the prime <math>p</math> is a [[full reptend prime]] in base 10, i.e., the decimal expansion of <math>1/p</math> has a repeating block of the maximum possible length <math>p - 1</math>. The condition holds for <math>p = 7</math>, and the corresponding decimal expansion of 1/7 is: | + | If 10 is a primitive root modulo a prime <math>p</math>, then the prime <math>p</math> is a [[full reptend prime]] in base 10, i.e., the decimal expansion of <math>1/p</math> has a repeating block of the maximum possible length <math>p - 1</math>. The condition holds for <math>p = 7</math> (note that <math>10 \equiv 3 \pmod 7</math> and 3 is a primitive root), and the corresponding decimal expansion of 1/7 is: |

<math>0 \cdot \overline{142857}</math> | <math>0 \cdot \overline{142857}</math> |

## Revision as of 19:48, 3 January 2012

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

## Contents

## Summary

### Factorization

The number 7 is a prime number.

### Properties and families

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

prime number | fourth 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 |
divide and check |

Mersenne number | , i.e., | plug and check | |

Mersenne prime (both a prime number and a Mersenne number) | same as for Mersenne number | combine above | |

safe prime (odd prime such that half of that minus one is also prime) | second safe prime | 5, 7, 11, 23, 47, 59, 83, 107, 167, [SHOW MORE]View list on OEIS |
plug and check is prime. |

regular prime | third regular prime | 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, [SHOW MORE]View list on OEIS |

## Structure of integers mod 7

### Discrete logarithm

Template:Discrete log facts to check against

3 is a primitive root mod 7, so we can take that as the base of the discrete logarithm. We thus get an explicit bijection from the additive group of integers mod 6 to the multiplicative group of nonzero congruence classes mod 7, given by . The inverse of that mapping is the discrete logarithm, i.e., the discrete logarithm of is means :

Congruence class mod 7 (written as smallest positive integer) | Congruence class mod 7 (written as smallest magnitude integer) | Discrete logarithm to base 3, written as integer mod 6 | Is it a primitive root mod 7 (if and only if the discrete log is relatively prime to 6)? | Is it s quadratic residue or nonresidue mod 7 (residue if discrete log is even, nonresidue if odd) |
---|---|---|---|---|

1 | 1 | 0 | No | quadratic residue |

2 | 2 | 2 | No | quadratic residue |

3 | 3 | 1 | Yes | quadratic nonresidue |

4 | -3 | 4 | No | quadratic residue |

5 | -2 | 5 | Yes | quadratic nonresidue |

6 | -1 | 3 | No | quadratic nonresidue |

Alternatively, we could take discrete logs to base 5, which is the other primitive root. This is simply the negative of the other discrete log:

Congruence class mod 7 (written as smallest positive integer) | Congruence class mod 7 (written as smallest magnitude integer) | Discrete logarithm to base 5, written as integer mod 6 | Is it a primitive root mod 7 (if and only if the discrete log is relatively prime to 6)? | Is it s quadratic residue or nonresidue mod 7 (residue if discrete log is even, nonresidue if odd) |
---|---|---|---|---|

1 | 1 | 0 | No | quadratic residue |

2 | 2 | 4 | No | quadratic residue |

3 | 3 | 5 | Yes | quadratic nonresidue |

4 | -3 | 2 | No | quadratic residue |

5 | -2 | 1 | Yes | quadratic nonresidue |

6 | -1 | 3 | No | quadratic nonresidue |

### Primitive roots

The *number* of primitive roots equals the number of generators of the additive group of integers modulo 6 (= 7 - 1) which is the Euler totient function of 6, which is 2. If is a primitive root, the primitive roots are and .

Explicitly, the primitive roots are 3 and 5 (= -2).

### Quadratic residues and nonresidues

Of the six congruence classes of invertible elements mod 7, three are quadratic residues and three are quadratic nonresidues. If is a primitive root, the quadratic nonresidues are , and the quadratic residues are . Alternatively, we can obtain the quadratic residues by taking the congruence classes of .

Explicitly, the quadratic residues are 1,2,4 and the quadratic nonresidues are 3,5,6.

## Curiosities

### Constructibility of regular 7-gon

7 is the smallest prime for which the regular -gon is *not* constructible by straightedge and compass. In fact, it is the smallest natural number for which the regular -gon is not constructible by straightedge and compass. This can be seen by noting that the cyclotomic extension of adjoining 7th roots is a degree six Galois extension and cannot be expressed using successive quadratic extensions.

### Significance of 10 being a primitive root

Template:Base 10-specific observation

If 10 is a primitive root modulo a prime , then the prime is a full reptend prime in base 10, i.e., the decimal expansion of has a repeating block of the maximum possible length . The condition holds for (note that and 3 is a primitive root), and the corresponding decimal expansion of 1/7 is:

The corresponding number:

has the property that it is a cyclic number, i.e., its product with any of the numbers from 1 to 6 is obtained by cyclically permuting its digits.