# odd number

An odd number^{} $n$ is an integer of the form $2m+1$, and as such it is not divisible by 2. In terms of congruences^{}, $n\equiv 1mod2$, and in its binary representation the least significant bit is 1. With the exception of 2, all prime numbers^{} are odd numbers.

The addition of two odd numbers gives an even number, or any even number of odd summands gives an even number as a result. For example, $11+13+17+19=60$.

But the multiplication of two odd numbers, or any even amount of odd multiplicands always gives an odd number. For example, $11\times 13\times 17\times 19=46189$.

A negative number raised to the power of an odd number gives a negative number. For example, ${(-1)}^{11}={(-1)}^{13}={(-1)}^{17}={(-1)}^{19}=-1$ (compare ${(-1)}^{10}={(-1)}^{12}={(-1)}^{16}={(-1)}^{18}=1$).

The sum of the first $n$ consecutive positive odd numbers is ${n}^{2}$.

The famous Gregory series^{} which gives a quarter of $\pi $ is an alternating sum of the reciprocals^{} of the odd numbers:

$$\frac{\pi}{4}=\sum _{i=0}^{\mathrm{\infty}}{(-1)}^{i}\frac{1}{2i+1}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\mathrm{\dots}$$ |

However, note that this series is not absolutely convergent.

Title | odd number |
---|---|

Canonical name | OddNumber |

Date of creation | 2013-03-22 17:42:30 |

Last modified on | 2013-03-22 17:42:30 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 7 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A51 |

Related topic | EvenNumber |

Related topic | SumOfOddNumbers |