odd number

An odd numberMathworldPlanetmathPlanetmath n is an integer of the form 2m+1, and as such it is not divisible by 2. In terms of congruencesMathworldPlanetmath, n1mod2, and in its binary representation the least significant bit is 1. With the exception of 2, all prime numbersMathworldPlanetmath 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, 11×13×17×19=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 n2.

The famous Gregory seriesMathworldPlanetmath which gives a quarter of π is an alternating sum of the reciprocalsMathworldPlanetmath of the odd numbers:


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