|
|
|
|
|
An odd number  is an integer of the form  , and as such it is not divisible by 2. In terms of congruences,
 , 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,
 .
But the multiplication of two odd numbers, or any even amount of odd multiplicands always gives an odd number. For example,
 .
A negative number raised to the power of an odd number gives a negative number. For example,
 (compare
 ).
The sum of the first  consecutive positive odd numbers is  .
The famous Gregory series which gives a quarter of  is an alternating sum of the reciprocals of the odd numbers:
However, note that this series is not absolutely convergent.
|
"odd number" is owned by PrimeFan.
|
|
(view preamble)
Cross-references: absolutely convergent, series, reciprocals, alternating sum, Gregory series, positive, consecutive, sum, negative number, even, multiplication, odd, even number, addition, prime numbers, least significant bit, representation, binary, congruences, terms, divisible, integer
There are 23 references to this entry.
This is version 4 of odd number, born on 2007-12-19, modified 2008-01-04.
Object id is 10152, canonical name is NumberOdd.
Accessed 779 times total.
Classification:
| AMS MSC: | 11A51 (Number theory :: Elementary number theory :: Factorization; primality) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
