An odd integer is any number equal to 2k + 1 for some integer k. The definition includes positive and negative values, but excludes zero.
The definition has two requirements
An odd integer is a number n that can be written as 2k + 1 for an integer k. Both parts matter: the formula gives the shape, and the word integer restricts the permitted values of k. This is the definition used in Barrus and Clark’s number theory text.
Integers include the negative counting numbers, zero, and the positive counting numbers. They do not include values such as 2.4 or one third. A number’s notation can look fractional while its value is an integer: 12/4 equals 3.
Generate some examples
Choose a value of k, double it, and add one:
| k | 2k + 1 | Odd integer |
|---|---|---|
| −4 | −8 + 1 | −7 |
| −1 | −2 + 1 | −1 |
| 0 | 0 + 1 | 1 |
| 6 | 12 + 1 | 13 |
As k increases by one, the resulting odd integer increases by two. This produces the sequence …, −7, −5, −3, −1, 1, 3, 5, 7, … . It continues in both directions.
The expression 2k − 1 describes the same set when k ranges over all integers. For example, 13 is both 2 × 6 + 1 and 2 × 7 − 1. The index changes; the number does not.
Why an integer cannot be both
Suppose one integer were even and odd. Then it would satisfy n = 2a and n = 2b + 1 for integers a and b. Subtracting gives 2(a − b) = 1, or a − b = 1/2. The difference between two integers cannot be one half. That contradiction rules out being both.
Every integer also falls into one of the two groups: dividing by two leaves the mathematical remainder zero or one. There is no third integer remainder between them.
Use the definition in a question
To test 57, solve k = (57 − 1) / 2 = 28. Because 28 is an integer, 57 is odd. For 58, the same calculation gives 28.5, so it fails the odd definition.
The calculator gives a quick classification for decimal integer input. Enter 57, not an expression such as (2*28)+1. For more context, read why zero is even or see how negative integers fit the definition.