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Odd Number

An odd number is an integer of the form n=2k+1, where k is an integer. The positive odd numbers are 1, 3, 5, 7, ... (OEIS A005408), and are also the gnomonic numbers. Odd numbers also include -1, -3, -5, .... Integers which are not odd are called even.

Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence n=1 (mod 2) holds for odd n. The oddness of a number is called its parity, so an odd number has parity 1, while an even number has parity 0.

The generating function for the positive odd numbers is

(x(1+x))/((x-1)^2)=x+3x^2+5x^3+7x^4+...

for |x|<1.

The product of an even number and an odd number is always even, as can be seen by writing

(2k)(2l+1)=2[k(2l+1)],

which is divisible by 2 and hence is even.

See also

Connell Sequence, Consecutive Numbers, Even Number, Gnomonic Number, Nicomachus's Theorem, Odd Number Theorem, Odd Perfect Number, Odd Prime, Parity

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References

Commission on Mathematics of the College Entrance Examination Board. Informal Deduction in Algebra: Properties of Odd and Even Numbers. Princeton, NJ, 1959.Merzbach, U. C. and Boyer, C. B. A History of Mathematics, 3rd ed. New York: Wiley, p. 49, 2011.Sloane, N. J. A. Sequence A005408/M2400 in "The On-Line Encyclopedia of Integer Sequences."

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Odd Number

Cite this as:

Weisstein, Eric W. "Odd Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OddNumber.html

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