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Date: 4-2-2020
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Date: 1-11-2019
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Date: 20-2-2020
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An odd number is an integer of the form , where
is an integer. The odd numbers are therefore ...,
,
, 1, 3, 5, 7, ... (OEIS A005408), which are also the gnomonic numbers. Integers which are not odd are called even.
Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence holds for odd
. 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 odd numbers is
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The product of an even number and an odd number is always even, as can be seen by writing
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which is divisible by 2 and hence is even.
REFERENCES:
Commission on Mathematics of the College Entrance Examination Board. Informal Deduction in Algebra: Properties of Odd and Even Numbers. Princeton, NJ, 1959.
Sloane, N. J. A. Sequence A005408/M2400 in "The On-Line Encyclopedia of Integer Sequences."
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