Parity

The parity of an integer is its attribute of being even or odd. Thus, it can be said that 6 and 14 have the same parity (since both are even), whereas 7 and 12 have opposite parity (since 7 is odd and 12 is even).

A different type of parity of an integer  is defined as the sum  of the bits in binary representation, i.e., the digit count , computed modulo 2. So, for example, the number  has two 1s in its binary representation and hence has parity 2 (mod 2), or 0. The parities of the first few integers (starting with 0) are therefore 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, ... (OEIS A010060), as summarized in the following table.

	binary	parity		binary	parity
1	1	1	11	1011	1
2	10	1	12	1100	0
3	11	0	13	1101	1
4	100	1	14	1110	1
5	101	0	15	1111	0
6	110	0	16	10000	1
7	111	1	17	10001	0
8	1000	1	18	10010	0
9	1001	0	19	10011	1
10	1010	0	20	10100	0

A generating function for parity is given by

(1)

The constant generated by interpreting the sequence of parity digits as a binary fraction  is called the Thue-Morse constant.

The parity function obeys the sum identity

(2)

for any . For example, for  and ,

(3)

REFERENCES:

Commission on Mathematics of the College Entrance Examination Board. Informal Deduction in Algebra: Properties of Odd and Even Numbers. Princeton, NJ, 1959.

Gardner, M. "Parity Checks." Ch. 8 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 71-78, 1984.

Sloane, N. J. A. Sequence A010060 in "The On-Line Encyclopedia of Integer Sequences."