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Evil number
Class of binary number From Wikipedia, the free encyclopedia
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In number theory, an evil number is a non-negative integer that has an even number of 1s in its binary expansion.[1] These numbers give the positions of the zero values in the Thue–Morse sequence, and for this reason they have also been called the Thue–Morse set.[2] Non-negative integers that are not evil are called odious numbers.
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Examples
The first evil numbers are:
- 0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39 ...[1]
Equal sums
The partition of the non-negative integers into the odious and evil numbers is the unique partition of these numbers into two sets that have equal multisets of pairwise sums.[3]
As 19th-century mathematician Eugène Prouhet showed, the partition into evil and odious numbers of the numbers from to , for any , provides a solution to the Prouhet–Tarry–Escott problem of finding sets of numbers whose sums of powers are equal up to the th power.[4]
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In computer science
In computer science, an evil number is said to have even parity.
References
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