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.

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 ${\displaystyle 0}$ to ${\displaystyle 2^{k}-1}$, for any ${\displaystyle k}$, provides a solution to the Prouhet–Tarry–Escott problem of finding sets of numbers whose sums of powers are equal up to the ${\displaystyle k}$th power.^[4]

In computer science

In computer science, an evil number is said to have even parity.