# Why are some numbers designated as “evil”?

There was an interesting question on AI about evil numbers. Some quick searching yielded a Wolfram page devoted to it.

I understand the definition of "evil" number, but not why the term was chosen.

• Why are some numbers called evil in recreational mathematics?
• It's related to the superstition that $666$ is the number of the antichrist. – Thomas Andrews Sep 18 '17 at 21:41
• The number 666 is associated with the Antichrist or the devil in Western popular culture. – user856 Sep 18 '17 at 21:42
• I don't know about the integer variant, however, the other definition of evil. (Since the numbers in that case which are not called "evil" are called "odious," it is an obviously joking definition. Odious means "extremely unpleasant; repulsive.") – Thomas Andrews Sep 18 '17 at 21:43
• I doubt many mathematicians consider them evil. Some mathematicians have chosen the label "evil," but that is just a label, as you say. – Thomas Andrews Sep 18 '17 at 21:57
• It's just some silliness, for fun and not to be taken seriously. You could just as well define a set of numbers with some relationship to 13 to be called "unlucky numbers". As for the "evil" and "odious", it's clearly just a play on the words "even" and "odd". – user856 Sep 18 '17 at 22:21

There are two definitions of "evil number" at that MathWorld page.

# Definition 1 (666)

## Definition:

A number $$x$$ is "evil" if, for some $$n$$, the first $$n$$ decimal digits of the fractional part sum to 666.

## Source:

MathWorld lists: Pegg, E. Jr. and Lomont, C. "Math Games: Evil Numbers." Oct. 4, 2004. However, their link is currently (permanently?) broken. That said, it seems to be mirrored at this page on Ed Pegg Jr.'s website.

### Why "evil"?

The opening quote of that article make the reference clear.

This calls for wisdom. If anyone has insight, let them calculate the number of the beast, for it is man's number. His number is 666. (Revelations 13:18)

This number from the Bible is commonly associated with the devil, or evil in general.

### Why defined?

Presumably, this was defined just for fun with the number's association with evil. Quoting the article: "The number 666 pops up a lot in recreational mathematics."

# Definition 2 (even)

## Definition:

A nonnegative integer $$x$$ is "evil" if it has an even number of ones in its binary expansion. It is called "odious" otherwise.

## Source:

MathWorld only references OEIS, whose only old reference for the sequence s the original copy of "A Handbook of Integer Sequences", which I do not think uses the name "evil numbers". However, I strongly suspect that while the numbers were considered earlier, the source for this name is the book Winning Ways for your Mathematical Plays, which contains the following two relevant lines:

...are what we call the odious numbers with an odd number of ones in their expansions, while...the evil ones, with an even number.

Every number is odious or evil according to the number of 1's in its binary expansion (odious for odd, evil for even).

For some evidence for this being the original source, Aviezri S. Fraenkel claims in The vile, dopey, evil and odious game players:

"Odious" and "evil" where[sic] coined by Elwyn Berlekamp, John Conway and Richard Guy while composing their famous book Winning Ways.

### Why "evil"?

Winning Ways is filled with wordplay-based definitions. As Rahul pointed out in a comment, this pun is just about "even" and "evil" sharing sounds/letters, and similarly for "odious" and "odd", with an extra bit of humor since they're both negative terms.

### Why defined?

There have been other references to this "evil/odious" distinction since, but in Winning Ways there are two specific applications, both related to the idea that under bitwise xor/nimber addition, the evil and odious numbers act like even and odd numbers do in the context of regular addition.

One application they include is to a combinatorial game they call "Mock Turtles", in which the Grundy value for the game specified by $$n$$ happens to be $$2n$$ or $$2n+1$$, whichever one is odious.

The other application is to present Kayles as an example for an analysis of octal games via the concept of rare and common Grundy values. In the case of Kayles, it turns out that the rare values are the evil ones, and the common values are the odious ones.

• Oh, so the negation of the "Parity Bit" in Program Status Word would be the Evil Bit! – SF. Sep 19 '17 at 10:15
• A great answer! – Klangen Feb 5 '18 at 9:43