Math games #666

Evil Numbers
Ed Pegg Jr. and Chris Lomont, October 4, 2004

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)

The number 666 pops up a lot in recreational mathematics. Mike Keith has a whole page devoted to this number of the beast. Among the many tidbits recorded by Mike are the following:

φ/2 + sin(666) = 0. (φ is the Golden Ratio)
φ(666) = 6·6·6. (Here, φ(n) is the Euler Totient Function)
666 = 16 - 26 + 36.
666 = 2² + 3² + 5² + 7² + 11² + 13² + 17²
The first 144 decimal digits of π sum to 666.
The first 146 decimal digits of φ sum to 666. (new, found by Ilan Honig)
Many of these items are also listed at the Beast Number entry for MathWorld. Ilan suggested the last item be added. Both pi and phi, π and φ, could be considered evil numbers. Evil numbers have the despicable property that if the digits following the decimal point are added one at a time, 666 will be hit exactly. It's like the game of 21, where you want to hit 21, without going over.

How common are evil numbers? This actually turns out to be a nice puzzle, which I put on mathpuzzle.com. (and answer below -- stop reading now if you wish to solve this on your own). There are 905 constants listed at the back of Steven Finch's marvelous book Mathematical Constants -- how many of them would be needed to make the list of evil constants complete? How much of a can of worms would it be to list all the evil constants?

A few additional evil constants were found in a search:

The first 132 decimal digits of the Ramanujan constant sum to 666. (262537412640768743.999999999999250072597 ...)
The first 137 decimal digits of the hard hexagon entropy constant sum to 666. (1.395485972479302735229500663566 ...)
The first 141 decimal digits of the first Stieltjes constant sum to 666. (-0.072815845483676724860586375874901319 ...)
The first 144 decimal digits of π sum to 666. (3.141592653589793238462643383279502884197169399375105820 ...)
The first 146 decimal digits of φ sum to 666. (1.618033988749894848204586834365638117720309179805762862 ...)
The first 146 decimal digits of 31/2 sum to 666. (1.7320508075688772935274463415058723669428052538103806 ...)
The first 153 decimal digits of the Glaisher-Kinkelin constant sum to 666. (1.282427129100622636875342568869791 ...)
The first 154 decimal digits of the Cube Line Picking constant sum to 666. (0.66170718226717623515583113324841 ...)
The first 155 decimal digits of 21/3 sum to 666. (1.2599210498948731647672106072782283505702514647015079 ...)
One can also sum the numbers in the continued fraction expansion.

The first 51 numbers in the CF of the Cube Line Picking constant sum to 666. (0, 1, 1, 1, 21, 1, 2, 1, 4, 10, 1, 2, 2, 1, ...)
The first 57 numbers in the CF of π sum to 666. (3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, ...)
The first 59 numbers in the CF of the Bloch constant sum to 666. (0, 2, 8, 2, 1, 1, 2, 102, 1, 4, 2, 2, 4, 1, 3, 3, 1, 1, 1, ...)
The first 144 numbers in the CF of the Gauss constant sum to 666. (0, 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, ...)
The first 174 numbers in the CF of the Landau Constant sum to 666. (0, 1, 1, 5, 3, 1, 1, 2, 1, 1, 6, 3, 1, 8, 11, 2, 1, 1, ...)
Evil numbers are easy to find, as one might expect with an arbitrarily defined property. Interestingly, π is doubly evil. Many numbers aren't evil. For example, e isn't evil. The important sums are 665 and 668 after 141 decimal digits -- e goes over 666 without hitting it. Chris Lomont notes that you can take every 15th digit of e, and show that e is somewhat evil. The farthest he had to reach for any constant was with Catalan's constant, which wasn't evil until he took every 28th digit.

It appears that most numbers are at least somewhat evil. With enough fiddling, almost anything is likely to be findable.

How common are evil numbers? (number of non-zero digits / sum of digits) gives a good estimate. For a base 10 number, the estimate gives 9/(1+2+3+4+5+6+7+8+9) = 20% = 1/5. Thus, any random number has a 1 in 5 chance of having this property.

http://www.mathpuzzle.com/MAA/27-Evil%20Numbers/mathgames_09_04_04.html