Beast Number

The occult ``number of the beast'' associated in the Bible with the Antichrist. It has figured in many numerological studies. It is mentioned in Revelation 13:18: ``Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is 666.''

The beast number has several interesting properties which numerologists may find particularly interesting (Keith 1982-83). In particular, the beast number is equal to the sum of the squares of the first 7 Primes

$\begin{displaymath} 2^2+3^2+5^2+7^2+11^2+13^2+17^2=666, \end{displaymath}$

(1)

satisfies the identity

$\begin{displaymath} \phi(666)=6\cdot 6\cdot 6, \end{displaymath}$

(2)

where $\phi$ is the Totient Function, as well as the sum

$\begin{displaymath} \sum_{i=1}^{6\cdot 6} i = 666. \end{displaymath}$

(3)

The number 666 is a sum and difference of the first three 6th Powers,

$\begin{displaymath} 666 = 1^6 - 2^6 + 3^6 \end{displaymath}$

(4)

(Keith). Another curious identity is that there are exactly two ways to insert ``+'' signs into the sequence 123456789 to make the sum 666, and exactly one way for the sequence 987654321,

$\displaystyle 666$	$\textstyle =$	$\displaystyle 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9$
			(5)
$\displaystyle 666$	$\textstyle =$	$\displaystyle 9 + 87 + 6 + 543 + 21$	(6)

(Keith). 666 is a Repdigit, and is also a Triangular Number

$\begin{displaymath} T_{6\cdot 6}=T_{36}=666. \end{displaymath}$

(7)

In fact, it is the largest Repdigit Triangular Number (Bellew and Weger 1975-76). 666 is also a Smith Number. The first 144 Digits of $\pi-3$ , where $\pi$ is Pi, add to 666. In addition $144=(6+6)\times(6+6)$ (Blatner 1997).

A number of the form which contains the digits of the beast number ``666'' is called an Apocalyptic Number, and a number having 666 digits is called an Apocalypse Number.

References

Bellew, D. W. and Weger, R. C. ``Repdigit Triangular Numbers.'' J. Recr. Math. 8, 96-97, 1975-76.

Blatner, D. The Joy of Pi. New York: Walker, back jacket, 1997.

Castellanos, D. ``The Ubiquitous $\pi$ .'' Math. Mag. 61, 153-154, 1988.

Hardy, G. H. A Mathematician's Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p. 96, 1993.

Keith, M. ``The Number of the Beast.'' http://users.aol.com/s6sj7gt/mike666.htm.

Keith, M. ``The Number 666.'' J. Recr. Math. 15, 85-87, 1982-1983.