666 is the occult "number of the beast," also called the "sign of the devil" (Wang 1994), 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 origin of this number is not entirely clear, although it may be as simple as the number containing the concatenation of one symbol of each type (exclude M = 1000) in Roman numerals: DCLXVI = 666 (Wells 1986).
The first few numbers containing the beast number in their digits are 666, 1666, 2666, 3666, 4666, 5666, 6660, ... (Sloane's A051003).
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
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(1) |
and satisfies the identity
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(2) |
where 
is the totient function, as well as the sum
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(3) |
which is the sum of numbers on a roulette wheel (Emanouilidis 1998). Emanouilidis (1998) also gives additional more obscure connections between 666 and the numbers on a roulette wheel. The number 666 is a sum and difference of the first three 6th powers,
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(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,
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(Keith). 666 is a repdigit, and is also a triangular number
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(7) |
In fact, it is the largest repdigit triangular number (Bellew and Weger 1975-76). 666 is also a smith number. Finally,
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(8) |
A number x in which the first n decimal digits of the fractional part
sum to 666 is known as an evil
number (Pegg and Lomont 2004).
"666" is the combination of the mysterious suitcase retrieved by Vincent Vega (John Travolta) and Jules Winnfield (Samuel L. Jackson) in Quentin Tarantino's 1994 film Pulp Fiction.
The number 666 can also be found in a number of words an phrases. For example, summing the ASCII character codes for INDONESIA gives 666.
There are (exactly) six 6s in 6666 (G. L. Honaker, Jr., pers. comm., Feb. 9, 2003).
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.
Wang (1994) showed that
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where
is the golden ratio, which can be combined to give
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(11) |
(Wang 1994; Livio 2002, p. 23).
Apocalypse Number, Apocalyptic Number, Bimonster, Evil Number, Monster Group, Repdigit, Roman Numerals
Bellew, D. W. and Weger, R. C. "Repdigit Triangular Numbers." J. Recr. Math. 8, 96-97, 1975-76.
Castellanos, D. "The Ubiquitous 
.
Eco, U. Foucault's Pendulum. San Diego: Harcourt Brace Jovanovich, p. 31, 1989.
Emanouilidis, E. "Roulette and the Beastly Number." J. Recr. Math. 29, 246-247, 1998.
Gardner, M. "Mathematical Games: A Fanciful Dialogue about the Wonders of Numerology." Sci. Amer. 202, 150-156, Feb. 1960.
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.
Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, pp. 22-23, 2002.
Pegg, E. Jr. and Lomont, C. "Math Games: Evil Numbers." Oct. 4, 2004. http://www.maa.org/editorial/mathgames/mathgames_10_04_04.html.
Sloane, N. J. A. Sequences A051003 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
Wang, S. C. "The Sign of the Devil... and the Sine of the Devil." J. Rec. Math. 26, 201-205, 1994.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.
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