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 (excluding ) in Roman numerals:  (Wells 1986).

The first few numbers containing the beast number in their digits are 666, 1666, 2666, 3666, 4666, 5666, 6660, ... (OEIS A051003).

"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. Various conspiracy theories, including the novel The Da Vinci Code (Brown 2003, p. 22), have suggested that the glass pyramid at the Louvre museum in Paris is dedicated to the Beast and therefore constructed of exactly 666 panes of glass. However, despite recent works of fiction perpetuating this myth, official and unofficial counts unambiguously show that the pyramid contains more than 670 panes. (The Louvre officially states it as 673 pieces, and subsequent independent counts have suggested there are 698.)

The number 666 can also be found in a number of words and phrases. For example, summing the ASCII character codes for INDONESIA gives 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

(1)

and satisfies the identity

(2)

where  is the totient function.

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,

(3)

(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,

			(4)
			(5)
			(6)

(Keith). 666 is a repdigit, and is also a triangular number

(7)

In fact, it is the largest repdigit triangular number (Bellew and Weger 1975-76). 666 is also a Smith number.

Sums involving 666 include

(8)

which is the sum of numbers on a roulette wheel, which 38 pockets labelled 0, 00, and 1-36 (Emanouilidis 1998). Another sum is

(9)

(E. Aström, pers. comm., Mar. 30, 2006), which is however rather trivial in the sense that  holds for any even number (C. Pacher, pers. comm., Jun. 9, 2006). Finally,

(10)

A number  in which the first  decimal digits of the fractional part  sum to 666 is known as an evil number (Pegg and Lomont 2004).

There are (exactly) six 6s in  (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

			(11)
			(12)

where  is the golden ratio, which can be combined to give

			(13)
			(14)
			(15)
			(16)

(Wang 1994; Livio 2002, p. 23; Barbosa, pers. comm., Sep. 24, 2006).

REFERENCES:

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

Brown, D. The Da Vinci Code. New York: Doubleday, 2003.

Castellanos, D. "The Ubiquitous ." Math. Mag. 61, 153-154, 1988.

De Geest, P. "The Number of the Beast 666." https://www.worldofnumbers.com/weblinks.htm.

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." https://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. https://www.maa.org/editorial/mathgames/mathgames_10_04_04.html.

Sloane, N. J. A. Sequence A051003 in "The On-Line Encyclopedia of Integer 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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