← 665 666 667 →
Cardinalsix hundred sixty-six
Ordinal666th
(six hundred sixty-sixth)
Factorization2 × 32 × 37
Greek numeralΧΞϚ´
Roman numeralDCLXVI
Greek prefixἑξακόσιοι ἑξήκοντα ἕξ
hexakósioi hexēkonta héx
Latin prefixsescenti sexaginta sex
Binary10100110102
Ternary2202003
Senary30306
Octal12328
Duodecimal47612
Chinese numeral六百六十六
Devanagari numeral६६६

666 (six hundred [and] sixty-six) is the natural number following 665 and preceding 667.

In Christianity, 666 is referred to in (most manuscripts of) chapter 13 of the Book of Revelation of the New Testament as the "number of the beast."

## In mathematics

666 is the sum of the first thirty-six natural numbers, which makes it a triangular number:

$\sum _{i=1}^{36}i=1+2+3+\cdots +34+35+36=666$ .

Since 36 is also triangular, 666 is a doubly triangular number. Also, 36 = 15 + 21 where 15 and 21 are triangular as well, whose squares (152 = 225 and 212 = 441) add to 666 and have a difference of 216 = 6 × 6 × 6.

The number of integers which are relatively prime to 666 is also 216, $\phi (666)=216$ ; and for an angle measured in degrees, $\sin {(666^{\circ })}=\cos {(216^{\circ })}=-{\tfrac {\varphi }{2))$ (where here $\varphi$ is the golden ratio).[a]

666 is also the sum of the squares of the first seven primes (22 + 32 + 52 + 72 + 112 + 132 + 172), while the number of twin primes less than 66 + 666 is 666.

A prime reciprocal magic square based on ${\tfrac {1}{149))$ in decimal has a magic constant of 666. The twelfth pair of twin primes is (149, 151), with 151 the thirty-sixth prime number.[b]

666 is a Smith number and Harshad number in base-ten. The 27th indexed unique prime in decimal features a "666" in the middle of its sequence of digits.[c]

The Roman numeral for 666, DCLXVI, has exactly one occurrence of all symbols whose value is less than 1000 in decreasing order (D = 500, C = 100, L = 50, X = 10, V = 5, I = 1).

## In religion

### Number of the beast

 Main article: Number of the beast

In the Textus Receptus manuscripts of the New Testament, the Book of Revelation (13:17–18) cryptically asserts 666 to be "the man's number" or "the number of a man" (depending on how the text is translated) associated with the Beast, an antagonistic creature that appears briefly about two-thirds into the apocalyptic vision. Some manuscripts[which?] of the original Koine Greek use the symbols χξϛ chi xi stigma (or χξϝ with a digamma), while other manuscripts spell out the number in words.

In modern popular culture, 666 has become one of the most widely recognized symbols for the Antichrist or, alternatively, the devil. Earnest references to the number occur both among apocalypticist Christian groups and in explicitly anti-Christian subcultures. References in contemporary Western art or literature are, more likely than not, intentional references to the Beast symbolism. Such popular references are therefore too numerous to list.

It is common to see the symbolic role of the integer 666 transferred to the numerical digit sequence 6-6-6. Some people take the Satanic associations of 666 so seriously that they actively avoid things related to 666 or the digits 6-6-6. This is known as hexakosioihexekontahexaphobia.

The Number of the Beast is cited as 616 in some early biblical manuscripts, the earliest known instance being in Papyrus 115.

## In other fields

1. ^ Other relevant trigonometric equalities include:
• sin(6°) − sin(66°) = sin(666°),
• cos(6°) − cos(66°) = cos(666°), and
• tan(666°) × tan(216) = −1.
Where, sin(666°) = cos(216)° =φ/2, also equivalently −cos(36°).
2. ^ 149 and 151 also generate a twin-prime sum equal to 300, which is the 24th triangular number.
3. ^ It is the number 11110...109877666779011...011111 (241 digits long), with an equal amount of digits (119) that precede and follow a central 666 string (that also contains all sixes in the number). It is the twenty-seventh indexed member within the sequence of unique primes (where 33 = 27).
1. ^ Revelation 13
2. ^ Beale, Gregory K. (1999). The Book of Revelation: A Commentary on the Greek Text. Grand Rapids, Michigan: Wm. B. Eerdmans Publishing. p. 718. ISBN 080282174X. Retrieved 9 July 2012.
3. ^ "The Numerology of the Beast". people.math.harvard.edu. Retrieved 2021-03-30.
4. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
5. ^ Sloane, N. J. A. (ed.). "Sequence A002817 (Doubly triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
6. ^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
7. ^ a b c Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books. pp. 145–146. ISBN 0-14-008029-5. OCLC 39262447. S2CID 118329153.
8. ^ Wang, Steve C. (1994). "The Sign of the Devil...and the Sine of the Devil" (PDF). Journal of Recreational Mathematics. Baywood Publishing. 26 (3): 201–205. ISSN 0022-412X. OCLC 938842643. Archived (PDF) from the original on 2023-07-12.
9. ^ Bogomolny, Alexander. "Beauty and the Beast - in Trigonometry". Cut The Knot (Interactive Mathematics Miscellany and Puzzles). Archived from the original on 2023-07-12. Retrieved 2023-07-11.
10. ^ Sloane, N. J. A. (ed.). "Sequence A024450 (Sum of squares of the first n primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-12.
11. ^ Caldwell, Chris K.; Honaker, Jr., G. L. "666". PrimeCurios!. PrimePages. Retrieved 2023-07-12.
66 + 666 = 47322, and the 666th pair of twin primes is (47147, 47149).
12. ^ Sloane, N. J. A. (ed.). "Sequence A077800 (List of twin primes {p, p+2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
13. ^
14. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-12.
15. ^ Sloane, N. J. A. (ed.). "Sequence A040017 (Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
16. ^ "Revelation 13:18". Stephanus New Testament. Bible Gateway. Retrieved 2006-06-22.
17. ^ Novum Testamentum Graece, Nestle and Aland, 1991, footnote to verse 13:18 of Revelation, page 659: "-σιοι δέκα ἕξ" as found in C [C=Codex Ephraemi Rescriptus]; for English see Metzger's Textual Commentary on the Greek New Testament, note on verse 13:18 of Revelation, page 750: "the numeral 616 was also read ..."
18. ^ "The Other Number of the Beast". Archived from the original on 2000-03-01.
19. ^ a b "666 – professors explain Roulette and Nero in detail; numberphile.com". Archived from the original on 2013-03-31. Retrieved 2013-04-06.
20. ^ Helwig, Andreas (1512). Antichristus Romanus. VVttenbergae, Typis Laurentij Seuberlichs.
21. ^ Baer, John (1980-09-20). "Six Won \$1.2 Million in Rigged Lottery, Pa. Says". Washington Post. ISSN 0190-8286. Retrieved 2021-03-18.
22. ^
23. ^ Brook, John Lee (June 2011). Blood In, Blood Out: The Violent Empire of the Aryan Brotherhood. SCB Distributors. ISBN 978-1-900486-80-4. OCLC 793002272. Retrieved 10 January 2018.
24. ^ "Aleister Crowley | Biography, Teachings, Reputation, & Facts | Britannica". www.britannica.com. Retrieved 2022-01-24.
25. ^ Mah, Adeline Yen (2009). China: Land of Dragons and Emperors. ISBN 978-0375890994. Retrieved 2013-12-07.
26. ^ "Know the Meaning of Numbers in Chinese Culture". au.ibtimes.com. Archived from the original on 2014-11-08. Retrieved 2014-10-30.
27. ^ "666 – Good day, bad day or just another day?". www.newsgd.com. Retrieved 2014-10-30.
28. ^
29. ^ "What you need to know about 666 Fifth Avenue". www.cbsnews.com. 19 June 2017. Retrieved 2021-07-29.