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666 (number)

From Wikipedia, the free encyclopedia
This article is about the numerical value. For the year 666 AD, seeAD 666. For other uses, see666 (disambiguation).
"1010011010" redirects here. For the song, see1010011010 (Cog song).
Natural number
← 665666 667 →
Cardinalsix hundred sixty-six
Ordinal666th
(six hundred sixty-sixth)
Factorization2 × 32 × 37
Divisors1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666
Greek numeralΧΞϚ´
Roman numeralDCLXVI
Greekprefixἑξακόσιοι ἑξήκοντα ἕξ
hexakósioi hexēkonta héx
Latinprefixsescenti sexaginta sex
Binary10100110102
Ternary2202003
Senary30306
Octal12328
Duodecimal47612
Hexadecimal29A16
Chinese numeral六百六十六
Devanagari numeral६६६

666 (six hundred [and] sixty-six ortriple six) is thenatural number following665 and preceding667.

InChristianity, 666 is referred to in most manuscripts of chapter 13 of theBook of Revelation of theNew Testament as the "number of the beast" or "number of (a) man".[1][2][3]

In mathematics

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666 is thesum of the first thirty-sixnatural numbers, which makes it atriangular number:[4]

i=136i=1+2+3++34+35+36=666{\displaystyle \sum _{i=1}^{36}i=1+2+3+\cdots +34+35+36=666}.

In fact, 666 is the largest triangular number that is also arepdigit.[5]

Since36 is a triangular number too, 666 is adoubly triangular number. Also, 666 is the sum of squares of two consecutivetriangular numbers.

T(15+21) = 152 + 212,

which can be re-written as:

T(T5+T6) = (T5)2 + (T6)2

TT5+TT3 = (T5)2 + (TT3)2.

The difference of 21² and 15² is:

216 =6 ×36 or6 ×6 ×6,

Which can be expressed:

216 =(T3) ×(T3 or(TT2) ×(TT2.

666 is acongruent number, as it is possible to make aright triangle withrationally numbered side lengths whose area is 666.[6]

The number of integers which arerelatively prime to 666 is216,ϕ(666)=216{\displaystyle \phi (666)=216};[7] and for anangle measured in degrees,sin(666)=cos(216)=φ2{\displaystyle \sin {(666^{\circ })}=\cos {(216^{\circ })}=-{\tfrac {\varphi }{2}}} (where hereφ{\displaystyle \varphi } is thegolden ratio).[8][9][a]

666 is also the sum of the squares of the first sevenprimes(22 + 32 + 52 + 72 + 112 + 132 + 172),[8][11] while the number oftwin primes less than66 + 666 is 666.[12]

Aprime reciprocal magic square based on1149{\displaystyle {\tfrac {1}{149}}} indecimal has amagic constant of 666. The twelfth pair of twin primes is (149,151),[13] with 151 the thirty-sixth prime number.[b]

666 is aSmith number andHarshad number in base ten.[14][15] The 27th indexedunique prime in decimal features a "666" in the middle of its sequence ofdigits.[16][c]

TheRoman 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).[8]

In Christian religion

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Number of the beast

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666 is often associated withSatan, who is illustrated here in an engraved interpretation byGustave Doré.
Main article:Number of the beast

In theTextus Receptus manuscripts of theNew Testament, theBook 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 withthe Beast, an antagonistic creature that appears briefly about two-thirds into the apocalyptic vision. Some manuscripts[which?] of the originalKoine Greek use the symbols χξϛchixistigma (or χξϝ with adigamma), while other manuscripts spell out the number in words.[17]

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

It is common to see the symbolic role of theinteger 666 transferred to thenumerical 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 ashexakosioihexekontahexaphobia.

There is a broad consensus in contemporary scholarship that the number of the beast refers to theRoman emperorNero, who reigned from 55 to 68 AD.[18][19] The Greek version of his name (Νέρων Καῖσαρ, "Neron Kaisar") transliterates into Hebrew as נרון קסר (NRON QSR), and, usinggematria, yields a numerical value of 666. His Latin name, "Nero Caesar", transliterated asנרו קסר (NRO QSR), produces the number616 instead, which is also cited as the Number of the Beast in some early biblical manuscripts.[20][21]

Other occurrences

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In other fields

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A 666 float in a Paris street parade

See also

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Notes

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  1. ^Other relevanttrigonometric equalities include:[10]
    • 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 to300, which is the 24thtriangular number.
  3. ^It is the number11110...109877666779011...011111 (241 digits long), with an equal amount of digits (119) that precede and follow a central 666string (that also contains all sixes in the number). It is the twenty-seventhindexed member within the sequence of unique primes (where 33 = 27).

References

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  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. Retrieved2021-03-30.
  4. ^Sloane, N. J. A. (ed.)."Sequence A000217 (Triangular numbers)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-07-11.
  5. ^Hercher, Christian; Fegert, Karl (6 March 2025)."Triangular Numbers With a Single Repeated Digit"(PDF).Journal of Integer Sequences.28. Retrieved28 April 2025.
  6. ^Math1089 (2023-05-20)."Exploring Mathematical Wonders of the Number 666".math1089.in. Retrieved2025-08-28.{{cite web}}: CS1 maint: numeric names: authors list (link)
  7. ^Sloane, N. J. A. (ed.)."Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-07-11.
  8. ^abcWells, 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.
  9. ^Wang, Steve C. (1994)."The Sign of the Devil...and the Sine of the Devil"(PDF).Journal of Recreational Mathematics.26 (3). Baywood Publishing:201–205.ISSN 0022-412X.OCLC 938842643.Archived(PDF) from the original on 2023-07-12.
  10. ^Bogomolny, Alexander."Beauty and the Beast - in Trigonometry".Cut The Knot (Interactive Mathematics Miscellany and Puzzles).Archived from the original on 2023-07-12. Retrieved2023-07-11.
  11. ^Sloane, N. J. A. (ed.)."Sequence A024450 (Sum of squares of the first n primes.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-07-12.
  12. ^Caldwell, Chris K.; Honaker, Jr., G. L."666".PrimeCurios!.PrimePages. Retrieved2023-07-12.
    66 + 666 = 47322, and the 666th pair of twin primes is (47147, 47149).
  13. ^Sloane, N. J. A. (ed.)."Sequence A077800 (List of twin primes {p, p+2}.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-07-11.
  14. ^Sloane, N. J. A. (ed.)."Sequence A006753 (Smith (or joke) numbers: composite numbers n such that sum of digits of n is equal to the sum of digits of prime factors of n (counted with multiplicity).)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-07-11.
  15. ^Sloane, N. J. A. (ed.)."Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-07-12.
  16. ^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))".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved2023-07-11.
  17. ^"Revelation 13:18".Stephanus New Testament. Bible Gateway. Retrieved2006-06-22.
  18. ^”Nero as the Antichrist.” Encyclopaedia Romana. University of Chicago.https://penelope.uchicago.edu/encyclopaedia_romana/gladiators/nero.html. “666 (or rather DCLXVI) signifies the Antichrist because that number signifies Nero.”
  19. ^Ehrman, Bart (2016). ‘666, The Number of the Beast.’ Bart Ehrman Blog.https://ehrmanblog.org/666-the-number-of-the-beast/..”
  20. ^Novum Testamentum Graece, Nestle and Aland, 1991, footnote to verse 13:18 of Revelation, page 659: "-σιοι δέκα ἕξ" as found inC [C=Codex Ephraemi Rescriptus]; for English see Metzger'sTextual Commentary on the Greek New Testament, note on verse 13:18 of Revelation, page 750: "the numeral 616 was also read ..."
  21. ^"The Other Number of the Beast".{{cite web}}:|archive-url= is malformed: timestamp (help)CS1 maint: url-status (link)
  22. ^"666 – professors explain Roulette and Nero in detail; numberphile.com". Archived fromthe original on 2013-03-31. Retrieved2013-04-06.
  23. ^Baer, John (1980-09-20)."Six Won $1.2 Million in Rigged Lottery, Pa. Says".Washington Post.ISSN 0190-8286. Retrieved2021-03-18.
  24. ^"Detailed Analysis - Mac/Sevendust-A - Viruses and Spyware - Advanced Network Threat Protection | ATP from Targeted Malware Attacks and Persistent Threats | sophos.com - Threat Center".www.sophos.com. Retrieved2022-01-24.
  25. ^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. Retrieved10 January 2018.
  26. ^"Aleister Crowley | Biography, Teachings, Reputation, & Facts | Britannica".www.britannica.com. Retrieved2022-01-24.
  27. ^Mah, Adeline Yen (2009).China: Land of Dragons and Emperors. Random House Children's Books.ISBN 978-0375890994. Retrieved2013-12-07.
  28. ^"Know the Meaning of Numbers in Chinese Culture".au.ibtimes.com. Archived fromthe original on 2014-11-08. Retrieved2014-10-30.
  29. ^"666 – Good day, bad day or just another day?".www.newsgd.com. Retrieved2014-10-30.
  30. ^RFC 7999
  31. ^"What you need to know about 666 Fifth Avenue".www.cbsnews.com. 19 June 2017. Retrieved2021-07-29.

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