666 (number)

← 665 666 667 →
List of numbers; Integers; ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	six hundred sixty-six
Ordinal	666th; (six hundred sixty-sixth)
Factorization	2 × 32 × 37
Divisors	1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666
Greek numeral	ΧΞϚ´
Roman numeral	DCLXVI
Greek prefix	ἑξακόσιοι ἑξήκοντα ἕξ; hexakósioi hexēkonta héx
Latin prefix	sescenti sexaginta sex
Binary	10100110102
Ternary	2202003
Senary	30306
Octal	12328
Duodecimal	47612
Hexadecimal	29A16
Chinese numeral	六百六十六
Devanagari numeral	६६६

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."^[1]^[2]^[3]

In mathematics

666 is the sum of the first thirty-six natural numbers, which makes it a triangular number:^[4]

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

.

Since 36 is also triangular, 666 is a doubly triangular number.^[5] 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, $\phi (666)=216$ ;^[6] and for an angle measured in degrees, $\sin {(666^{\circ })}=\cos {(216^{\circ })}=-{\tfrac {\varphi }{2}}$ (where here $\varphi$ is the golden ratio).^[7]^[8]^[a]

666 is also the sum of the squares of the first seven primes (2² + 3² + 5² + 7² + 11² + 13² + 17²),^[7]^[10] while the number of twin primes less than 6⁶ + 666 is 666.^[11]

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),^[12] with 151 the thirty-sixth prime number.^[b]

666 is a Smith number and Harshad number in base ten.^[13]^[14] The 27th indexed unique prime in decimal features a "666" in the middle of its sequence of digits.^[15]^[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).^[7]

In Christian religion

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.^[16]

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.^[17]^[18]

Other occurrences

In the Bible, 666 is the number of talents of gold Solomon collected each year (see 1 Kings 10:14 and 2 Chronicles 9:13).
In the Bible, 666 is the number of Adonikam's descendants who return to Jerusalem and Judah from the Babylonian exile (see Ezra 2:13).
Using gematria, Neron Caesar transliterated from Greek into Hebrew short-form spelling, נרון קסר, produces the number 666. The Latin spelling of "Nero Caesar" transliterated into Ktiv haser Hebrew, נרו קסר, produces the number 616. Thus, in the Bible, 666 may have been a coded reference to Nero, who was the Roman emperor from 55 to 68 AD.^[19] Though historic protestants such as Andreas Helwig in 1612 proposed the application of the Isopsephy principle to the papal name Vicarius Filii Dei.^[20]

In other fields

The Number of the Beast, the 1982 album by English heavy metal band Iron Maiden, references 666 in its title and the album's title song.
Is the magic sum, or sum of the magic constants of a six by six magic square, any row or column of which adds up to 111.
Is the sum of all the numbers on a roulette wheel (0 through 36).^[19] This is a corollary of the fact that the number is a Triangular number, as mentioned earlier.
Was a winning lottery number in the 1980 Pennsylvania Lottery scandal, in which equipment was tampered to favor a 4 or 6 as each of the three individual random digits.^[21]
Was the original name of the Macintosh SevenDust computer virus that was discovered in 1998.^{[citation needed]} It is also the name of an extension that SevenDust can add to an uninfected Macintosh.^[22]
The number is a frequent visual element of Aryan Brotherhood tattoos.^[23]
Aleister Crowley adopted the title "the Beast 666".^[24] As such, 666 is also associated with him, his work, and his religious philosophy of Thelema.
Molar mass of the high-temperature superconductor YBa₂Cu₃O₇.
In Chinese numerology, the number is considered to be lucky and is often displayed in shop windows and neon signs.^[25]^[26] In China, 666 can mean "everything goes smoothly" (the number six has the same pronunciation as the character 溜, which means "smooth".^[27]
Is commonly used by ISPs to blackhole traffic using BGP communities.^[28]
666 Fifth Avenue in New York City, which was bought for $1.8 billion in 2007, was the most expensive real estate deal in New York's history.^[29]

Notes

^
Other relevant trigonometric equalities include:^[9]
- $sin(6°) - sin(66°) = sin(666°)$ ,
- $cos(6°) - cos(66°) = cos(666°)$ , and
- $tan(666°) \times tan(216°) = -1$ .
Where, sin(666°) = cos(216°) = −⁠φ/2⁠, also equivalently −cos(36°).
^ 149 and 151 also generate a twin-prime sum equal to 300, which is the 24th triangular number.
^ 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).

References

^ Revelation 13
^ 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.
^ "The Numerology of the Beast". people.math.harvard.edu. Retrieved 2021-03-30.
^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
^ Sloane, N. J. A. (ed.). "Sequence A002817 (Doubly triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
^ 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.
^ ^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.
^ 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.
^ 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.
^ 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.
^ 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).
^ 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.
^ 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).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
^ 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.
^ 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.
^ "Revelation 13:18". Stephanus New Testament. Bible Gateway. Retrieved 2006-06-22.
^ 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 ..."
^ "The Other Number of the Beast". Archived from the original on 2000-03-01.
^ ^a ^b "666 – professors explain Roulette and Nero in detail; numberphile.com". Archived from the original on 2013-03-31. Retrieved 2013-04-06.
^ Helwig, Andreas (1512). Antichristus Romanus. VVttenbergae, Typis Laurentij Seuberlichs.
^ Baer, John (1980-09-20). "Six Won $1.2 Million in Rigged Lottery, Pa. Says". Washington Post. ISSN 0190-8286. Retrieved 2021-03-18.
^ "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. Retrieved 2022-01-24.
^ 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.
^ "Aleister Crowley | Biography, Teachings, Reputation, & Facts | Britannica". www.britannica.com. Retrieved 2022-01-24.
^ Mah, Adeline Yen (2009). China: Land of Dragons and Emperors. Random House Children's Books. ISBN 978-0375890994. Retrieved 2013-12-07.
^ "Know the Meaning of Numbers in Chinese Culture". au.ibtimes.com. Archived from the original on 2014-11-08. Retrieved 2014-10-30.
^ "666 – Good day, bad day or just another day?". www.newsgd.com. Retrieved 2014-10-30.
^ RFC 7999
^ "What you need to know about 666 Fifth Avenue". www.cbsnews.com. 19 June 2017. Retrieved 2021-07-29.

External links

Weisstein, Eric W. "Beast Number". MathWorld.
Media related to 666 (number) at Wikimedia Commons

[10] Other relevant trigonometric equalities include:^[9]
$sin(6°) - sin(66°) = sin(666°)$ ,

$cos(6°) - cos(66°) = cos(666°)$ , and

$tan(666°) \times tan(216°) = -1$ .
Where, $sin(666°) = cos(216°) =$ −⁠φ/2⁠, also equivalently $-cos(36°)$ .

[2] $sin(6°) - sin(66°) = sin(666°)$ ,

[3] $cos(6°) - cos(66°) = cos(666°)$ , and

[4] $tan(666°) \times tan(216°) = -1$ .

[14] 149 and 151 also generate a twin-prime sum equal to 300, which is the 24th triangular number.

[18] 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

[Beale,_718-2] 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.

[triang-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.

[Wells-7] 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. 26 (3). Baywood Publishing: 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.

[11] 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.

[12] 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).

[13] 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.

[15] 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).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.

[16] 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.

[17] 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.

[gateway-stephanus-19] "Revelation 13:18". Stephanus New Testament. Bible Gateway. Retrieved 2006-06-22.

[20] 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 ..."

[21] "The Other Number of the Beast". Archived from the original on 2000-03-01.

[numberphile-22] "666 – professors explain Roulette and Nero in detail; numberphile.com". Archived from the original on 2013-03-31. Retrieved 2013-04-06.

[23] Helwig, Andreas (1512). Antichristus Romanus. VVttenbergae, Typis Laurentij Seuberlichs.

[24] Baer, John (1980-09-20). "Six Won $1.2 Million in Rigged Lottery, Pa. Says". Washington Post. ISSN 0190-8286. Retrieved 2021-03-18.

[25] "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. Retrieved 2022-01-24.

[Brook-2011-26] 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.

[27] "Aleister Crowley | Biography, Teachings, Reputation, & Facts | Britannica". www.britannica.com. Retrieved 2022-01-24.

[China666-28] Mah, Adeline Yen (2009). China: Land of Dragons and Emperors. Random House Children's Books. ISBN 978-0375890994. Retrieved 2013-12-07.

[29] "Know the Meaning of Numbers in Chinese Culture". au.ibtimes.com. Archived from the original on 2014-11-08. Retrieved 2014-10-30.

[30] "666 – Good day, bad day or just another day?". www.newsgd.com. Retrieved 2014-10-30.

[31] RFC 7999

[32] "What you need to know about 666 Fifth Avenue". www.cbsnews.com. 19 June 2017. Retrieved 2021-07-29.

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