666 (number) - Definition and Overview

There is also a house music group named 666. For the year, see AD 666.

666 (Six hundred sixty-six in American English, Six hundred and sixty-six elsewhere) is the Number of the Beast in the Christian Bible, in the Book of Revelation.

Mathematics

Like any number, 666 has many mathematical properties, some more interesting than others:

• 666 = 2 × 3² × 37 is the 333rd even number and the 544th composite number. (The 666th composite number is 806 = 2 × 13 × 31). 666 is the 260th non square-free number: {4, 8, 9, 12, 16, 18, 20, 24, ..., 648, 650, 652, 656, 657, 660, 664, 666, ...}. The 666th non square-free number is 1701.

• 666 is a composite number (being quite evidently divisible by 2, 3 and 111). The 666th prime is 4973. 666 is not a lucky number. The 666th lucky number is 5559.

• 666 has 12 divisors: {1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666}. The sum of its divisors is 1482. The first numbers in the interval [1, 10⁶] that have the same sum of divisors are 666, 1017, 1481. The sum of its proper divisors is 816>666, so 666 is the 162nd abundant number. 666 has 3 distinct prime divisors {2, 3, 37} (ω(666) = ω(2 × 3² &times 37) = ω(2) + ω(3²) + ω(37) = 1+1+1=3) and a total of 4 prime divisors (Ω(666)=...=4).

• It is a Smith number.

• By Euler's totient function φ(666) = 216. 666 is the 14th number with this value of φ: {247, 259, 327, 333, 351, 399, 405, 436, 494, 518, 532, 648, 654, 666, ...} But there is no number whose value of φ is 666, making it a nontotient.

• The harmonic mean of the digits of 666 is an integer: 3/(1/6 + 1/6 + 1/6) = 6. 666 is the 54th number with this property.

• 666 is the 25th repdigit.

• 666 is the 76th palindromic number in base 10 and it is also the sum of two consecutive palindromic primes: 666 = 313 + 353. It is the 62nd palindromic composite number.

• 666 is the sum of the squares of the first seven primes.

• The sum of all the integers from 1 to 36 inclusive is 666. This means 666 is the 36th triangular number. (The 666th triangular number, on the other hand, (1 + 2 + 3 + ... + 664 + 665 + 666) is 222111.)

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• A prime reciprocal magic square based on 1/149 in base 10 has a magic total of 666.

• <math>6^{6^6}<math> or 6⁴⁶⁶⁵⁶ is 36,306 digits long. Its first 150 digits are:

26591197721532267796824894043879185949053422002699

24300660432789497073559873882909121342292906175583

03244068282650672342560163577559027938964261261109...

The sum of the digits 1 through 146 of this number is 666, as is the sum of its digits 2 through 148 and its digits 2 through 149.

• It is also curious that the Roman numeral representation of the number 666 (DCLXVI) uses once each of the Roman numeral symbols with values under 1,000, and that they are in exact reverse order of their respective values (D = 500, C = 100, L = 50, X = 10, V = 5, I = 1). Curious indeed that the Roman Empire was the most prominent empire in the world at the time this number originally became symbolically attached to "the Devil".

• On the Windows operating system, the key sequence Alt+666 will create a capital U umlaut (Ü).

• 666 is also the value of the frequency of the Bodenseesender, a big AM radio station in South Germany, which is receivable at night in whole Europe, in Kilohertz