The **beast number** or **sign of the devil** is equal to 666.^{[1]} It is associated with Satan, evil, and bad luck in

Western numerology.

The number's origin is in Revelation 13:18 from the Christian Bible: "This calls for wisdom. Let the person who has insight calculate the number of the beast, for it is the number of a man. That number is 666."^{[2]} Some translations give 616.

A number with 666 digits is called an apocalypse number, and a number of the form 2^{n} containing 666 in its decimal expansion is called an apocalyptic number.

## Properties Edit

Its prime factorization is 2 × 3^{2} × 37.

A popular phenomenon in recreational mathematics is discovering "unusual" properties of 666 (mostly coincidental). Here are some of them:

- 666 is the sum of the squares of the first seven primes: 2
^{2}+ 3^{2}+ 5^{2}+ 7^{2}+ 11^{2}+ 13^{2}+ 17^{2}= 666 - 666 = DCLXVI, using all of the Roman numerals except for M in descending order. For the same reason, the six Japanese yen coins sum to ¥666.
- 666 is the increasing and decreasing sum of the first six perfect cubes: 1
^{3}+ 2^{3}+ 3^{3}+ 4^{3}+ 5^{3}+ 6^{3}+ 5^{3}+ 4^{3}+ 3^{3}+ 2^{3}+ 1^{3}= 666 - φ(666) = 6 · 6 · 6, where φ represents Euler's totient function.
- 666 is the alternating sum of the first three sixth powers: 1
^{6}- 2^{6}+ 3^{6}. - 666 is the 36th (6 · 6th) triangular number.
- Thus, it is the sum of the numbers on the American roulette wheel. 00, 0, 1, 2, 3, ... 35, 36.
- It is the largest repdigit triangular number.
^{[3]}

- 222,111 is the 666th triangular number, and the 222,111
^{th}triangular number is 24,**666**,759,216. - There are exactly six digit 6's in the decimal expansion of 666
^{6}: 87,2**66**,0**6**1,345,**6**23,**6**1**6**.- Similarly, there are 2 · 6 (12) digit 6's in the decimal expansion of 666
^{66}: 2,235,111,150,895,389,301,249,381,342,232,951,352,130,031,828,**6**40,**6**47,**6**1**6**,754,238,557,881,1**6**3,071,89**6**,851,147,531,724,784,530,814,582,097,445,922,881,737,078,370,5**6**7,5**6**2,092,223,091,453,**6**15,905,548,102,178,073,175,097,305,**6**70,733,077,**6**83,179,094,01**6**.

- Similarly, there are 2 · 6 (12) digit 6's in the decimal expansion of 666
- There are three ordered pandigital sums that produce 666, two increasing and one decreasing: 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9 = 9 + 87 + 6 + 543 + 21 = 666
- The first 144 (12
^{2}) digits of pi add up to 666. - The sine of the angle 666° multiplied by -2 is equal to the golden ratio.
- The sum of the first nine digits of pi in groups of three is: 314 + 159 + 265 = 666 + 66 + 6
- Similarly, if you reverse the digits, you get: 562 + 951 + 413 = 666 + 666 + 666 - 66 - 6
- Interestingly, by turning 666 upside down, you get 999, the UK official emergency number.

It is also the prize for 4 correct final digits in the German lottery *Super 6* in euros.

## Sources Edit

## See also Edit

- 777
- Legion's number of the first and second kind
- Leviathan number
- Hyper-leviathan number