# 666

This number is a composite.

The sum of the squares of the first seven prime numbers:
2^{2} + 3^{2} + 5^{2} + 7^{2} + 11^{2} + 13^{2} + 17^{2} = 666.

There are 6 * 6 * 6 integers which are relatively prime to 666.

666 is the sum of two consecutive palindromic primes.

Add 666 to 18691113008663 and get the next consecutive prime.

The binomial coefficient C(37, 2) is 666.

The fraction 666/999 is the ratio of the smallest even and odd primes. [Trigg]

666 can be expressed in different ways as a sum of different primes using all ten digits, i.e., 666 = 2 + 5 + 83 + 109 + 467 = 2 + 5 + 83 + 167 + 409 = 2 + 5 + 89 + 103 + 467 = 2 + 5 + 89 + 107 + 463. Note that it is the smallest even number and the smallest palindrome with this property. [Capelle]

The numbers 666*6*910+1, 666*(6+6)*910+1 and 666*(6+6+6)*910+1 are all prime. Hence, their product is a "beastly" Carmichael number. [Patterson]

666 is the largest three-digit repdigit such that the product of itself and all truncations of itself plus one is prime. I.e., 666*66*6 + 1 is prime. [Opao]

The largest prime factor of 666 is 37. Written in Roman Numerals it reads "DCLXVI" which is the 6 smallest Roman Numerals in sequence. If you add together the numerical position on the English alphabet of each of these letters, the largest prime factor of the sum is also 37. [Schuler]

The first beastly gap between consecutive twin prime pairs occurs between {774131, 774133} and {774797, 774799}. [Opao]

10^2300-167 is the largest prime of 2300 digits and 10^2300+499 the smallest prime of 2301 digits. A beastly gap thus originates (167+499=666) for the first time around some 10^k axis. Note that 167 and 499 are prime as well as the following combinations 2300167, 2301499 and last but not least 102300167. [De Geest]

π(6*6*6) = 47 is the only prime number p such that the
sum of the digits of 666^{p} is
equal to 666. If we define S(n) as the sum of the digits of
n, we can write that
S(666^{π(6*6*6)}) = 666. [Capelle]

The 666th prime to contain the beast number is an emirp. [Gaydos]

The sum of the first 666 prime numbers is *evilly* divisible by
66659. [Honaker]

666 = (3*5)^{2} +
(3*7)^{2}, written with all the
single-digit primes. Note that 666, which is formed from
the triangular number 6 repeated 3 (the only prime
triangular number) times, is the smallest triangular number
n = a^{2} +
b^{2} such that a, b and a+b are
triangular numbers. [Capelle]

There are exactly 666 twin primes less than 6^{6} + 666. [Wesolowski]

The only three-digit repdigit that yields twin primes of the form aaa + a*a*a +/- 1. [Silva]

If we define S(n) as the sum of the digits of n, then we
can write that
S((prime(6+6+6))^{6}) =
S(prime(6+6+6)) + S(prime(666)) + S(prime(6*6*6)), where
all the terms are prime. Note that 666 = (6+6+6) *
S((prime(6+6+6))^{6}). [Capelle]

666+666+666 +/- 1 is a pair of twin primes. [Silva]

The smallest "beast quadruplet" (666-digits) is also the smallest "beast quintuplet." (10^665+ 2969689524331 + d, where d = -4,0,2,6,8.) [Luhn]

66 can be expressed as the sum of two distinct primes in exactly 6 ways.