The Number of the Beast

The number 666 is cool. Thanks to the Book of Revelation (Chapter 13, verse 18, to be exact), it has been endowed with all sorts of evil connotations. Strangely enough, there are still people who take such silliness seriously...but enough of that. What's on display here is some of the interesting mathematical properties of this infamous number.

The number 666 is a simple sum and difference of the first three 6th powers:

666 = 1^6 - 2^6 + 3^6.

It is also equal to the sum of its digits plus the cubes of its digits:

666 = 6 + 6 + 6 + 6³ + 6³ + 6³.

There are only five other positive integers with this property. Go ahead, find them, and prove they are unique!

666 is related to (6² + n²) in the following interesting ways:

666 = (6 + 6 + 6) · (6² + 1²)
666 = 6! · (6² + 1²) / (6² + 2²)

The sum of the squares of the first 7 primes is 666:

666 = 2² + 3² + 5² + 7² + 11² + 13² + 17²

There are exactly two ways to insert '+' signs into the sequence 123456789 to make the sum 666, and exactly one way for the sequence 987654321:

666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9
666 = 9 + 87 + 6 + 543 + 21

A standard function in number theory is phi(n), which is the number of integers smaller than n and relatively prime to n. Remarkably,

phi(666) = 6·6·6.

The nth triangular number is given by the formula T(n) = (n)(n+1)/2. 666 is the 36th triangular number - in other words,

T(6·6) = 666.

For some further factoids relating the triangular (and other polygonal) numbers to 6's, see the first of my conjectures.

Define pi(n,d) as the d consecutive digits of pi starting at the nth digit after the decimal point. Then we can make the following pretty statement:

pi(666,3) = 7·7·7.

as well as the following one, which contains nothing but 6's and 3's (plus two 666's):

pi(666*3.663663663...,3) = 666.

Finally, we close with an observation that makes a commentary on the folly of attaching a meaning to the number 666. If the letter A is defined to be equal to 36 (=6·6), B=37, C=38, and so on, then:

The sum of the letters in the word SUPERSTITIOUS is 666.

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