The Number of the Beast
Mike Keith
The number 666 is cool. Made famous by the Book of Revelation (Chapter 13, verse 18, to be exact), it has also been studied extensively by mathematicians because of its many interesting properties. Here is a compendium of mathematical facts about the number 666. Most of the well-known "chestnuts" are included, but many are relatively new and have not been published elsewhere.
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The number 666 is a simple sum and difference of the first three 6th powers:
666 = 16 - 26 + 36.
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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. Exercise: find them, and prove they are the only ones!
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666 is related to (6² + n²) in the following interesting ways:
666 = (6 + 6 + 6) · (6² + 1²)
666 = 6! · (6² + 1²) / (6² + 2²)
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The sum of the squares of the first 7 primes is 666:
666 = 2² + 3² + 5² + 7² + 11² + 13² + 17²
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The sum of the first 144 (= (6+6)·(6+6)) digits of pi is 666.
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16661 is the first beastly palindromic prime, of the form 1[0...0]666[0...0]1. The next one after 16661 is
1000000000000066600000000000001
which can be written concisely using the notation 1 013 666 013 1, where the subscript tells how many consecutive zeros there are. Harvey Dubner determined that the first 7 numbers of this type have subscripts 0, 13, 42, 506, 608, 2472, and 2623 [see J. Rec. Math, 26(4)].
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A very special kind of prime number [first mentioned to me by G. L. Honaker, Jr.] is a prime, p (that is, let's say, the kth prime number) in which the sum of the decimal digits of p is equal to the sum of the digits of k. The beastly palindromic prime number 16661 is such a number, since it is the 1928'th prime, and
1 + 6 + 6 + 6 + 1 = 1 + 9 + 2 + 8.