Everything you wanted to know about 666 but were afraid to ask!
The number 666, first made famous by the Book of Revelation (chapter 13, verse 18), has also been studied 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.
The number 666 is a simple sum and difference of the first three 6th powers: 666 = 16 - 26 + 36. It is also equal to the sum of its digits plus the cubes of its digits: 666 = 6 + 6 + 6 + 6 + 6 + 6. There are only five other positive integers with this property. Exercise: find them, and prove they are the only ones!
666 is related to (6 + n) in the following interesting ways: 666 = (6 + 6 + 6) q (6 + 1) 666 = 6! q (6 + 1) / (6 + 2) The sum of the squares of the first 7 primes is 666: 666 = 2 + 3 + 5 + 7 + 11 + 13 + 17 The sum of the first 144 (= (6+6) q (6+6)) digits of pi is 666: 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)]. 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. The triplet (216, 630, 666) is a Pythagorean triplet, as pointed out to me by Monte Zerger. This fact can be rewritten in the following nice form: (6 q 6 q 6) + (666 - 6 q 6) = 666 The sequence of palindromic primes begins 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, etc. Taking the last two of these, we discover that 666 is the sum of two consecutive palindromic primes: 666 = 313 + 353. A well-known remarkably good approximation to pi is 355/113 = 3.1415929... If one part of this fraction is reversed and added to the other part, we get 553 + 113 = 666. [from Martin Gardner's "Dr. Matrix" columns]
The Dewey Decimal System classification number for "Numerology" is 133.335. If you reverse this and add, you get 133.335 + 533.331 = 666.666 [Wang, J. Rec. Math, 26(3)] The number 666 is related to the golden ratio! (If a rectangle has the property that cutting off a square from it leaves a rectangle whose proportions are the same as the original, then that rectangle's proportions are in the golden ratio. Also, the golden ratio is the limit, as n becomes large, of the ratio between adjacent numbers in the Fibonacci sequence.) Denoting the Golden Ratio by t, we have the following identity, where the angles are in degrees: sin(666) = cos(6 q 6 q 6) = -t/2 which can be combined into the lovely _expression: t = - (sin(666) + cos(6 q 6 q 6) ) 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 Smith number is an integer in which the sum of its digits is equal to the sum of the digits of its prime factors. 666 is a Smith number, since 666 = 2 q 3 q 3 q 37 while at the same time 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7. The following fact is quite well known, but still interesting: If you write the first 6 Roman numerals, in order from largest to smallest, you get 666: DCLXVI = 666.
The previous one suggests a form of word play that was popular several centuries ago: the chronogram. A chronogram attaches a numerical value to an English phrase or sentence by summing up the values of any Roman numerals it contains. (Back then, U,V and I,J were often considered the same letter for the purpose of the chronogram, however I prefer to distinguish them.) What's the best English chronogram for 666? My offering is a statement about, perhaps, what you should do when you encounter the number 666: Expect The Devil. Note that four of the six numerals are contained in the last word.
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 q 6 q 6. A polygonal number is a positive integer of the form P(k,n) = n((k - 2)n + 4 - k)/2 where k is the 'order' of the polygonal number (k=3 gives the triangular numbers, k=4 the squares, k=5 the pentagonal numbers, etc.), and n is its index. A repdigit polygonal number is a polygonal number that also happens to be a repdigit. Finally, define the wickedness of a polygonal number as n/k.
Now, an amazing fact: 666 is conjectured to be the most wicked repdigit polygonal number. The alphametic below has a unique solution (i.e., there is only one way to replace letters with digits so that the addition sum is correct): SIX SIX SIX +BEAST =SATAN [by Monte Zerger] Note that 1998 (a recent year) = 666 + 666 + 666. Not only that, but if we set A=3, B=6, C=9, etc., we find, amazingly, that NINETEEN NINETY EIGHT = 666 Frank Fiederer points out that the age of the United States in 1998 is also related to 666, since 1998 - 1776 = 666/3. Finally, we close with an observation that makes a commentary on the folly of attaching a specific meaning to the number 666. If the letter A is defined to be equal to 36 (=6 q 6), B=37, C=38, and so on, then: The sum of the letters in the word SUPERSTITIOUS is 666. Happy 666!
As a special bonus to whoever writes post 667: do you know what that number signifies? The Neighbor of the Beast!
