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Message: <blockquote data-quote="Novindu" data-source="post: 17331" data-attributes="member: 718">The Number of the Beast Mike KeithThe 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.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. Exercise: find them, and prove they are the only ones! 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 The sum of the first 144 (= (6+6)·(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 1000000000000066600000000000001which 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, and1 + 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:(6·6·6)² + (666 - 6·6)² = 666²There are only two known Pythagorean triangles whose area is a repdigit number:(3, 4, 5) with area 6(693, 1924, 2045) with area 666666It is not known whether there are any others, though a computer search has verified that there are none with area less than 10^40.  [see J. Rec. Math, 26(4), Problem 2097 by Monte Zerger]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 get553 + 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 get133.335 + 533.331 = 666.666[from G. L. Honaker, Jr.] There are exactly 6 6's in 666^6. There are also exactly 6 6's in the previous sentence![by P. De Geest, slight refinement by M. Keith] The number 666 is equal to the sum of the digits of its 47th power, and is also equal to the sum of the digits of its 51st power. That is,666^47 =    5049969684420796753173148798405564772941516295265408188117632668936540446616033068653028889892718859670297563286219594665904733945856666^51 =    993540757591385940334263511341295980723858637469431008997120691313460713282967582530234558214918480960748972838900637634215694097683599029436416and the sum of the digits on the right hand side is, in both cases, 666. In fact, 666 is the only integer greater than one with this property. (Also, note that from the two powers, 47 and 51, we get (4+7)(5+1) = 66.)The number 666 is one of only two positive integers equal to the sum of the cubes of the digits in its square, plus the digits in its cube. On the one hand, we have666^2 = 443556666^3 = 295408296while at the same time,(4^3 + 4^3 + 3^3 + 5^3 + 5^3 + 6^3) + (2+9+5+4+0+8+2+9+6) = 666.The other number with this property is 2583.We can state properties like this concisely be defining Sk<img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite23" alt="(n)" title="Thumbs down    (n)" loading="lazy" data-shortname="(n)" /> to be the sum of the kth powers of the digits of n. Then we can summarize items #13, #14, and #2 on this page by simply writing:666     = S2(666) + S3(666)     = S1(666^47)       = S1(666^51)     = S3(666^2) + S1(666^3)</blockquote>

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