QUOTE(Nobody @ Oct 13 2004, 07:43 PM)

Why did you use '666-Microchip'?

Can you not find anything good in this?

Sure we can, we get 666 in our bodies!!! Well actually it really doesn't matter what number it is 666, 3.14, 360, 180, 90, 69, 142857, 13, 1..... Anyways getting any number of any sort injected into your skin for identity purposes would suck whether it be 666 or any number at all.

But 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.

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 + 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

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:

(6·6·6)² + (666 - 6·6)² = 666²

There are only two known Pythagorean triangles whose area is a repdigit number:

(3, 4, 5) with area 6

(693, 1924, 2045) with area 666666

It is not known whether there are any others, though a computer search has verified that there are none with area less than 1040. [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 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

There are exactly 6 6's in 6666. There are also exactly 6 6's in the previous sentence!

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,

66647 = 5049969684420796753173148798405564772941516295265

4081881176326689365404466160330686530288898927188

59670297563286219594665904733945856

66651 = 9935407575913859403342635113412959807238586374694

3100899712069131346071328296758253023455821491848

0960748972838900637634215694097683599029436416

and 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 have

6662 = 443556

6663 = 295408296

while at the same time,

(43 + 43 + 33 + 53 + 53 + 63) + (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(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(66647)

= S1(66651)

= S3(6662) + S1(6663)

[P. De Geest and G. L. Honaker, Jr.] Now that we have the Sk(n) notation, define SP(n) as the sum of the first n palindromic primes. Then:

S3( SP(666) ) = 3 · 666

where the same digits (3, 666) appear on both sides of the equation!

[By Carlos Rivera] The number 20772199 is the smallest integer with the property that the sum of the prime factors of n and the sum of the prime factors of n+1 are both equal to 666:

20772199 = 7 x 41 x 157 x 461, and 7+41+157+461 = 666

20772200 = 2x2x2x5x5x283x367, and 2+2+2+5+5+283+367 = 666.

Of course, integers n and n+1 having the same sum of prime factors are the famous Ruth-Aaron pairs. So we can say that (20772199, 20772200) is the smallest beastly Ruth-Aaron pair.

[By G. L. Honaker, Jr.] The sum of the first 666 primes contains 666:

2 + 3 + 5 + 7 + 11 · · · + 4969 + 4973 = 1533157 = 23 · 66659

[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(6·6·6) = -t/2

which can be combined into the lovely expression:

t = - (sin(666) + cos(6·6·6) )

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 = 2·3·3·37

while at the same time

6 + 6 + 6 = 2 + 3 + 3 + 3 + 7.

Consider integers n with the following special property: if n is written in binary, then the one's complement is taken (which changes all 1's to 0's and all 0's to 1's), then the result is written in reverse, the result is the starting integer n. The first few such numbers are

2 10 12 38 42 52 56 142 150 170 178 204 212 232 240 542 558 598 614...

For example, 38 is 100110, which complemented is 011001, which reversed is 100110. Now, you don't really need to be told what the next one after 614 is, do you?

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.

Might I say what a remarkable number!!!

Now the coolest thing about 666 is that 666=Evil is a bunch of bullshit.............

TNO.........

One night I ventured to my wooden rocking chair right outside my back porch, and I looked to the sky in the break of twilight, staring into the dark mirror of space, filled with sands of star dust, colossal supernovas and red giants, strings of wormholes and distant planets beyond my grasp. A place where tiny enigmatic hidden black holes tear through the fabric of space and time which, who knows; may lead into other dimensions and perhaps alternate realities. And as I look up at this heavens mirror which my hands can only reach so far through this looking glass, I felt an odd sensation of something looking back staring into my own eyes, staring into a whole other universe of which I new nothing about until I saw my reflection within it, and in that moment I then knew that the key which open the portal doors to parallel worlds was myself. "The Eyes of the Universe, Chris Landrum"