Re: six six seven


jupiter49@xxxxxxx wrote:

Six six six, who the hell needs that fairy tale when we've been living
it here at least since Kennedy took the bullet to the head?

You can't ignore 666. It is the number of Ronald Wilson Reagan!!!!
(666) also:


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.

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

[from G. L. Honaker, Jr.] There are exactly 6 6's in 6666. 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,

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

.

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