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.
The
early ones include some of the old, well-known
"chestnuts", but many of the later ones are 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 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. After seeing this,
I immediately noticed that there are 6 6's in
that statement as well! [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
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 the last two items (as well as
the second one on this page) as: 666 = S1(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 (20772119, 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.
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)
= 6·6·6. The nth triangular number is given by
the formula T(n) = (n)(n+1)/2. 666 is the 36th
triangular number - in other words, T(6·6) = 666.
In 1975 Ballew and Weger proved (see J. Rec. Math,
Vol. 8, No. 2): 666 is the largest triangular
number that's also a repdigit (A repdigit is a
number consisting of a single repeated non-zero
digit, like 11 or 22 or 555555.)
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, the amazing fact: 666 is conjectured
to be the most wicked repdigit polygonal number.
Since 666 = P(3,36), n/k = 12. I recently showed
by computer calculation that there are no counterexamples
to this conjecture less than 1050. It seems quite
certain that this is true but so far no one has
proved it. Whilst on the subject of polygonal
numbers, we can find among them some rather beastly
configurations. One of the more striking is the
following: If one arranges a group of people in
a solid 3010529326318802-sided polygon with 666
people on each side, there will be a total of
666666666666666666666 persons in all. Or, more
simply, P(3010529326318802, 666) = 666666666666666666666.
Define
pi(n,d) as the d consecutive digits of pi starting
at the nth digit after the decimal point. Then
we can make the following pretty statement: pi(666,
3) = 7·7·7. as well as the following one, which
contains nothing but 6's and 3's (and two 666's):
pi(666 · 3.663663663..., 3) = 666.
One
day, as I was staring at the number 666, I saw
two (evil?) eyes peering between the digits, like
so: 6o6o6. This seemed to imply that the number
60606 might worthy of further contemplation. Indeed,
note the following facts: 60606 = 7 x 13 x 666.
60606 = P(7,156) - i.e., 60606 is a 7-gonal number.
(Note that this can be written entirely using
the evocative numbers 6, 7, and 13, by saying
60606 = P(7, (6+6)·13)).
In addition we can make a statement using only
7's: 60606 is the 7th palindromic 7-gonal number.
60606 has exactly 6 prime factors. 60606+1 is
a prime number. Not only that, but it's a prime
(p) for which the period length of the decimal
expansion of its reciprocal (1/p) attains the
maximum possible value of p-1.
In other words: 1/(60606 + 1) has period length
60606. 60606 is, just like 666, the sum of two
consecutive palindromic primes (both of which
contain the evil eyes!): 60606 = 30203 + 30403.
[Thanks to G. L. Honaker, Jr., Jud McCranie and
Patrick De Geest for these.]
While
we're on the subject of numbers closely related
to 666... in July 2000 I snapped a picture of
my car's odometer which suggests that it might
be worthwhile to explore the double-beast number
(666666). Besides the obvious 666666 = 1001 x
666, Patrick De Geest points out that 666666 is
a palindrome in both base 10 and base 16 (hexadecimal
- get it?), where its value is A2C2A. He also
notes that in base 31 it is MBMB, which just like
666666 (made of two 666's) is formed by concatenating
two identical parts (MB).
Perhaps
MB = Multiple Beast? [found by Jud McCranie] It
is a theorem that every positive integer occurs
as the period length of the reciprocal of some
prime. So, the obvious question arises: what's
the smallest prime with period length 666?
The
answer was found in June 1998: p = 902659997773
is the smallest prime whose reciprocal has period
length 666. The first 666 digits after the decimal
point of 1/p (which then repeat) are: 000000000001107836840523732794015856393629176199911567364459
553453849096605279881838076680979988886781773038423114524370
500571392445408560228574284480352437836776725525116619485115
892576776519141738094220028289530945207260114524370499463555
604884827434558428086723261636865158160657066031266795971496
637303661413240039402749172168836999999999998892163159476267
205984143606370823800088432635540446546150903394720118161923
319020011113218226961576885475629499428607554591439771425715
519647562163223274474883380514884107423223480858261905779971
710469054792739885475629500536444395115172565441571913276738
363134841839342933968733204028503362696338586759960597250827
831163
Note:
if you turn the prime p upside down, there's a
666 inside, slightly to the left of the middle,
and if you turn the single period of 1/p upside
down, there's a 66666666666 inside, slightly to
the left of the middle! [sent in by P. De Geest]
The
smallest prime number with a gap of 666 (that
is, such that the prime following it is larger
than it by exactly 666) is 18691113008663 Note
the three sixes!
Define
a dottable fraction as one in which dots (representing
multiplication) can be interspersed in both the
numerator and denominator to produce an expression
that's equal to the original fraction. The noteworthy
dottable fraction 666 = 6·6·6 64676 6·46·76 has
a numerator of 666 and has 666 embedded in the
denominator!
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 (=6·6),
B=37, C=38, and so on, then: The sum of the letters
in the word SUPERSTITIOUS is 666.
