The Number of the Beast 666

## What is so interesting about 666?

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The fear of the number 666 is termed as Hexakosioihexekontahexaphobia . People who suffer from this phobia naturally avoid all things related
to the number 666. So whenever such people encounter or interact with number 666, they visualize some thought connected with bad luck. All
these are superstitious views.

## Why 666 is famous as Unlucky?

• The Number of the Beast 666, is described in the New Testament book of Revelation (13:18) i.e. 18th verse of the 13th chapter of the King
James Bible as, “Here is wisdom. Let him that hath understanding count the number of the beast; for it is the number of a man; and his
number is Six hundred three score and six.” Some say the Beast is Satan, or the Antichrist. Because of number 666’s close association with
the Beast, many people think it is the devil’s number and consider it unlucky.

## What is the effect of such superstitions?

• People avoiding buildings/houses with 666 in the address. Former President Ronald Reagan and his wife, Nancy after moving to their
post-presidential home in Los Angeles changed the address of their house from from 666 St. Cloud Road to 668 St. Cloud Road.

• Some people avoid bills that contain \$6.66 or \$66.6 or \$666 either by adding or deleting some items.

• Some people don't like to travel on vehicles with number containing 666 .

• People avoid bus/highway routes connected with number 666. U.S. Highway Route 666 was renumbered to US Route 491 after controversy
over the supposed reference to the Biblical beast and rampant sign theft which persuaded the government to change the number in 2003.

• Its not unusual to find people changing their credit cards or debit cards with numbers containing 666.

## What is so interesting about 666?

Leaving aside the superstitions views, let us see how fascinating the number 666 from mathematical point of view is.

• The sum of the squares of the first (6 + 6/6) =7 primes is 666:

22 + 32 + 52 + 72 + 112 + 132 + 172 = 666

• The sum of cubes of ascending and descending numbers is 666.

13 + 23 + 33 + 43 + 53 + 63 + 53 + 43 + 33 + 23 + 13 = 666

• 666 is 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. These are 12, 30, 870, 960 and 1998.

• The number 666 is a simple sum and difference of the first three 6th powers:

666 = 16 - 26 + 36

• The Right Triangle of the Beast:

(216 × 216) + (630 × 630) = 666 × 666

• 666 is the sum of two consecutive palindromic primes i.e.

666 = 313 + 353

• 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 i.e.

666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9

666 = 9 + 87 + 6 + 543 + 21

• 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.

• 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.

66647 =
5049969684420796753173148798405564772941516295265
4081881176326689365404466160330686530288898927188
59670297563286219594665904733945856

and sum of digits of 5, 0, 4, 9, 9, ……, 5, 8, 5 and 6 is 666.

66651 =
9935407575913859403342635113412959807238586374694
3100899712069131346071328296758253023455821491848
0960748972838900637634215694097683599029436416

and sum of digits of 9, 9, 3, 5, 4, ……, 6, 4, 1 and 6 is 666.

• 666 is the sum of the first 144 decimals digits (digits after the decimal point) of Pi.

3.141592653589793238462643383279502884197169399375105820974944
592307816406286208998628034825342117067982148086513282306647
093844609550582231725359......

and sum of digits of 1, 4, 1, 5, 9, ……, 5, 3, 5 and 9 is 144.

• Some Curious Identities:

 666 = 36 - 34 + 33 – 32 666 + 6 + 6 + 6 = 53+ 63 + 73 666 = 64 – (63 + 63 + 63) + 6 + 6 + 6 666 = 21 × 32 + 23 × 34 666 = 07 + 19 - 25 - 38 + 46 + 52 + 61 + 74 + 80 + 93

• The number 20772199 is the smallest integer n, 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.

• The sum of the first 666 primes contains 666 as shown below: 2 + 3 + 5 + 7 + 11 · · · + 4969 + 4973 = 1533157 = 23 · 66659

• 16661 is the first beastly palindromic prime, of the form 1[0...0]666[0...0]1. The next one after 16661 is 1000000000000066600000000000001

• 666 is a Harshad number, one which is divisible by the sum of its digits: 666/(6 + 6 + 6) = 37.

• 666 is the largest triangular number that's also a repdigit number. (A repdigit is a number consisting of a single repeated non-zero digit, like 111 or 222 or 3333 etc.)

• 666 is a Smith number, since 666 = 2 × 3 × 3 × 37 and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7. 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.

• phi(666) = 6·6·6. Where phi(n) is the number of integers smaller than n and relatively prime to n.

### Some interesting facts

• Former U.S. president Ronald Wilson Reagan had six letters in his first, middle, and last name, concatenation of these gives 666.

• Bus Route 666 in Moscow was changed to 616 by the Moscow Transport Department in 1999.

• US Route 666 known as ‘The Highway of the Beast’ was renamed Route 491 in 2003, after controversy in connection with reference to the beast number.

• The six smallest basic Roman numerals are D (500), C (100), L (50), X (10), V (5), and I (1). The number DCLXVI in the Roman
numeral system is equivalent to 666 in decimal system.

D = 500

C = 100

L = 50

X = 10

V = 5

I = 1

Total = 666

• After completing his second term as American president in 1989 and leaving the white house, Ronald Wilson Reagan along
with Nancy Reagan moved to a new home 666 St. Cloud Road in Los Angeles. They changed it to 668 St. Cloud Road to
avoid the number 666. The change appears likely due to superstition about the number 666.

• In China the number is considered to be lucky and has nothing to do with the devil. In China, 666 mean smooth going.

• The numbers on a roulette wheel added together is equal to 666.

• One Seat in European Union Parliament remained unallocated and unoccupied atleast for some time in the past. The number of that seat is 666.

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 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1997
 Paul Hoffman, Archimedes Revenge, Penguin Books, 1991, Chapter 1.
 Martin Gardner, The Magic Numbers of Dr. Matrix, , Prometheus Books, 1985, Chapter 2.
 C.K Caldwell and G. L. Honaker, Jr., Prime Curios, CreateSpace, 2009.
Clifford A. Pickover, A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality, ? John Wiley & Sons; 1st edition (5 August 2005)