Origin and History of the number 666
Where does the number 666 come from? Many have read of it in the Bible or even heard of it from various sources, but very few know where it actually comes from or why the book of Revelation speaks about this number with such negative connotations. Here is a brief overview of the history of the number 666, where it comes from, and why the book of Revelation speaks so negatively of this number.
The number 666 comes directly from the worship practices of the city of Babylon about the time of the Bible prophet Daniel. The people of Babylon worshiped gods that were associated with the sun, moon, visible planets of our solar system, and certain of the stars involved with the practice of astrology. The Babylonians were the principle developers of astrology, as we know it today. In their worship system, they had 37 supreme gods, and one of these, the god associated with the sun, was supreme over all the other gods. These people believed that numbers in some way had power over the gods they worshiped. However, they had to create numbers that they could assign to the gods so that they could have power over them. To do this, they counted their gods and assigned a count number to each of the 36 lesser supreme gods, and then added up all of these numbers (from one to 36) and assigned the sum to the sun god. The first god they counted got the number one assigned to it, and the second god they counted got the number two assigned to it, and so on up to all 36 lesser supreme gods. Now, if you have not guessed it by now, the sum of the numbers from one to 36 totals 666, which they assigned to the god associated with the sun. They did the calculation like this:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
+ 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22
+ 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32
+ 33 + 34 + 35 + 36 = 666
However, they went further. They feared these gods and worried that one of them might strike them down sometime, so they made amulets with a 6 x 6 matrix of the numbers one through 36 on them. Today we call these numeric matrixes magic squares. The design of the amulets was to achieve a magic purpose, and they evidently believed that wearing a particular god's number would protect one from a specific god striking them down using his power of magic. Now, the amulets had to be as powerful as possible, so in order to increase its power, they arranged these numbers in such a way that the sum of any given row, column, or diagonal is 111, and therefore the sum of all six rows or all six columns was 666. This was supposed to provide extra protection, including from the god associated with the sun, since this god's sum was present on the amulet.
Below is an example of one such arrangement of numbers in a 6x6 matrix of numbers. Note that any given column, row, or diagonal adds up to 111. The ancient Babylonians wrote these numbers out on a small clay tablet, and after drying and baking it to make the writing permanent, they would hang it on a support and wear it around their neck. As long as they carried the amulet with them, they thought that the amulet provided protection.
6 32 3 34 35 1
7 11 27 28 8 30
19 14 16 15 23 24
18 20 22 21 17 13
25 29 10 9 26 12
36 5 33 4 2 31
The practice of creating amulets with this arrangement of numbers as part of astrology appears to have continued well past the time of Jesus. Archeologists have found amulets with Latin Inscriptions, so we know that the Romans were indulging in this practice.
Therefore, the 666 number came about because of the pagan worship practices of the Babylonians and their practice of astrology.
You may look for a copy of the book by Roy Allan Anderson called Unfolding the Revelation, in which explains this topic in detail. That book is (or at least has been) published by Pacific Press Publishing Association of Nampa, Idaho. Another book of good information about this is called the Trail of the Serpent, by Murl Vance and published by Oriental Watchman Publishing House of Pune, India. This book is no longer in print; however, you may find a copy of it by checking out some of the online book auctions.
Nevertheless, you may ask, why does the Bible speak about this number in such serious tones? Before responding to this question, another issue requires an answer.
What About Manuscript's Of Revelation That Have The Numbers 606 Or 616?
There are several ancient manuscripts of Revelation that show numbers such as 616 or 606 instead of 666. This very likely was due to a scribal error of some sort that somehow propagated down through additional copies as the scribe copied the manuscripts This is especially likely given that the original meaning of the number 666 was concerned with the numbers magical properties. The numbers 616 or 606 had no magical meaning of any significance, so that by itself should inform the reader that these were due to copy errors. Remember that people copied things by hand before the invention of the printing press. They did not have copy machines in those days. If you wanted, a copy of something, had the money, and could read, you hired somebody to copy it for you, or you did it yourself. Those were the only choices available. Of course, if you did not have the money, there was only one choice then - you did not get a copy of your own and had to wait until somebody was willing to read it to you! If they did not read it to you, then you were simply out of luck.
Why does the book of Revelation Speak against Babylon and the Number 666?
When the Medes and Persians conquered Babylon (539 B.C,) they came with their own religious practices and gods, so had no need of the priests of the Babylonian religious system. Although the Persians were somewhat sympathetic towards the Babylonian god Marduk and did accommodate the priests of this god for a while, it appears from history that eventually they simply fired the whole crew of priests and installed their own! Sounds like things that happen in Washington when a new political party comes into power. What scholars do know is that eventually the Babylonian priests left Babylon for a new location, probably because they were out of a job, though the exact reason why they left Babylon is unclear.
Therefore, these Babylonian priests, being out of a livelihood (or whatever the real reason was), began to look for more fertile ground to work. The Babylonian priests found it in the city of Pergamum, which was a city in what is now Western Turkey, with no port or major industry at that time except education. It was a city of the universities of its day, where students came to learn medicine and law, and the Babylonian priests added their religious schooling there after leaving Babylon. There was also a major library there, with some 200,000 volumes.
Some of the Babylonian priests went instead to Egypt, where they taught their religious practices to the Egyptians. The Egyptians readily picked up these Babylonian religious concepts and further extended and developed some of the ideas that are in astrology. They also picked up the idea of 37 supreme gods from the Babylonians. One source the author has run into claimed that the Egyptians saw the numbers 3, 6, and 7 as having magical properties, and that was one reason they liked the idea of 37 supreme gods (it had both 3 and 7 in it). By adding up the numbers 1 to 36, they could reach the magical 666. Also, they could multiply 37 by 3 or any multiple of it, and reach magical numbers such as 111, 222, 333, 444, 555, 666 (the product of 3 x 6 x 37), 777, 888, and so on.
Of course, we use the base ten numbering system, so three and seven do occur in the number 37. Whether the Babylonians and Egyptians at that time used the same numbering system is something the author does not know. However, it seems very unlikely that they used the base ten numbering system, so they likely would have seen 37 for its apparent magical properties in being able to generate numbers such as 111, 222, 666, etc., and not for containing the digits 3 and 7. Therefore, they may have extended this idea. Historically we know that the Egyptians developed several ideas that are a part of modern day astrology. Now, back to what happened at Pergamum.
The teaching of the Babylonian religious practices by the Babylonian priests and their direct descendants went on in the city of Pergamum for centuries afterwards. They taught astrology and making the amulets with the numbers 1 to 36 on them in the special arrangements they commonly used. They built a large complex as part of their worship system, which you can view a model of by clicking here. This continued until about the year 133 B.C, at which time the last king of the Attalid Empire, who had his capital at Pergamum, died, and in his will, he gave his kingdom to the Romans. After putting down a small rebellion, the Romans easily took over the Attalid Empire by 129 B.C. Then, or sometime relatively soon after this, the priests who were still teaching the Babylonian religious practices saw their opportunity and went to Rome. The Romans often adopted the religious practices of other cultures, something that went a long way towards helping their empire survive as long as it did. The Babylonian priests rightly calculated that the Romans would be very willing to learn and follow their teachings.
Therefore, Babylonian priests moved to Rome and set up shop there. Soon all of Rome filled with their religious teachings and practices. It eventually became so pervasive that people called Rome the new Babylon. It is for this reason that many teach that references to Babylon in Revelation is in fact a code word for Rome. Of course, some will dispute this, but for more information see is Babylon a symbol for Rome?
Eventually, when the Christian Church came along, the practices and beliefs of the Babylonian religion followed right into the church. Historians have noted that at one point it seemed almost the entire city of Rome converted virtually overnight to Christianity, but what actually happened is the Babylonian religion was simply brought into the church and the three supreme gods of the Babylonian religion were simply renamed as God the Father, God the Son, and God the Holy Spirit. Other lesser gods became saints of the church. The pagan Babylonian religious practices came into the church right along with it. This made it easy for the followers of the Babylonian religion to "convert" to Christianity, but it certainly was not a genuine conversion. Because of the movement of pagan religious practices into the Christian church, Revelation condemns Babylon. Since the practice of writing out the sum of the numbers one through 36 on amulets continued and was brought into the church, this is used to help identify who the beast is - the church which imported the pagan religion into the church against God's wishes. God condemns pagan religious practices, but in spite of that, they brought these practices into the church anyway.
History supports this. The pagan Babylonian priests had a chief priest who held the title Pontifex Maximus (translated to Latin and essentially meant that he was head pagan priest). You have no doubt heard of the word "Pontiff" applied to the pope. History explains where that term comes from. Originally, the king of the Attalid Empire held the title of Pontiff, and he passed it on to the Babylonian Priests before he died and they went to Rome. This title was given to Julius Caesar in 63 B.C, thereby making him the supreme priest of the Babylonian Religion and its gods. (This shows that the Babylonian priests arrived in Rome before the time of Julius Caesar or this event could not have happened.) This meant that the Babylonian religion literally moved into Rome and took it over, becoming head of the Roman government. This title passed from emperor to emperor, all of whom served as chief priest of the Babylonian Religion from then on, until, in 376 A.D, the Roman Emperor Gratian became the first since Julius Caesar to refuse the title Pontifex Maximus. Before Gratian refused the title Pontifex Maximus, Pope Callistus I passed a decree, which he called himself the Pontifex Maximus or the bishop of bishops. To read more about Pope Callistus I, see Pope Callistus I from Catholic Encyclopedia Article (219-223 A.D. as Pope). When Pope Callistus I took the title of Pontifex Maximus this made him head of the Babylonian religion. Here, you have the man that Catholic Church claims as one of the early popes. Callistus I was the head of the Christian Church in Rome, and took the title as head of the pagan Babylonian Religion and its collection of pagan gods. Therefore, did the Babylonian Religion move into the Christian Church? History says most definitely YES.
Some might dispute that Church leaders incorporated pagan religious practices into the Christian Church of that time, but even Catholic historians admit to it. Cardinal Newman, in his Development of Christian Doctrine, pages 272, 273, says that the Church incorporated many pagan religious practices into the church. Cardinal Newman claims that the church sanctified them and that made it safe to bring these practices into the church. Here is what historians have to say about the influx of Babylonian religious practices into the Catholic Church, - Go about 1/4 of the way down the page and look for the section titled "Ecclesiastical characteristics" While some of these practices may have been relatively innocuous, what do you believe God thinks of the less innocuous ones? Consider that many Catholic Pilgrims every year kiss the feet of the Roman god Saturn, thinking they are kissing the image of Peter (see Links below on Historical Pictures of Christianity). Do you think God approves of this? These people are kissing the feet of a pagan god idol. God condemns idolatry, the making of images for worship, and even bowing down before images as part of an act of worship. Yet many images are in Catholic Churches and the Church teaches people to bow down before them. The worshiper may be praying to someone else while bowing before the image, but the act of bowing down before the image, regardless of what or whom one is actually worshipping at the time God condemns in Exodus 20:5.
Ultimately, this traces back to the Babylonian religious practices. The Roman Catholic Church does this in direct violation of the command by God not to do this. Many do so ignorantly, and God will be fair and just for these individuals, but God still does not approve bowing to an image. In ancient Israel, they too brought pagan practices into their "church" (synagogue), but in spite of the evident belief among them that this made these practices safe to bring into the synagogue, God condemned this practice. Why would it be any different for the Christian Church today? Has God changed (consider Malachi 3:6 - For I am the LORD, I CHANGE NOT therefore ye sons of Jacob are not consumed.)
Who is the supreme authority in this universe, God or the pope? It appears the Catholic Church thinks the Pope is because it long ago dropped the commandment against the making of, or worship of images, from the list of the Ten Commandments. However, a long time ago God told Isaiah.
Isa 48:11 For mine own sake, even for mine own sake, will I do it: for how should my name be polluted? and I will not give my glory unto another.
Look at that phrase again, "I will not give my glory unto another". It seems unlikely this is speaking of the light that surrounds the presence of God, of that we may be sure. Verses around this text speak of the great power of God as creator and his ability to foretell the future, so it strongly suggests that it is speaking of his power, his authority, and most of all, his position. The Popes have usurped that authority and claimed they have the right to interpret or change the Law of God (see link below about the authority they have claimed for themselves). When the Catholic Church claims it has the authority to change the Law of God, did this give the Church an authority equal with God? Consider again, about what God says, "I will not give my glory unto another." Something we need think about very carefully.
Many individuals in the Catholic Church are genuine Christians who will be in heaven, but it is also true that as a religious system, God condemns it because it is a conglomeration of paganism and Christianity. So, do not mistake the destiny of the individual member of that church for the destiny of the system of which they are a part. They are not necessarily synonymous. God loves all people everywhere and wants to save everyone, but not all will choose salvation and will be lost as a result.
Nevertheless, God is now calling all genuine Christians out of this system so that they will not have to suffer the same fate that is going to come to the system as a whole. The end of this world is near and God tells us in the Bible that this system, along with all other forms of evil in this world, is soon to be the subject of the judgments of God. To prevent the destruction of good people along with those who do evil, God is calling those who will follow him to come out of Babylon.
Rev. 18:1-4 And after these things I saw another angel come down from heaven, having great power; and the earth was lightened with his glory.
And he cried mightily with a strong voice, saying, Babylon the great is fallen, is fallen, and is become the habitation of devils, and the hold of every foul spirit, and a cage of every unclean and hateful bird.
For all nations have drunk of the wine of the wrath of her fornication, and the kings of the earth have committed fornication with her, and the merchants of the earth are waxed rich through the abundance of her delicacies.
And I heard another voice from heaven, saying, Come out of her, my people, that ye be not partakers of her sins, and that ye receive not of her plagues.
Note that Revelation 17 portrays a woman that symbolizes Babylon and it says she is the mother of harlots, which clearly indicates she has daughters. There are many who believe that a woman in Revelation symbolizes a church, so this woman of Revelation 17 symbolizes a corrupt church. The Catholic Church is the only church that claims it is the mother church, so, the daughters must be the protestant churches that will corrupt themselves by joining with Babylon at the end and by doing the same thing the mother did - commit fornication with the kings of the earth of this world. Therefore, this call applies to the protestant churches as well.
The only way to salvation is through Jesus, for Acts 4:12, says, speaking of Jesus, "Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved." Note that Peter, whom the Catholic Church claims was the first pope, spoke these last words. It also claims the right to determine salvation for other people. Nevertheless, here Peter himself declares it is Jesus alone who saves. The Pope or the church cannot save you. Many of God's people are in the Catholic and Protestant churches and He is calling all to return to worshipping him the way He approves. Look to Jesus and His word (the Bible) for direction and help for this because He is the sole source of salvation. He will help you if you sincerely ask Him and are willing to follow where He leads. Jesus gave His life for you, so He is very willing to help. Consequently, you can trust Him to lead you in the right path. Please choose to follow Him wherever He leads. You will not be disappointed in the end.
Now, the Bible speaks of the number 666 is very serious tones. Probably the most serious warning in the Bible is in connection with the mark of the beast and the number 666 (see Revelation 14). Nevertheless, since you should now understand what Babylon is, how it is identified (through its description in the Bible and the number 666), and know that it is about to be destroyed. You should be able to understand that God is trying to save the lives of those caught up in this system by giving a warning of what is about to happen so that people may get out to safety. As a result, if God is calling you today, please follow Him wherever He leads you. You will be glad you followed Him.
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
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
[from Patrick Capelle]
666 is a divisor of 123456789 + 987654321.
[from Patrick Capelle] 666 can be expressed just using the digit 3, as
666 = p(3·3·3) + p(p(3·3·3))
where p(i) means the ith prime number (that is, p(1) = 2, p(2) = 3, p(3) = 5, etc.)
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? Here's one that's pretty apt:
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, and is equal to the sum of the numbers from 1 to n.
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.)
[From Patrick Capelle] 666 is the sum of the squares of two consecutive triangular numbers:
666 = 152 + 212
which can also be elegantly written as
T(6·6) = T(5)2 + T(6)2.
But also note that T(5) + T(6) = T(8). Indeed, 666 is the smallest triangular number of the form a2 + b2 with a+b also triangular.
The doubly-triangular numbers are those numbers of the form T(T(n)), where T(n) are the triangular numbers defined in the previous item. The sequence of doubly-triangular numbers begins
1, 6, 21, 55, 120, 321, 406, 666, 1035
so we see that 666 is the eighth doubly-triangular number (i.e., T(T(8)) = 666).
The nth doubly-triangular number is, among other things, the number of ways to paint the vertices of a square using a set of n colors, where the colors are distinct but rotations and reflections of a given colored square are considered the same. So there are 666 distinct ways of painting the vertices of a square with a set of eight colors.
[from Monte Zerger] 6 (= T(3)), 66 (= T(11)), and 666 (= T(36)) are all triangular numbers in base 10. These three numbers are also triangular in two other bases: 49 and 2040:
(6)49 = 6 = T(3)
(66)49 = 300 = T(24)
(666)49 = 14706 = T(171)
(6)2040 = 6 = T(3)
(66)2040 = 12246 = T(1564)
(666)2040 = 24981846 = T(7068)
[from Monte Zerger] 6666 = 87266061345623616, which contains 6 6's. In addition, the digits of 6666 can be split into two sets in two different ways, both of which sum up to the same value, 36 (= 6 x 6).
The first eight and last nine digits both sum to 36:
8 + 7 + 2 + 6 + 6 + 0 + 6 + 1 = 6 x 6 = 3 + 4 + 5 + 6 + 2 + 3 + 6 + 1 + 6
while the 6's and non-6's also add up to 36:
6 + 6 + 6 + 6 + 6 + 6 = 6 x 6 = 8 + 7 + 2 + 0 + 1 + 3 + 4 + 5 + 2 + 3 + 1
Finally, note that 6666 is almost pandigital - the only digit it's missing is an upside-down 6 (i.e., 9).
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, an amazing fact:
666 is conjectured to be the most wicked repdigit polygonal number.
Since 666 = P(3,36), its wickedness value is n/k = 12. I recently showed by computer calculation that there are no counterexamples to this conjecture less than 1050. See my paper here for more details. 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. See the paper link in the previous item for more like this.
Define PI(n,d) as the d consecutive decimal digits of the number π (3.14159265358979...) starting at the nth digit after the decimal point. Then we can make the following pretty statement:
PI(666, 3) = 7·7·7 (since the digits at that position are "343", or 7 cubed)
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.
Inserting zeros between the sixes in 666 gives the number 60606, which has a few interesting properties of its own:
60606 = 7 x 13 x 666 = 91 x 666 = T(13) x T(36) - i.e., 60606 is the product of two triangular numbers.
60606 = 7 x 37 x (13 x 18), which is interesting in that Rev 13:18 is the place where 666 is mentioned.
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, Monte Zerger, and Patrick De Geest for these.]
While we're on the subject of numbers closely related to 666...in July 2000 I snapped the following picture of my car's odometer
which suggests that it might be worthwile to explore the double-wide-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 (hex-adecimal - 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 could be read as 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
Observe that 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!
[from Simon Whitechapel] A mathematically important number sequence is:
7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, ...
which is the sequence of primes p whose reciprocal in base 10 has maximum period p-1. The last one, 1/149 with period 148, has the following digits after the decimal point (which then repeat):
0067114093959731543624161073825503355
7046979865771812080536912751677852348
9932885906040268456375838926174496644
2953020134228187919463087248322147651
As luck would have it, the sum of these is 666. If these 148 numbers (the first 148 digits of 1/149) are written as the top row of a 148x148 square grid, and then the digits of 2/149 as the second row, then 3/149 and so on, the result is a 148x148 pseudo-magic square, in which every row and column sums to 666.
[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! Also, Patrick Capelle points out that this prime (1869113008663) and the following one (1869113008663 + 666 = 18691113009329) both have the same digit sum, 53 (also a prime).
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 a denominator of the form 6x6y6.
Here's another one (actually, two) based on a fraction [by Manley Perkel and Mike Keith]. The fraction 1666/6664 (which has a 666 in both numerator and denominator) has two interesting properties:
(1) The numerical value of the fraction (0.25) is the same as the numerical value of the fraction you get by "canceling" (i.e., erasing or removing) the 666 from both the numerator and denominator.
(2) The value of the fraction is the same as the value you get by splitting the fraction in half and multiplying the two parts together; that is,
1666 = 16 . 66
6664 66 64
A fraction like this is known as a fractured fraction.
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.