Mathematical Beauty of 666

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"Everywhere there is number, there is beauty."
(PROCLUS, 410-485 Ante J.-C.)

Mathematical beauty of 666

Notice

666 and basic integers

1.1. Generalities on 6, 666	1.9. 666 and number 135
1.2. 666 and number 36	1.10. 666 and number 153
1.3. 666 and number 39	1.11. 666 and number 180
1.4. 666 and number 72	1.12. 666 and number 216
1.5. 666 and number 108	1.13. 666 and number 360
1.6. 666 and number 120	1.14. 666 and number 810
1.7. 666 and number 123	1.15. 666 and number 846
1.8. 666 and number 126

666 and prime numbers

666 and opposite numbers (reciprocals)

666 and powers

666 and transcendent numbers (irrationals)
                  5.1.      666, anti-beast (999) and number π
                  5.2.      666, anti-beast (999) and golden number φ
                  5.3.      666, anti-beast (999) and number e

666 and isomorphous triplets

666 and palindromic numbers

NOTICE

This review of the elegant mathematical properties of the 666 is primarily inspired by the electronic publications of Gerard Colombat, S. Desrosiers, Eli Eshoh, Mike Keith and Naohiro. We present also certain new properties which were discovered as well as references to biblical numerology. The results exposed here can be better appreciated by means of the Microsoft Calculator available under this link (display of 32 digits) or in your computer under the path Start/Programs/Accessories/Calculator. We suggest also our links to a powerful on line calculator, MegaCalculator and its Instructions.

1. 666 and basic integers
1.1 Generalities on numbers 6 and 666
By nature, 6 is the very first "perfect number", quite special: it is at the same time equal to the sum and the product of its own dividers (see also this Perfect numbers page). Moreover, its dividers comprise a remarkable property associated with the specific number of their respective dividers. Indeed number 6 has 4 dividers (1, 2, 3 and 6). Its three clean dividers have in their turn respectively 1, 2 and 2 dividers: 1 is its single divider, 2 has 2 and 1 for dividers and 3 has 3 and 1. Such as it comes to light from the relation below, these numbers are bound by square (2) and cubic (3) powers, which evokes triple 6 (= 2 x 3).

1 + 2 + 3

1 x 2 x 3

(1 + 2 + 2 + 4)²

1³ + 2³ + 2³ + 4³

4 + 6 + 8

6 + 6 + 6

666	=	(3 x 37 x 3) + (3 x 37 x 3)
666	=	2 x 3² x 37
666	=	18 x 37
666	=	(6 + 6 + 6) x 37

The "Trinity of the Beast" is consigned in the following relation:

6 x 37 x 3003003

666 666 666

Here is another remarkable identity due to Mike Keith in Dottable Fractions (July 1998), where triple 6 appears in all the numerators and the denominators of the two fractions:

666 ÷ 6x6x6

64676 ÷ 6x46x76

3.08 333 333 333 ...

Number 666 has 11956824258286445517629485 partitions, letting appear triple 6. Patrick De Geest reports that this number can be written in the form of product comprising also triple 6:

p(666)	=	11956824258286445517629485
p(666)	=	5 x 11 x 709 x 306624548231476997503

The attentive observation of the number of partitions of the 666 made it possible to Enoch Haga to note the absence in this result of two basic digits (0 and 3) and to establish the following relation [which curiously does not take account of the occurrences after the 666, 29485 being completely excluded]:

11 + 95 + 68 + 24 + 25 + 82 + 86 + 44 + 55 + 176

666

More simply, triple 6 is obtained by the sum of 3 terms comprising 3 times 6. The relation which results from this could receive a double and contradictory interpretation: first of all, one can see there the expression of a perfect universe (as figure 6), created in 6 days, starting from the infinite vacuum (000), by a basically trinitary God; but one can perceive also there the satanic snake which, under the hypocritical appearance of divine trinity (triple 6) will success to corrupt the human being and will try to destroy (000) the perfection.

6 + 60 + 600

666

Even our own genetic or nuclear structure would comprise 666 reducible numerical combinations!... In nuclear physics, the scientists Hans Jensen (1907-1973) and Maria Gœppert-Mayer (1906-1972), indeed established a "magic" numerical sequence of 7 numbers (all even numbers), them even associated with the number of protons or neutrons in the core of many elements and which confer a better nuclear stability to them. The discoveries of these two authors were published in 1955 in a common book, "Elementary Theory of Nuclear Shell Structure", N.Y.: John Wiley & Sons, 269 pages. Eight years afterwards, they shared the Nobel Prize of physics for 1963. These "magic numbers" are:
2 - 8 - 20 - 28 - 50 - 82 - 126
Their symmetrical or mirror numbers are thus:
2 - 8 - 02 - 82 - 05 - 28 - 621

However, Gerard Colombat (in The magic of numbers) noted that the sum of 6 of these mirror-numbers is very right equal to 666:

2 + 8 + 02 + 05 + 28 + 621

666

1.2 666 and number 36
According to the checkup of S. Desrosiers, the word beast is mentioned 36 times in the Apocalypse whereas the name of Satan is used 36 times in the New Testament. Moreover, number 36 is employed only 2 times in the Bible [Joshua, 7/5 and 2^nd Chronicles 16/1].

The number of the Beast from the Apocalypse, 666, is a triangular number of which the root or secret value is 36. Such a number is of form T_n = ½ [n(n+1)], n being a natural entirety.
666 = T₃₆ = T_6x6 = 36(36 + 1)/2 = 18 x 37 (18 = 36÷2)

In addition, the sum of the first 36 numbers is equal to 666.

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

For Pythagore, 36 is the number of Large Tetraktis (or Large Quaternary), obtained by the sum of the first four odd numbers [1 + 3 + 5 + 7 = 16] and of the first four even numbers [2 + 4 + 6 + 8 = 20], is 36 = 16 + 20. It is problablement this internal duality of the number 36 which confers to him the capacity of generating at the same time the beast and the anti-beast, and this, of multiple ways. In addition, the sum of the figures of number 36 gives 9, that is to say the basic MAXIMUM figure... another sign of the power of the 36.

36=3!x6 (Triple 6...) 36=1³+2³+3³ (Trinity...) 36=2²x(1!+2!+3!) "Mirror" of 36 = 1/36 = 0.27 777 777 777 ... ⇒ 777 (1+6+7+9+6+1+6)⁴ = 36⁴ = 1 679 616
36 ⇒ 3+6=9 1/9= 0.111 111 111 ... ⇒111x6=666 111x7=777 111x9=999
2/7 = 0.2 857 142 857 142 ...	⇒ 857 + 142 = 999 857 ⇒ 8+5+7 = 20 142 ⇒ 1+4+2 = 7	⇒ 20/7=2. 857 142 857 142 ... ⇒ 857 + 142 = 999
2/77=0.0 259 740 259 740 ...	⇒ 259 + 740 = 999 740/259 = 2.857 142 857 142 ... 259 ⇒ 2 + 5 + 9 = 16 740 ⇒ 7 + 4 + 0 = 11	⇒ 857+142=999 ⇒ 16/11=1. 454 545 454 545 ... ⇒ 454+545=999
2/777=0.00 257 400 257 400 ...	⇒ 257+400 = 657 257 ⇒ 2 + 5 + 7 = 14 400 ⇒ 4 + 0 + 0 = 4	⇒ 7/65=0.107 692 307 692 307 ... ⇒ 692 + 307=999 ⇒ 65/7=9.2 857 142 857 142 ... ⇒ 857 + 142=999 ⇒ 4/14=0.2 857 142 857 142 ... ⇒ 857 + 142=999
27/7=3.857 142 857 142 ...	⇒ 857 + 142=999	⇒ 999
27/77=0.350 649 350 649 ...	⇒ 350 + 649=999 ⇒ 649/350=1.8 542 857 142 857 142 ...	⇒ 857 + 142=999
27/777 =0.0347 490 347 490 ...	⇒ 347 + 490 = 837 347 ⇒ 3 + 4 +7 = 14 490 ⇒ 4 + 9 + 0 = 13	⇒ 83/7=11. 857 142 857 142 ... ⇒ 857 + 142=999 ⇒14/13=1.0 769 230 769 230 ... ⇒ 769 + 230=999 ⇒13/14=0.92 857 142 857 142 ... ⇒ 857 + 142=999

1.3 666 and number 39
The partition of the square of 39 conduct to the 666. Number 39, employed 4 times in the Bible, is related to the punishment. According to Judaïc Mishna, there are 39 types of prohibited works during the Sabbath and any individual who transgressed the laws was to receive 39 blows of whips.

39 x 39 = 39²

1521

15² + 21²

666

15 + 21 = 36 = Numerical root or secret value of 666 666 = 36(36 + 1)/2
1/15 = 0.0 666 666 666 ...	⇒ 666
1/21 = 0.0 476 190 476 190 ...	⇒ 476 + 190 = 666
15/21 = 0.714 285 714 285 ...	⇒ 714 + 285 = 999

" Five [5] times at the hands of the Jews
I received forty lashes minus one."
(2nd Epistle in Corinthians, 11 / 24),
says Holy Paul about 39 blows of whip
which he received five times from the Jews.

11 ⇒ 1+1=2 24 ⇒ 2+4=6
39 ⇒ 3+9=12 12 ⇒ 1+2=3
1÷39=0.0 256 410 256 410 ...
5÷39=0.128 205 128 205 ...

5÷12 = 0.41 666 666 666 ...
5÷3= 1.666 666 666 ...

11÷24=0.458 333 333 333 ...
2÷6=0.333 333 333 ...

⇒ 256+410=666
⇒ 128 + 205=333
⇒ 333x2=666
⇒ 666
⇒ 666
⇒ 333x2=666
⇒ 333x2=666

1.4 666 and number 72
The 666 is also related to number 72:

666	=	(2² x 6² x 999) ÷ 6³
666	=	(2 x 72 x 999) ÷ (3 x 72)
666	=	(144000 - 144) ÷ 216

In addition, 72 intervenes like product in the numbers observed in the preceding table:

72 = 36 x 2 = 6² x 2	144 = 72 x 2 = 6² x 2²
216 = 72 x 3 = 6³	144 000 = 72 x 2000
1/72 = 0.013 888 888 888 ...	⇒ 888 and 13
1/13 = 0.0769 230 769 230 ...	⇒ 769 + 230 = 999

However this number 72, very invaluable and thus rare in the Bible, contains protection against the beast (888 or Jesus, the 13 or Last Supper and 999 or the anti-beast). The number 72 is quoted only 4 times in the Bible, in particular in NUMBERS, 31/38.

"Thirty-six [36] thousand,
oxen of which seventy-two
[72 = 2 x 36] fell as tax
to the LORD;"
(Numbers, 31 / 38).

36 ⇒ 3+6=9 72 ⇒ 7+2=9
31 ⇒ 3+1=4 38 ⇒ 3+8=11
4 ÷ 11 = 0.363 636 363 636 ...
36 ÷ 72 ≈ 9 ÷9 = 1
1/36 = 0.02 777 777 777 ...
1/72 ≈ 1/9 = 0.111 111 111 ...

⇒ 363 + 636 = 999
⇒ "tribute for the Eternal [1]"
⇒ 777
⇒ 111 x 6 = 666
⇒ 111 x 7 = 777
⇒ 111 x 8 = 888
⇒ 111 x 9 = 999

1.5 666 and number 108

666 = 72 ÷ 0.108 108 108 ...

72 ÷ 666 = 0. 108 108 108 ... = 120 ÷ 1110

0.6 x 1110 = 666 = (72 ÷ 120) x 1110

120 ÷ 72 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

1.6 666 and number 120 Number 120 is equal to factorial 5 and its numerical root (sum of its figures) is 3, predicts triple 6 ...

120	=	5! = 5x4x3x2x1
120	⇒	1 + 2 + 0 = 3

Number 120 is a triangular number of which the root or secret value is 15. Let us recall that such a number is form T_n = ½ [n(n+1)], n being a natural entirety.
120 = T₁₅ = 15(15 + 1)/2 = 15 x 8

Combination of number 120 and its secret value 15 conducts to the number of the beast, triple 6:

120	⇒	1+2+0=3	⇒	3 "times" 6 666
15	⇒	1+5=6	⇒	3 "times" 6 666

666 = 80 ÷ 0.120 120 120 ...

80 ÷ 666 = 0. 120 120 120 ... = 133.333 333 333 ... ÷ 1110

0.6 x 1110 = 666 = (80 ÷ 133.333 333 333 ...) x 1110

133.333 333 333 ... ÷ 80 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

NOTE: 133.3 ÷ 1110 = 0.120 09 0090 0900 900 900 900 ...
⇒ 09 + 0090 + 0900 = 999
80÷0.120120120120 = 666.000000000666000000000666

Like 672, number 120 is a "tri-perfect" number; it is equal to three times the sum of its 16 clean dividers (see also Perfect numbers):

Sum of the 16 dividers of number 120
1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120	=	360 = 3 x 120

It is important to note that 360 is the symbol of the circle. Describing the other properties and the symbolism of number 120, S. Desrosiers recalls that it

«représente la puissance et la gloire du Christ-Roi. C'est la limite d'âge fixée par Dieu pour l'homme. (Gn 6,3). Le Christ Jésus aurait eu 120 disciples, incluant les apôtres. Le Pape Paul VI fixa à 120 la limite de "cardinaux-électeurs" de moins de 80 ans. L'Arche de Gilgamesh, le Noé mésopotamien, mesurait 120 coudés des trois côtés, formant un cube. Ces dimensions auraient été fixées par Anu, Père des dieux. Il est écrit dans le Talmud que Noé disposa de cent vingt ans pour construire son arche. La vie du globule rouge du sang est de 120 jours ...»
[Our translation]
«represents the power and the glory of the Christ-King. It is the limit age fixed by God for the man. (Gn 6,3). Jesus Christ would have had 120 disciples, including the apostles. The Pope Paul VI fixed at 120 the limit of "cardinal-voters" of less than 80 years. The Arch of Gilgamesh, the mesopotamian Noah, measured 120 cubits on the three sides, forming a cube. These dimensions would have been fixed by Anu, Father of the gods. It is written in Talmud that Noah had one hundred twenty years to build his arch. The life of the red globule of blood is 120 days ...»

In addition, the number of the beast is exactly equal to the sum of the whole of circular shifts of 120.

Circular shifts of 120:
012, 021, 102, 120, 201 and 210

⇒

666=(012+021+102+120+201+210)

The circular shifts of 120 are generators of the anti-beast.

Trinitary-circular shifts of 120: 012, 021, 102, 120, 201 et 210 Generators of the anti-beast 999
1÷12=0.08 333 333 333 ...	⇒ 3x333=999
1÷21=0.047619 047619 047619 ...	⇒ 3x(476+190)÷2=999 ⇒ 3x(47+619)÷2=999 ⇒ 476+190=666 47+619=666
1÷102=0.0 0980392156862745 0980392156862745 ... 980+392+156+862+745 = 3135 98+03+92+15+68+62+74+50=462 31÷35=0.8 85714285 7142 857142 857142 ... 35÷31=1.129032258064516 129032258064516 129032258064516 ... 4÷62=0.0645161290322580 64516129032258 064516129032258 ... 2÷46=1/23=0. 0434782608695652173913 0434782608695652173913 ... 434+782+608+695+652+173+913=4257 57÷42=1.357 14285714 2857 142857 142857 ... 425÷7=60.714285714 2857 142857 142857 ... 42÷57=0.736842105263157894 736842105263157894 ... Note: 1÷19=0.052631578947368421 052631578947368421 ...	⇒ 142+857=999 571+428=999 ⇒ 285+714=999 ⇒ 129+032+258+064+516=999 ⇒ (645+161+290+322+580)÷2=999 ⇒ 142+857=999 571+428=999 ⇒ 285+714=999 ⇒ (736+842+105+263+157+894)÷3=999
1÷120=0.008 333 333 333 ...	⇒ 3x333=999
1÷201=0.00 4975 1243 7810 9452 7363 1840 7960 199 49÷75=0.65 333 333 333 ... 12÷43=0.279069767441860465116279069767441860465116 ... 279+069+767+441+860+465+116 = 2997 10÷78=0.128205 128205 128205 ... 94÷52=1.80 7692307 69230 769230 769230 ... 63÷73=0.863013698 6301369 86301369 86301369 ... 73÷63=1.158730 158730 158730 ...	⇒ 3x333=999 ⇒ 2997÷3=999 ⇒ 3x(128+205)=999 ⇒ 769+230=999 ⇒ 692+307=999 ⇒ 76+923=999 ⇒ 863+136=999 ⇒ 630+369=999 ⇒ 301+698=999 ⇒ 15+873=888 158+730=888
1÷210=0.0 0476190 47619 047619 047619 ...	⇒ 3x(476+190)÷2=999 ⇒ 3x(47+619)÷2=999 ⇒ 476+190=666 47+619=666

Microsoft calculator available under this link (display of 32 digits)
or under the computer path Start/Programs/Accessories/Calculator.

1.7 666 and number 123

666 = 82 ÷ 0.123 123 123 ...

82 ÷ 666 = 0. 123 123 123 ... = 136.666 666 666 ... ÷ 1110

0.6 x 1110 = 666 = (82 ÷ 136.666 666 666 ...) x 1110

136.666 666 666 ... ÷ 82 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

NOTE: 136.7 ÷ 1110 = 0.123 153 153 153 ...

The number of the beast is expressed according to the whole of the circular shifts of 123.

Circular shifts of 123:
123, 312, 231, 132, 213 et 321

⇒

666 = (123+132+213+231+312+321)÷2=(1332)÷2

According to Raymond Abellio (in its book The Bible, quantified document), the number of the beast would be the "key-number of the life", because being the sum of two series of which an ordinate [123 + 231 + 312 = 666] and the other disordered [132 + 321 + 213 = 666]. It would be the manifestation of the Trinity on three plans: material, animist and spiritual. However, number 123 and its circular shifts are truly generators of anti-beast.

Trinitary-circular shifts of 123: 123, 312, 231, 132, 213 et 321 Generators of anti-beast 999
1/123=0.00 813 00 813 00 813 ...	⇒ 813 + 00 = 813 8÷13=0.6 153 846 153 846 ...	⇒ 153 + 846 = 999
1/312=0.00 320 512 820 512	⇒ 320 + 512 + 820 = 1652 16÷52= 0.307 692 307 692 ...	⇒ 307 + 692 = 999
1/231=0.00 432 900 432 900 ...	⇒ 432 + 900 = 1332=2 x 666 32÷13=2.46 153 846 153 846 ...	⇒ 153 + 846 = 999
1/132=0.00 757 575 757 575 ...	⇒ 757 + 575 = 1332=2 x 666 32÷13 =2.46 153 846 153 846 ... ⇒ 2÷13=0.153 846 153 846...	⇒ 153 + 846 = 999 ⇒ 153 + 846 = 999
1/213=0.00 469 483 568 075	⇒ 469 + 483 = 952 9÷52=0.17 307 692 307 692 ... 52÷9=5.777 777 777 ... ⇒ 2÷13=0.153 846 153 846... ⇒ 3÷21=0.142 857 142 847 ...	⇒ 307 + 692 = 999 ⇒ 777 ⇒ 153 + 846 = 999 ⇒ 142 + 857 = 999
1/321=0.00 311 526 479 750	⇒ 311 + 526 = 837 83÷7=11. 857 142 857 142 ... ⇒ 3÷21=0.142 857 142 847 ... 311 ⇒ 3+1+1 = 5 526 ⇒ 5+2+6 = 13 5÷13=0.3 846 153 846 153 ...	⇒ 857 + 142 = 999 ⇒ 142 + 857 = 999 ⇒ 846 + 153 = 999

Number 123 would be measurements of the arms of the Glorious Cross recommended by Jesus. Relative with the properties of number 123, S. Desrosiers written:

«Dans l'un de ses messages, Jésus recommande qu'une Croix Glorieuse soit élevée à la limite du territoire de Dozulé, sur la Haute-Butte. Les bras de la Croix doivent se dresser de l'orient à l'occident, chacun mesurant 123 mètres, et sa hauteur, 6 x 123 mètres. Or, 123 x 6 = 738 mètres. En observant une carte de la ville de Jérusalem mentionnant les courbes de niveau, on s'aperçoit que 738 mètres correspondent exactement à l'altitude du Calvaire, où a été crucifié le Christ à Jérusalem, en Israël. De plus, 7+3+8 = 18, ce qui fait 3 x 6 = 6+6+6, symbole du péché d'iniquité. La Croix Glorieuse, qui est aussi Jésus ressuscité, se couvre ainsi du péché d'iniquité, symbolisé par les mensurations 666, afin de racheter le péché du monde.»
[Our translation]
«In one of his messages, Jesus recommends that a Glorious Cross should be built in extreme limits of the territory of Dozule, on High-Ridges. The arms of the Cross must draw up from East to Occident, each one measuring 123 meters, and its height, 6 x 123 meters. However, 123 x 6 = 738 meters. By observing a card of the city of Jerusalem mentioning the level lines, one realizes that 738 meters correspond exactly to the altitude of the Martyrdom, where Christ was crucifixed in Jerusalem, Israel. Moreover, 7+3+8 = 18, which makes 3 x 6 = 6+6+6, symbol of the sin of iniquity. The Glorious Cross, which is also Jesus risen from dead, thus covers sin of iniquity, symbolized by measurements 666, in order to repurchase the sin of the world.»

1.8 666 and number 126

666 = 84 ÷ 0.126 126 126 ...

84 ÷ 666 = 0. 126 126 126 ... = 140 ÷ 1110

0.6 x 1110 = 666 = (84 ÷ 140) x 1110

140 ÷ 84 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

1.9 666 and number 135

666 = 90 ÷ 0.135 135 135 ...

90 ÷ 666 = 0. 135 135 135 ... = 150 ÷ 1110

0.6 x 1110 = 666 = (90 ÷ 150) x 1110

150 ÷ 90 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

1.10 666 and number 153
Number 153 is the symbol of God and of the Church (according to Cyrille of Alexandria), it is also the symbol of the mission of Christ in the New Testament. S. Desrosiers (Numbers: Symbolism and Properties) recalls in its excellent document that the ministry of Christ lasted 918 days from its baptism to the festival of Passover. However 918 = 153 x 6. Desrosiers also underlines that «the word "beasts" appears 153 times in the biblical version of King James. The word "command" returns 153 times in the Old Testament». It is also the number of fish indicated in biblical miraculous fishing:

"So Simon Peter went over and
dragged the net ashore full
of one hundred fifty-three [153]
large fish. Even though there
were so many, the net was
not torn." (John 21/11).

21 ⇒ 2+1=3 11 ⇒ 1+1=2
21 ÷ 11 = 1.90 90 90 90 90 ...
2 ÷ 3 = 0.666 666 666 ...

11÷21 = 0.523 809 523 809 ...

⇒ 90 + 909 = 999
⇒ 666
⇒ 523 + 809 = 1332
1332÷2=666
1332÷3=444
1332÷4=333
⇒666, 888,999

The 153^th verse of the Genesis relates to Noah's Ark: "This is how you shall build it: the length of the ark shall be three hundred cubits, its width fifty cubits, and its height thirty cubits." (Genesis 6/15). God having taken the decision to destroy the human race, as consigned to the 151^th verse of the Genesis: "He said to Noah: "I have decided to put an end to all mortals on earth; the earth is full of lawlessness because of them. So I will destroy them and all life on earth." (Genesis 6/13). Number 153 is used only 2 times in the Bible.

666 = 102 ÷ 0.153 153 153 ...

102 ÷ 666 = 0. 153 153 153 ... = 170 ÷ 1110

0.6 x 1110 = 666 = (102 ÷ 170) x 1110

170 ÷ 102 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

NOTE: 136.7 ÷ 1110 = 0.123 153 153 153 ...

Triple 6 is expressed according to the whole of the circular shifts of number 153.

Circular shifts of 153:
153, 351, 135, 513, 315 et 531

⇒

666 = (135+153+315+351+513+531)÷3=1998÷3

In addition, number 153, characteristic of the miracles of Jesus, as its circular shifts are truly generators of anti-beast.

Trinitary-circular shifts of 153: 153, 351, 135, 513, 315 et 531 Generators of anti-beast 999
1/153=0.00 653 594 771 241 ...	⇒ 653 + 594 = 1247 124÷7=17.7 142 857 142 857 ... 653 ⇒ 6+5+3=14 594 ⇒ 5+9+4=18 18÷14=1.2 857 142 857 142 ... 14÷18=0.777 777 777 777 ...	⇒ 142 + 857 = 999 ⇒ 857 + 142 = 999 ⇒ 777
1/351=0.00 284 900 284 900 ...	⇒ 284 + 900 = 1184 84÷11= 7.636 363 636 363 ... ⇒ 1÷35= 0.02 857 142 857 142 ... 284 ⇒ 2+8+4=14 900 ⇒ 9+0+0=9 9÷14=0.642 857 142 857 142 ...	⇒ 636 + 363 = 999 ⇒ 857 + 142 = 999 ⇒ 857 + 142 = 999
1/135=0.00 740 740 740 ...	⇒ 740 + 740 = 1480 80÷14=5.7 142 857 142 857 ... ⇒ 1÷35= 0.02 857 142 857 142 ... ⇒ 5÷13=0.3 846 153 846 153 ... 740 ⇒ 7 + 4 + 0 = 11 1÷11=0.0 909 090 909 090 ...	⇒ 142 + 857 = 999 ⇒ 857 + 142 = 999 ⇒ 846 + 153 = 999 ⇒ 909 + 090 = 999
1/513=0.00 194 931 773 879 ...	⇒ 194 + 931 = 1266 12÷66 = 0.181 818 181 818 ... ⇒ 5÷13 = 0.3 846 153 846 153 ... 194 ⇒ 1+9+4=14 931 ⇒ 9+3+1=13 14÷13=1.0 769 230 769 230 ... 13÷14=0.92 857 142 857 142 ...	⇒ 181 + 818 = 999 ⇒ 846 + 153 = 999 ⇒ 769 + 230 = 999 ⇒ 857 + 142 = 999
1/315=0.00 317 460 317 460 ...	⇒ 317 + 460 = 777 7÷77=0.909 090 909 090 ... 1÷777=0.00 128 700 128 700 ... ⇒ 128 + 700 = 828 8÷28=0.2 857 142 857 142 ... 317 ⇒ 3+1+7=11 460 ⇒ 4+6+0=10 10÷11=0.909 090 909 090 ... 128 ⇒ 1+2+8=11 700 ⇒ 7+0+0=7 11÷7=1.57 142 857 142 857 ... 7÷11=0.636 363 636 363 ...	⇒ 909 + 090 = 999 ⇒ 857 + 142 = 999 ⇒ 909 + 090 = 999 ⇒ 142 + 857 = 999 ⇒ 636 + 363 = 999
1/531=0.00 188 323 917 137	⇒ 188 + 323 = 511 5÷11=0. 454 545 454 545 ...	⇒ 454 + 545 = 999

On the mathematical level, number 153 comprises remarkable properties. It is equal to the sum of the cubes of its three components. Consequently, number 153 is thus qualified as a number of Armstrong (still called "a more perfect number" or PDI, "Perfect Digital Invariant"), i.e. a number with n figures which is exactly the sum of n^th powers of its n components. Only three other numbers with 3 digits, 370, 371 and 407, have this property. However, the following sequence, produced by Sloane Neil J A. [A005188], 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... is made up only of numbers of Armstrong. Actually, a number of Armstrong is only one particular case of the general category of narcissistic numbers, i.e. any number equal to an expansion based on the proper figures composing this last.

153	=	(1) ³ + (5) ³ + (3) ³ 1 + 125 + 27
370	=	(3) ³ + (7) ³ + (0) ³ 27 + 343 + 0
371	=	(3) ³ + (7) ³ + (1) ³ 27 + 343 + 1
407	=	(4) ³ + (0) ³ + (7) ³ 64 + 0 + 343

Number 153 is a triangular number of which the root or secret value is 17.

153 = T₁₇ = 17(17 + 1)/2 = 17 x 9

In addition, the sum of the first 17 numbers is equal to 153.

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17

153

However, the triangular root of 153, that is number 17, is determining to obtain triple 6. Indeed, the sum of the squares of the seven prime numbers ranging between 1 and 17 gives 666.

666

2² + 3 ² + 5 ² + 7 ² + 11 ² + 13 ²+ 17 ²

Relative with the numerical power of 153, it is important to note that the total sum of its three components is equal to the basic MAXIMUM figure [9]. A scenario identical applies to the 666 as well as to its triangular root 36, whose total sum of its two components is also 9.

(36 + 1) x (17 + 1)

666

37 x 18

666 = Triangulation of 36 36 ⇒ 3 + 6 = 9	153 = Triangulation of 17
666 = T₃₆ = 36(36 + 1)/2	153 = T₁₇ = 17(17 + 1)/2
666 ⇒ 6 + 6 + 6 = 18 18 ⇒ 1 + 8 = 9	153 = 17 x 9 153 ⇒ 1 + 5 + 3 = 9

The sum of the first five factorials is equal to 153.

1! + 2! + 3! + 4! + 5!
1 + 2 + 6 + 24 + 120

153

The number of the beast is a function of factorials of all the components of numbers 153 and 123.

666	=	(1! x 5! x 3!) - 3!(1! + 2! + 3!)
666	=	(1! x 5! x 3!) - 6(1! + 2! + 3!)
666	=	6! - 6(1! + 2! + 3!)

With

6= 1+2+3

6 = 1 x 2 x 3 = 3!

6 = √(1³+2³+3³) = √36

6! = 1! x 5! x 3!

There is a direct link between number 231 (circular shift of 123 and divine or crowned number in the Jewish Cabal) and number 153, such as had noted S. Desrosiers. "La somme des diviseurs - parties aliquotes - du nombre 231, symbole du Dieu Trin, donne comme résultat 153" [Our translation: "The sum of the dividers - aliquot parts - of number 231, symbol of Trin God, gives like result 153 "]:

Dividers of number 231:
1, 3, 7, 11, 21, 33 et 77

⇒ 1 + 3 + 7 + 11 + 21 + 33 + 77 = 153

Number 231 comprises 7 dividers of which largest is 77: generating sign of anti-beast 777.

Gerard Colombat (In connection with number 153) cumulatively associates number 153 with 13, that he describes as "number of the rebellion", and with 1989 (year of the fall of the Berlin Wall). By breaking up 1989 into 3 triangular numbers, he establishes a link with the 666 according to relations hereafter:

13 x 153

1989

153 + 153 + 315 + 666 + 666

13 x 153 - 3(3² x 7²)

666

1989 - 1323

More vigorously, on the basis of links identified between numbers 888 and 153, Gerard Colombat concludes:

«Jésus qui a dit qu'il était l'Alpha et l'Oméga a aussi le nombre 801 pour symbolique puisque ce nombre correspond à la valeur alphanumérique de ces deux lettres grecques soit 800 + 1. Et la décomposition de 801 aboutit au même résultat 153.
Et tous les nombres divisibles par 3 aboutissent à ce même résultat de 153 selon le même processus et l'on retrouve ainsi la valeur " 1/3 " résiduelle présente dans les Ecritures.
Le nombre 666 aboutit donc au même résultat [que 888, 999, 777 et tout autre nombre divisible par 3], 153 le nombre Christique Victorieux par excellence, symbole de résurrection car tout conduit et ramène au NOM Unique qui a été donné aux hommes.»
[Our translation]
«Jesus who said that he was the Alpha and the Omega has also number 801 for symbolic code since this number corresponds to the alphanumeric value of these two Greek letters, that is 800 + 1. And the decomposition of 801 leads to same result 153.
And all numbers divisible by 3 lead to this same result of 153 according to same process and one thus finds the residual value "1/3" contained in the holy Writings.
Number 666 thus leads to the same result [as 888, 999, 777 and any other number divisible by 3], 153 the Christ Victorious Number by excellence, symbol of resurrection, for anything leads and brings back to the Single NAME which was given to the men.»

666 ⇒	[6] ³ + [6] ³ + [6] ³
	216 + 216 + 216	=648
648 ⇒	[6] ³ + [4] ³ + [8] ³
	216 + 64 + 512	=792
792 ⇒	[7] ³ + [9] ³ + [2] ³
	343 + 729 + 8	=1080
1080 ⇒	[1] ³ + [0] ³ + [8] ³ + [0] ³
	1 + 0 + 512 + 0	=513
513 ⇒	[5]³ + [1] ³ + [3] ³
	125 + 1 + 27	=153
153 ⇒	[1] ³ + [5] ³ + [3]³
	1 + 125 + 27	= 153

777 ⇒	[7] ³ + [7] ³ + [7]³
	343 + 343 + 343	= 1029
1029 ⇒	[1] ³ + [0] ³ + [2]³ + [9]³
	1 + 0 + 8 + 729	= 738
738 ⇒	[7]³ + [3]³ + [8] ³
	343 + 27 + 512	= 882
882 ⇒	[8] ³ + [8]³ + [2] ³
	512 + 512 + 8	= 1032
1032 ⇒	[1]³ + [0] ³ + [3]³ + [2]³
	1 + 0 + 27 + 8	= 36
36 ⇒	[3] ³ + [6] ³
	27 + 216	= 243
243 ⇒	[2] ³ + [4] ³ + [3]³
	8 + 64 + 27	= 99
99 ⇒	[9]³ + [9]³
	729 + 729	= 1458
1458 ⇒	[1] ³ + [4]³ + [5] ³ + [8] ³
	1 + 64 + 125 + 512	= 702
702⇒	[7]³ + [0] ³ + [2]³
	343 + 0 + 8	= 351
351⇒	[3]³ + [5] ³ + [1]³
	27 + 125 + 1	= 153
153 ⇒	[1] ³ + [5] ³ + [3]³
	1 + 125 + 27	= 153

888 ⇒	[8] ³ + [8] ³ + [8]³
	512 + 512 + 512	= 1536
1536 ⇒	[1] ³ + [5] ³ + [3]³ + [6]³
	1 + 125 + 27 + 216	= 369
369 ⇒	[3]³ + [6]³ + [9] ³
	27 + 216 + 729	= 972
972 ⇒	[9] ³ + [7]³ + [2] ³
	729 + 343 + 8	= 1080
1080 ⇒	[1]³ + [0] ³ + [8]³ + [0]³
	1 + 0 + 512 + 0	= 513
513 ⇒	[5] ³ + [1] ³ + [3]³
	125 + 1 + 27	= 153
153 ⇒	[1] ³ + [5] ³ + [3]³
	1 + 125 + 27	= 153

999 ⇒	[9] ³ + [9] ³ + [9]³
	729 + 729 + 512	= 2187
2187 ⇒	[2] ³ + [1] ³ + [8]³ + [7]³
	8 + 1 + 512 + 343	= 864
864 ⇒	[8]³ + [6]³ + [4] ³
	512 + 216 + 64	= 792
792 ⇒	[7] ³ + [9]³ + [2] ³
	343 + 729 + 8	= 1080
1080 ⇒	[1]³ + [0] ³ + [8]³ + [0]³
	1 + 0 + 512 + 0	= 513
513 ⇒	[5] ³ + [1] ³ + [3]³
	125 + 1 + 27	= 153
153 ⇒	[1] ³ + [5] ³ + [3]³
	1 + 125 + 27	= 153

Applied to the "number of the rebellion" 13, this iterative process leads on the cyclic terminal 133.

13 ⇒	[1] ³ + [3] ³
	1 + 27	= 28
28 ⇒	[2] ³ + [8]³
	8 + 512	= 520
520 ⇒	[5]³ + [2]³ + [0] ³
	125 + 8 + 0	= 133
133 ⇒	[1] ³ + [3]³ + [3] ³
	1 + 27 + 27	= 55
55 ⇒	[5]³ + [5] ³
	125 + 125	= 250
250 ⇒	[2] ³ + [5] ³ + [0]³
	8 + 125 + 0	= 133
133 ⇒	[1] ³ + [3]³ + [3] ³
	1 + 27 + 27	= 55
55 ⇒	[5]³ + [5] ³
	125 + 125	= 250
250 ⇒	[2] ³ + [5] ³ + [0]³
	8 + 125 + 0	= 133

However, number 133 is itself another winner against the beast, for it generates 999 in multiple ways.

133 ⇒ 1 + 3 + 3 = 7 ≈ Universal Anti-Beast
1/7 = 0.142 857 142 857 ...	⇒ 142 + 857 = 999
1/33 = 0.0 30 303 03 03 ...	⇒ 3 x (30 + 303) = 999
1/13 = 0.0769 230 769 230 ...	⇒ 769 + 230 = 999
3/13 = 0.230 769 230 769 ...	⇒ 230 + 769 = 999
13/3 = 4.333 333 333 ...	⇒ 3 x 333 = 999

1.11 666 and number 180

666 = 120 ÷ 0.180 180 180 ...

120 ÷ 666 = 0. 180 180 180 ... = 200 ÷ 1110

0.6 x 1110 = 666 = (120 ÷ 200) x 1110

200 ÷ 120 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

1.12 666 and number 216
The 666, whose secret value is 36, has also a link with number 216 (multiple of 36).

6 x 36	=	216	=	6³
1³ x 2³ x 3³	=	216	=	3³ + 4³ + 5³
6+[36x6]+6+[36x6]+6+[36x6]	=	666	=	6+[216]+6+[216]+6+[216]
6 + 6³+ 6 + 6³ + 6 + 6³	=	666	=	6+[6x6x6]+6+[6x6x6]+6+[6x6x6]

666 = 144 ÷ 0.216 216 216 ...

144 ÷ 666 = 0. 216 216 216 ... = 240 ÷ 1110

0.6 x 1110 = 666 = (144 ÷ 240) x 1110

240 ÷ 144 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

Triple 6 is expressed according to the whole circular shifts of number 216.

Circular shifts of 216:
216, 612, 621, 126, 261 et 162

⇒

666=(126+162+216+261+612+621)÷3=(1998)÷3

Number 216 and its circular shifts are truly generators of the anti-beast.

Trinitary-circular shifts of 216: 216, 612, 621, 126, 261 et 162 Generators of the anti-beast 999
1/216=0.00 462 962 962 962 ...	⇒ 462 + 962 = 1424 24÷14=1.7 142 857 142 857 ... ⇒ 6÷21=0.2 857 142 857 142 ...	⇒ 142 + 857 = 999 ⇒ 857 + 142 = 999
1/612=0.00 163 398 692 810	⇒ 163 + 398 = 561 1÷56= 0.17 857 142 857 142 ...	⇒ 857 + 142 = 999
1/621=0.00 161 030 595 813	⇒ 161 + 030 = 191 1÷91=0.0 109 890 109 890 ... ⇒ 6÷21=0.2 857 142 857 142 ... ⇒ 1÷62=0.0 161 290 322 580 645 161+290+322+580+645=1998	⇒ 109 + 890 = 999 ⇒ 857 + 142 = 999 ⇒ 1998 ÷2 = 999
1/126=0.00 793 650 793 650 ...	⇒ 793 + 650 = 1443 43÷14 = 3.07 142 857 142 857 ... ⇒ 1÷26 = 0.03 846 153 846 153 ... 793 ⇒ 7+9+3=19 650 ⇒ 6+5+0=11 19÷11=1.727 272 727 272 ...	⇒ 142 + 857 = 999 ⇒ 846 + 153 = 999 ⇒ 727 + 272 = 999
1/261=0.00 383 141 762 452	⇒ 383 + 141 = 524 4÷52=0.0 769 230 769 230 ... 383 ⇒ 3+8+3=14 141 ⇒ 1+4+1=6 6÷14=0.42 857 142 857 142 ... ⇒ 1÷26 = 0.03 846 153 846 153 ...	⇒ 769 + 230 = 999 ⇒ 857 + 142 = 999 ⇒ 846 + 153 = 999
1/162=0.00 617 283 950 617	⇒ 617 + 283 = 900 1÷900=0.00 111 111 111 ... 617 ⇒ 6+1+7=14 283 ⇒ 2+8+3=13 14÷13=1.0 769 230 769 230 ... 13÷14 = 0.92 857 142 857 142 ... ⇒ 1÷62=0.0 161 290 322 580 645 161+290+322+580+645=1998	⇒ 9 x 111 = 999 ⇒ 769 + 230 = 999 ⇒ 857 + 142 = 999 ⇒ 1998 ÷2 = 999

1.13 666 and number 360

666 = 240 ÷ 0.360 360 360 ...

240 ÷ 666 = 0. 360 360 360 ... = 400 ÷ 1110

0.6 x 1110 = 666 = (240 ÷ 400) x 1110

400 ÷ 240 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

Trinitary-circular shifts of 360: 036, 063, 306, 360, 603 et 630 Generators of the anti-beast 999
1/036=0.0 2 777 777 777 ... ⇒ 777	⇒ 2 x 777 = 1554 15÷54=0.2 777777777 ... ⇒ 777 4÷155=0.0 258064516129032 258064516129032 ... 1554÷7=222 1554÷14=111	258+064+516+129+032=999 ⇒ 4x222=888 ⇒ 9x111=999
1/063=0. 015873015873 015873 ... ⇒ 158+730=888 873+15=888 ⇒ 587+301=888	1÷888=0.001 12612 6 126126 ... 8÷88=0.0909090 90 90 90 ... 8÷888=0.00900900900 900 ...	⇒ 126+261+612=999 ⇒ 90+909=999 ⇒ 900+90+9=999
1/306=0.0 0326797385620915 0326797385620915 ... ⇒ 03+26+79+73+85+62+09+15=352	2÷35=0.0 57142857 142857 ... 3÷52=0.05 7692307 692307 ... 1÷352=0.002840 9090 90 90	⇒ 571+428=999 714+285=999 ⇒ 857+142=999 ⇒ 76+923=999 769+230=999 ⇒ 692+307=999 ⇒ 90+909=999
1/360=0.00 2 777 777 777 ... ⇒ 777	⇒ 2 x 777 = 1554 15÷54=0.2777777777 ... ⇒ 777 4÷155=0.0 258064516129032 258064516129032 ... 1554÷7=222 1554÷14=111	258+064+516+129+032=999 4x222=888 9x111=999
1/603=0.00165837479270315091 2106135986733 ... 6 triplets ⇒ 165+837+479+270+315+091=2157	21÷57=0.368421052631578947 368421052631578947 ... 368+421+052+631+578+947=2997 57÷21=2.71428571 42 857 142 ... 215÷7=30.7142857142857142 ...	⇒ 2997÷3=999 ⇒ 714+285=999 571+428=999 ⇒ 857+142=999
1/630=0.0 015873015873 015873 015873 ... ⇒ 158+730=888 873+15=888 ⇒ 587 + 301 = 888	1 ÷ 888=0.001 12612 6126126 ... 8÷88=0.0909090 90 90 90 ... 8÷888=0.00900900900 900 ...	⇒ 126+261+612=999 ⇒ 90+909=999 ⇒ 900+90+9=999

1.14 666 and number 810

666 = 540 ÷ 0.810 810 810 ...

540 ÷ 666 = 0. 810 810 810 ... = 900 ÷ 1110

0.6 x 1110 = 666 = (540 ÷ 900) x 1110

900 ÷ 540 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

Naohiro showed that the 666 is closely related to the dividers of mumber 810. These 19 last are:

1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 27 , [√810 = 28,46] , 30 , 45 , 54 , 81 , 90 , 135 , 162 , 270 , 405 .

60	=	1 + 3 + 5 + 9 + 15 + 27
606	=	30 + 54 + 90 + 162 + 270
36	=	2 + 6 + 10 + 18
36	=	3! x 6

666

45 + 81 + 135 + 405

666

(1 + 3 + 5 + 9 + 15 + 27) + (30 + 54 + 90 + 162 + 270)

1.15 666 and number 846

666 = 564 ÷ 0.846 846 846 ...

564 ÷ 666 = 0. 846 846 846 ... = 940 ÷ 1110

0.6 x 1110 = 666 = (564 ÷ 940) x 1110

940 ÷ 564 = 10/6 = 5/3 = 1.666 666 666 ... ⇒ 666

2. 666 and prime numbers
A prime number is a positive entirety which has only two dividers, 1 and itself.
Click on this link to see the 10 000 first prime numbers; the 10 000^th is 104 729.

666	=	888 - 222
666	=	(131+127+113+109+107+103+101+97) - (113+109)
Sum of 6 prime numbers	=	(Sum of 8 continuous prime numbers) - (Sum of 2 continuous prime numbers)
666	=	(131+127+107+103+101+97)

3. 666 and opposite numbers (reciprocals)

4. 666 and powers

666

1⁶ - 2⁶ + 3⁶

666

6 + 6 + 6 + 6³ + 6³ + 6³

666

(6 + 6 + 6) x (6² + 1²)
(18) x (36 + 1)
18 x 37

666

6! x (6² + 1²) / (6² + 2²)
720 x (36 + 1) / (36 + 4)
26 640 / 40
[Note: 6! = 1 x 2 x 3 x 4 x 5 x 6]

666

2² + 3² + 5² + 7² + 11² + 13² + 17²

666

1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 5³ + 4³ + 3³ + 2³ + 1³

(In the book of Martin Gardner, "Puzzles from other Worlds", chap. 31)

666

(37² + 37) - (3⁶ + 3³) + (2³ + 4^(3/2))
(1369 + 37) - (729 + 27) + (8 + 8)

Monte Zerber noted that the triplet (216, 630, 666) is a triplet of Pythagore:

(666)²	=	(630)² + (216)²
443 556	=	396 900 + 46 656
(666)²	=	(666 - 6 x 6)² + (6 x 6 x 6)²

216 ⇒		2 + 1 + 6 = 9
630 ⇒		6 + 3 + 0 = 9
666 ⇒	6 + 6 + 6 = 18 ⇒	1 + 8 = 9
46 656 ⇒	4 + 6 + 6 + 5 + 6 = 27 ⇒	2 + 7 = 9
396 900 ⇒	3 + 9 + 6 + 9 + 0 + 0 = 27 ⇒	2 + 7 = 9
443 556 ⇒	4 + 4 + 3 + 5 + 5 + 6 = 27 ⇒	2 + 7 = 9

Numbers 666 and 2583 are the two only positive entireties equal to the sum of cubes of the basic digits of their square, plus the sum of figures of their cube:

666²	=	443 556	666³	=	295 408 296
666	=	(4³ + 4³ + 3³ + 5³ + 5³ + 6³) + (2 + 9 + 5 + 4 + 0 + 8 + 2 + 9 + 6)

666 ⇒	6 + 6 + 6 = 18 ⇒	1 + 8 = 9
2583 ⇒	2 + 5 + 8 + 3 = 18 ⇒	1 + 8 = 9

2583²	=	6 671 889	2583³	=	17 233 489 287
2583	=	(6³+6³+7³+1³+8³+8³+9³) + (1+7+2+3+3+4+8+9+2+8+7)

According to Patrick De Geest, in "The Number of the Beast 666", 666 is equal to the sum of figures of its forty-seventh power, and is also the sum of digits of its 51^th power. That is to say:

666⁴⁷ = 5049969684420796753173148798405564772941516295265 4081881176326689365404466160330686530288898927188 59670297563286219594665904733945856

666⁵¹ = 9935407575913859403342635113412959807238586374694 3100899712069131346071328296758253023455821491848 0960748972838900637634215694097683599029436416

and with the sum of the figures on the right side equalizes to 666, in both two cases. In fact, 666 is the only greater integer with this property. Moreover, combining the two powers, 47 and 51, we obtain (4 + 7) x (5 + 1) = 66.
That is, 666 is equal to the sum of the cubes of the digits in its square (666² = 443556, and the sum of the cubes of these digits is 4³ + 4³ + 3³ + 5³ + 5³ + 6³ = 621) plus the sum of the digits in its cube (666³ = 295408296, and 2+9+5+4+0+8+2+9+6 = 45, and 621+45 = 666).

5.       666 and transcendent numbers (irrationals π, φ and e)
5.1.       666, the anti-beast (999) and number π
In mathematics, Pi is a real number noted π which is defined as the ratio of the circumference C of a circle to its diameter d = 2r (where r is the radius of the circle).
By definition:        π ≡ C ÷ d = C ÷ 2r

We show below the first 547 decimals of π:

3.1415926535897932384626433832795028841971693993751058209749445
  9230781640628620899862803482534211706798214808651328230664709
  3844609550582231725359408128481117450284102701938521105559644
  6229489549303819644288109756659334461284756482337867831652712
  0190914564856692346034861045432664821339360726024914127372458
  7006606315588174881520920962829254091715364367892590360011330
  5305488204665213841469519415116094330572703657595919530921861
  1738193261179310511854807446237996274956735188575272489122793
  81830119491298336733624406566430860213949463952247371907021

In the constant π, Michael Steuben discovered that the sum of the first 144 decimals is right equal to 666. It is important to announce here that 144 = (6 + 6) x (6 + 6) [D. Blatner^[2], 1997]. Also in π, this string 666 is at position 2440 while counting starting from the first digit after the decimal point (see Patrick De Geest). The prime palindromic number 727 follows immediately.

In addition, Eli Eshoh notes that the number ranging between the 1997^th and the 2001^th decimal of π is 90099 ⇒ 900 + 99 = 999. Always relative with π , Eshoh underlines three other curious relations. Initially, by interpreting the decimal point like a sign of multiplication and by considering per pairs the first ten decimals of π, he clarifies the link with number 666. Then, he raises the relation between the roots 3 and 4 of π and the famous number 1998.

π = 3.1415926536 ...

⇒

3x(14+15+92+65+36)

3 x 222=666

³√π = 1, 464 591 887 56 ...	⇒ 464 + 591 + 887 + 56 = 1998 ⇒ (464 + 591 + 887 + 56) ÷ 3 = 666 ⇒ (464 + 591 + 887 + 56) ÷ 2 = 999
⁴√π = 1, 331 335 36 ...	⇒ 331 + 335 = 666

π ≈ 3.141592573467

1.09999901 x 1.19999911 x 1.39999931 x 1.69999961

(Results based on works of Castellanos^[1])

π¹⁸ = 888 582 403,07126

For a definition of these concepts of Trigonometry (cosine, sine, theorem of Pythagore, etc.) see the excellent article of Thibaut BERNARD as well as the documents of Franck DUFFAUD and RICORDEL

5.2. 666, the anti-beast (999) and the golden number φ
By definition: φ ≡ ½(1 + √5) = 1.61803398875 ...

6(1 + φ) ÷ 5

3.141640786 ... ≈ π

(Results based on works of Dixon^[3])

By fixing φ = 1,618 033 989
³²√φ = 1,0 151 515 151 515 ...	⇒ 151 + 515 = 666
⁴⁴√φ = 1,0 10 99 666	⇒ 666
φ¹⁶ = 2206,99955	⇒ 999
φ¹⁸ = 5777,99983	⇒ 999
φ⁷¹ = 6,88846551 x 10¹⁴	⇒ 888
φ¹⁴³ = 7,6777252 x 10²⁹	⇒ 777
φ¹⁴⁹ = 1,3777118 x 10³¹	⇒ 777

5.3. 666, the anti-beast (999) and number e
By definition, the number e ≡ 2,7 182 8 182 8 459 045 ... ⇒ 718 + 281 = 999
The first decimals of the number e comprise twice times the sequence 18 / 28, which returns to twice times the couple 182 / 8, to finally lead to the palindrome number 8281828.

√e = 1,64 872127 0700 128 ...

⇒

872 + 127 = 999
8 + 1 = 9 = 7 + 2

³⁸√e = 1,02 666511

e¹⁶ = 8886110,520507872

e¹²² = 9,6366656 x 10⁵²

6. 666 and isomorphous triplets

111	=	666 ÷ 6
222	=	666 ÷ 3
333	=	666 ÷ 2
444	=	666 - (6 x 6 x 6) - 6 = 666 - 216 - 6
555	=	666 - 6 x 3 x 6 + 3 = 666 - 108 + 3
666	=	6x111 = 3x222 = 2x333 = 2x999÷3
777	=	666 + 6 x 3 x 6 + 3 = 666 + 108 + 3
888	=	666 + (6 x 6 x 6) + 6 = 666 + 216 + 6
999	=	3x666÷2

4 + 6 + 8

6 + 6 + 6

7. 666 and palindromic numbers
By definition, a palindrom is a word whose reading is identical as well in regular direction as in opposite direction. A good example is radar (guide for what follows!). In the same way, a palindromic number is a symmetrical number whose reading remains invariant in all the directions. The 666 is thus a famous triangular palindromic number.

Almost all integers (except the mysterious 196) lead to a palindromic number at the end of a simple iterative procedure of opposite-sum:

  Choose any number N                   39
  Reverse its digits                            93      [93 is the "mirror number" of 39]
  Make the sum of all                       39 + 93 = 132
  Continue the procedure until obtaining a palindrom
  Reverse the figures                        232
  Make the sum of all                      132 + 231 = 363 = palindrom

The application of the procedure to 69 gives the palindromic number 4884, after 4 iterations:

    69 + 96 = 165
              165 + 561 = 726
                          726 + 627 = 1353
                                      1353 + 3531 = 4884

Our famous triple 6 is deduced from 135 after only one and single iteration!:

135 + 531

666

John Walker, during three continuous years (from August 12, 1987 to May 24, 1990) applied this procedure to the number 196 [196 is the smallest number not generating a palindrom] without never obtaining a palindrom, and this, after 2,415,836 iterations!

The prime palindromic numbers are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, ... [see Sloane Neil J A., "An On-Line Version of the Encyclopedia of Integer Sequences", sequence A002385; Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematical Entertains New York: Dover, 1964, p. 228].

Patrick De Geest established that the sum of the 666 smaller prime palindromic numbers is the number 2391951273. In 1998, G.L. Honaker Jr. noticed that the digits of the aforementioned sum check the following remarkable equality [which he describes as "Trinity of the Beast"]:

2³ + 3³ + 9³ + 1³ + 9³ + 5³ + 1³ + 2³ + 7³ + 3³	=	666 + 666 + 666
The truly year of this discovery is 1998	=	3 x 666

There is no reason to panic for this hinge year 1998 is related as well to the beast (666) as with the anti-beast (999):

3 x 666

1998

999 x 2

666 ⇒ 6 + 6 + 6 = 18	18 ⇒ 1 + 8 = 9
999 ⇒ 9 + 9 + 9 = 27	27 ⇒ 2 + 7 = 9
1998 ⇒ 1 + 9 + 9 + 8 = 27	27 ⇒ 2 + 7 = 9
Divine omnipresence and omnipotence: final victory of triple 9

However, such as it comes to light from the next relations [noticed by Eli Eshoh and Gerard Colombat (The Magic of Numbers)], the mystery of the curious link between triple 6 and number 1998 [satanic year according to some] remains whole:

666⁴=196 741 925 136	⇒ 196+741+925+136=1998
666⁶=87 266 061 345 623 616	⇒ 87+266+061+345+623+616=1998
666⁷ = 58 119 196 856 185 328 256	⇒ 58+119+196+856+185+328+256=1998

Gerard Colombat (In connection with number 216) established another link between number 216 and triple 666 or 1998. Indeed, circular shifts of the digits composing number 216 lead to six possible combinations: 216, 612, 621, 126, 261 and 162. However the total sum of these permuted values of number 216 is equal to triple 666, that is to say 1998. We discovered that the total sum of the permuted values of number 153 is also equal to 1998.

Circular shifts of 216:
216, 612, 621, 126, 261 et 162

⇒

126+162+216+261+612+621=1998

Circular shifts of 153:
153, 351, 135, 513, 315 et 531

⇒

135+153+315+351+513+531=1998

Eli Eshoh also raises links between 1998 and the transcendent numbers e and π (cube root).

e = 2.718 281 828 ...	⇒ 718 + 281 = 999 ⇒ 2 x (718 + 281) = 2 x 999 = 1998
π = 3.14 15 92 65 36 ... ³√π = 1.464 591 887 56 ...	⇒ 464 + 591 + 887 + 56 = 1998

In addition, I discovered that 1998 are associated with number 62, like sum of the first 5 series of "trinitary" decimals generated by his reverse.

1÷62 = 0.0 161 290 322 580 645

161 + 290 + 322 + 580 + 645 = 1998
⇒ 1998 ÷2 = 999

The first prime palindromic number, of the form 1[0... 0]666[0... 0]1 is:

16661

G.L. Honaker, Jr noted that not only 16661 is the 1928^th prime number, but there is moreover equality of the numerical sums of the two numbers:

1 + 6 + 6 + 6 + 1 = 1 + 9 + 2 + 8

The next prime palindromic number, after 16661, is

1000000000000066600000000000001

which can be written with concision by using notation 1 0₁₃ 666 0₁₃ 1, where the subscript indicates the number of consecutive zeros. Harvey Dubner determined that the 7 prime numbers of this type have as subscripts 0, 13, 42, 506, 608, 2472, and 2623 [in J. Rec. Math. 26(4)].

In addition let us note the double masterly presence of the number 13 [Jesus and his apostles]. In spite of all the evil superstition which surrounds him, the number 13 is a spontaneous generator of the anti-beast:

1/13 = 0.0769 230 769 230 ...	⇒ 769 + 230 = 999
2/13 = 0.153 846 153 846 ...	⇒ 153 + 846 = 999

More generally, the division of any integer by 13 leads to the anti-beast (except of course the multiples of 13).

G.L. Honaker, Jr. [February 28, 1999] also identified a beautiful prime palindromic number, composed of 59 digits and containing the 666:

31300000000000000000000000006660000000000000000000000000313

This other splendid prime palindromic number, with 81 digits and with triple 6 in the center, is limited by 9230329 which is precisely the 666^th prime palindromic number:

923032900000000000000000000000000000000666000000000000000000000000000000009230329

In addition, this number comprises on the whole 64 zeros, that is to say two series of 32 surrounding the 666. However:

666 x 32 = 21312 a beautiful palindromic number
666 x 64 = 42624 another splendid palindromic number

Number 9230329 by itself only conducts G.L. Honaker, Jr. to another definition of his "Trinity of the Beast":

9+2+3+0+3+2+9 = 28 (the second perfect number, after the 6)
28 = 1³ + 3³ (... another trinity of the beast?)

Note that the central number, 303, is the number of prime entireties less than or equal to 1999; 1999 being itself the first prime number of form 666n+1 !!!

The 666 is the sum of two consecutive prime palindromic numbers:

666

313 + 353

The 666 is also the sum of three consecutive even palindromic numbers:

666

212 + 222 + 232

Triple 6 is also written by multiple ways like difference or sum of isomorphous palindroms:

666	=	999 - 333	333 + 333	=	666
666	=	888 - 222	222 + 444	=	666
666	=	777 - 111	111 + 555	=	666

These isomorphous palindroms are all multiples at the same time of 3 and 37:

111=37x3=37x3x1	222=37x6=37x3x2	333=37x9=37x3x3
444=37x12=37x3x4	555=37x15=37x3x5	666=37x18=37x3x6
777=37x21=37x3x7	888=37x24=37x3x8	999=37x27=37x3x9

In addition to its other special properties, the palindromic number 21312 is worth 32 times the number of the beast (Patrick De Geest, August 12, 1999). The sum of the whole digits composing this palindromic number [which does not comprise any 0] is equal to the number of its dividers.

21312 ÷ 32

666

2⁶ x 3² x 37	=	21312	=	2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 37
21312 ⇒ 2 + 1 + 3 + 1 + 2 = 9

Patrick De Geest also established that the smallest prime number containing exactly eleven times figure 6 is palindromic:

1,666,666,666,661
[See sequence of Sloane A037065]

Notes

^[1] Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.
Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148-163, 1988.
See also The best approximation to pi with primes Back up to the text
^[2]Blatner, D. The Joy of Pi. New York: Walker, back jacket, 1997. Back up to the text
^[3] Dixon, R. Back up to the text

666: Bivalent Myth