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Evokation
  
 
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AVATAR 4 AVATAR

 

THE GARDEN OF THE GOLDEN FLOWER

Longfield Beatty 1996

Page 207 / 208

"And the next quotation is "relayed" from Budge (op. Cit., p. 521), having come from Papyrus No. 10188 (Brit. Mus.) There have been some omissions in order to reinforce as much as possible the particular aspect of it which is our immediate concern."
"...from Papyrus No. 10188"

" THE LAMENT OF THE SISTERS "
( Isis and Nepthys over the dead Osiris)


"Beautiful Youth, come to thy exalted house at once: we see thee not.

"Hail, beautiful boy, come to thy house, draw nigh after
thy separation from us

"Hail Beautiful Youth, Pilot of Time, who groweth
except at this hour.
"Holy image of his Father, mysterious essence proceeding from Tem.

"The Lord! How much more wonderful is he than his
Father, the first-born son of the womb of his mother.

"Come back to us in thy actual form; we will embrace
thee. Depart not from us, thou Beautiful Face, dearly beloved
one, the image of Tem, Master of Love.

"Come thou in peace, our Lord, we would see thee.

"Great Mighty One among the Gods, the road that thou
travellest cannot be described.

"The Babe, the Child at morn and at eve, except when
thou encirclest the heavens and the earth with thy bodily form.

"Come, thou Babe, growing young when setting, our
Lord, we would see thee.

"Come in peace, Great Babe of His Father, thou art
established in thy house.

"Whilst thou travellest thou art hymned by us, and
life springeth up for us out of thy nothingness. O our Lord,
come in peace, let us see thee.

"Hail Beautiful Boy, come to thy exalted house.; let thy
back be to thy house. The Gods are upon their thrones.
Hail ! come in peace, King.

"Babe! How lovely it is to see thee! Come, come to us,
O Great One, glorify our love.

"O ye gods who are in Heaven.
O ye gods who are in Earth.
O ye gods who are in the Tuat.
O ye gods who are in the Abyss.
O ye gods who are in the service of the Deep.
We follow the Lord, the Lord, of Love!"

 


BRAHMA
"If the red slayer think he slays,
Or if the slain think he is slain
They know not well the subtle ways
I keep and pass and turn again."
R.W. Emerson

 

 

WHY SMASH ATOMS

A. K. Solomon 1940

Page 77

"ONCE THE FAIRY TALE HERO HAS PENETRATED THE RING OF FIRE ROUND

THE MAGIC MOUNTAIN

HE IS FREE TO WOO THE HEROINE IN HER CASTLE ON THE MOUNTAIN TOP"

 

 

THE TRUE AND INVISIBLE ROSICRUCIAN ORDER

Paul Foster Case 1981

Page 108

" Concerning the Invisible, Magical Mountain and the Treasure therein Contained."

 

 

THE MAGIC MOUNTAIN

Thomas Mann 1875-1955

Page 466

"Had not the normal, since time was, lived on the achievements of the abnormal? Men consciously and voluntarily descended into disease and madness, in search of knowledge which, acquired by fanaticism, would lead back to health; after the possession and use of it had ceased to be conditioned by that heroic and abnormal act of sacrifice. That was the true death on the cross, the true Atonement."

 

8
SON OF MAN
-
-
-
3
SON
48
21
3
2
OF
21
12
3
3
MAN
28
10
1
8
SON OF MAN
97
43
7
-
-
9+7
4+3
-
8
SON OF MAN
16
7
7
-
-
1+6
-
-
8
SON OF MAN
7
7
7

 

 

FOUR QUARTETS

BURNT NORTON

T.S. Eliot 1943

Page 15/16

"AT THE STILL POINT OF THE TURNING WORLD. NEITHER FLESH NOR FLESHLESS;

NEITHER FROM NOR TOWARDS; AT THE STILL POINT, WHERE THE DANCE IS,

BUT NEITHER ARREST NOR MOVEMENT. AND DO NOT CALL IT FIXITY,

WHERE PAST AND FUTURE ARE GATHERED.

NEITHER MOVEMENT FROM NOR TOWARDS,

NEITHER ASCENT NOR DECLINE. EXCEPT FOR THE POINT, THE STILL POINT,

THERE WOULD BE NO DANCE AND THERE IS ONLY THE DANCE"

 

 

LORD OF THE DANCE

I danced in the morning when the world was begun.
And I danced in the moon and the stars and the sun,
I came down from heaven and I danced on the earth;
At Bethlehem I had my birth.

Dance then wherever you may be;
I am the Lord of the Dance said he,
And I'll lead you all wherever you may be,
And I'll lead you all in the dance, said he.

I danced for the scribe and the Pharisee,
but they wouldn't dance and they wouldn't follow me.
I danced for the fishermen, for James and John-
They came with me and the dance went on.

Dance then wherever you may be;
I am the Lord of the Dance said he,
And I'll lead you all wherever you may be,
And I'll lead you all in the dance, said he.

I danced on the Sabbath and I cured the lame;
The holy people said it was a shame
They whipped and they stripped and they hung me on high
They left me there on a cross to die

Dance then wherever you may be;
I am the Lord of the Dance said he,
And I'll lead you all wherever you may be,
And I'll lead you all in the dance, said he.

I danced on a Friday when the sky turned black-
It's hard to dance with the devil on your back.
They buried my body and they thought I'd gone;
But I'm the dance and I still go on

Dance then wherever you may be;
I am the Lord of the Dance said he,
And I'll lead you all wherever you may be,
And I'll lead you all in the dance, said he.

They cut me down and I leapt up high
I am the life that will never, never die
I'll live in you if you'll live in me-
I am Lord of the dance said he.

 

8
DANCE
27
18
9

 

I

AM

THE DANCE AND THE DANCE

GOES

ON

 

4
NINE
-
42
-
24
-
6
-
6
3
-
SIX
-
52
-
25
-
7
7
4
NINE
-
42
-
24
-
6
-
6
3
-
SIX
-
52
-
25
-
7
7
4
NINE
-
42
-
24
-
-6
-
6
3
-
SIX
-
52
-
25
-
7
7
4
NINE
-
42
-
24
-
6
-
6
3
-
SIX
-
52
-
25
-
7
7
4
NINE
-
42
-
24
-
6
-
6
3
-
SIX
-
52
-
25
-
7
7
4
NINE
-
42
-
24
-
6
-
6
3
-
SIX
-
52
-
25
-
7
7
4
NINE
-
42
-
24
-
6
-
6
3
-
SIX
-
52
-
25
-
7
7
4
NINE
-
42
-
24
-
6
-
6
3
-
SIX
-
52
-
25
-
7
7
4
NINE
-
42
-
24
-
6
-
6
3
-
SIX
-
52
-
25
-
7
7
36
-
-
378
468
216
225
54
63
99
3+6
-
-
3+7+8
4+6+8
2+1+6
2+2+5
5+4
6+3
9+9
9
-
-
18
18
9
9
9
9
18
-
-
-
1+8
1+8
-
-
-
-
1+8
9
-
-
9
9
9
9
9
9
9

 

 

15
THE RAINBOW LIGHT
-
-
-
3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2
15
THE RAINBOW LIGHT
171
81
9
1+5
-
1+7+1
8+1
-
6
THE RAINBOW LIGHT
9
9
3

 

 

-
SUNS ENERGY
-
-
-
4
SUNS
73
28
1
6
ENERGY
74
38
2
10
SUNS ENERGY
147
66
3
1+0
-
1+4+7
6+6
-
1
SUNS ENERGY
12
12
3
-
-
1+2
1+2
-
1
SUNS ENERGY
3
3
3

 

 

-
-
-
-
-
LIGHT SOUND
-
-
-
L
=
3
-
5
LIGHT
56
29
2
S
=
1
-
5
SOUND
73
28
1
-
-
4
-
10
LIGHT SOUND
129
57
3
-
-
-
-
1+0
-
1+2+9
5+7
-
-
-
4
-
1
LIGHT SOUND
12
12
3
-
-
-
-
-
-
1+2
1+2
-
-
-
4
-
1
LIGHT SOUND
3
3
3

 

 
-
-
-
LIGHT SOUND
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
29
-
-
-
56
29
29
-
-
-
-
-
-
-
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
28
-
-
-
73
28
19
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
LIGHT SOUND
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
4
5
6
7
8
9
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
1+0
-
1+2+9
5+7
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
12
12
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
3
3
3
-
1
2
6
4
5
6
7
8
9

 

 
-
-
-
LIGHT SOUND
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
LIGHT SOUND
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
4
5
6
7
8
9
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
1+0
-
1+2+9
5+7
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
12
12
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
3
3
3
-
1
2
6
4
5
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 
-
-
-
LIGHT SOUND
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
-
5
LIGHT
56
29
2
-
-
-
-
-
-
-
-
-
-
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
10
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
I
=
9
2
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
LIGHT SOUND
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
-
5
LIGHT
56
29
2
-
1
2
6
4
5
6
7
8
9
S
=
1
-
5
SOUND
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
4
-
10
LIGHT SOUND
129
57
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
1+0
-
1+2+9
5+7
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
12
12
3
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LIGHT SOUND
3
3
3
-
1
2
6
4
5
6
7
8
9

 

 

2
RA
19
10
1
5
EGYPT
73
28
1

 


Amun (also Amon, Ammon, Amen) is the ancient Egyptian god of the sun and air. He is one of the most important gods of ancient Egypt who rose to prominence at Thebes at the beginning of the period of the New Kingdom (c. 1570-1069 BCE).29 Jul 2016

Amun, god of the air, was one of the eight primordial Egyptian deities. Amun's role evolved over the centuries; during the Middle Kingdom he became the King of the deities and in the New Kingdom he became a nationally worshipped god. He eventually merged with Ra, the ancient sun god, to become Amun-Ra.

Amun - Wikipedia

en.wikipedia.org › wiki › Amun

Amun was a major ancient Egyptian deity who appears as a member of the Hermopolitan Ogdoad. Amun was attested from the Old Kingdom together with his ...

Greek equivalent: Zeus

Symbol: two vertical plumes, the ram-headed ...

Major cult center: Thebes
Consort: Amunet; Wosret; Mut

munet · igh Priest of Amun · Precinct of Amun-Re · ategory:Amun

 

A
=
1
-
4
AMUN
-
-
-
-
-
-
-
-
A
1
1
1
-
-
-
-
-
M
13
4
4
-
-
-
-
-
U
21
3
3
-
-
-
-
-
N
14
5
5
A
=
1
-
4
AMUN
49
13
13
-
-
-
-
-
-
4+9
1+3
1+3
A
=
1
-
4
AMUN
13
4
4
-
-
-
-
-
-
1+3
-
-
A
=
1
-
4
AMUN
4
4
4

 

AMEN THE NAME

 

A
=
1
-
6
AMUN RA
-
-
-
-
-
-
-
-
RA
19
10
1
A
=
1
-
6
AMUN
-
-
-
-
-
-
-
-
A
1
1
1
-
-
-
-
-
M
13
4
4
-
-
-
-
-
U
21
3
3
-
-
-
-
-
N
14
5
5
A
=
1
-
6
AMUN RA
68
23
14
-
-
-
-
-
-
6+8
2+3
1+4
A
=
1
-
6
AMUN RA
14
4
5
-
-
-
-
-
-
1+4
-
-
A
=
1
-
6
AMUN RA
5
5
5

 

 

A
=
1
-
6
AMUN RA
-
-
-
-
-
-
-
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RE
23
14
5
A
=
1
-
6
AMUN
-
-
-
-
-
-
-
-
A
1
1
1
-
-
-
-
-
M
13
4
4
-
-
-
-
-
U
21
3
3
-
-
-
-
-
N
14
5
5
A
=
1
-
6
AMUN RE
72
27
18
-
-
-
-
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7+2
2+7
1+8
A
=
1
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6
AMUN RE
9
9
9
-
-
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A
=
1
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6
AMUN RE
9
9
9

 

 

A
=
1
-
4
ATEN
-
-
-
-
-
-
-
-
A
1
1
1
-
-
-
-
-
TEN
39
12
3
A
=
1
-
4
ATEN
40
13
13
-
-
-
-
-
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4+0
1+3
1+3
A
=
1
-
4
ATEN
4
4
4

 

 

A
=
1
-
4
ATEN
-
-
-
-
-
-
-
-
A
1
1
1
-
-
-
-
-
T
20
2
2
-
-
-
-
-
E
5
5
5
-
-
-
-
-
N
14
5
5
A
=
1
-
4
ATEN
40
13
13
-
-
-
-
-
-
4+0
1+3
1+3
A
=
1
-
4
ATEN
4
4
4

 

 

ESOTERIC = O SECRET I = ESOTERIC

ESOTERIC 6 SECRET 9 ESOTERIC

ESOTERIC = 9 SECRET 6 = ESOTERIC

ESOTERIC = I SECRET O = ESOTERIC

 

RA - IN - BOW

RAINBOW

 

7
RAINBOW
-
-
-
-
R+A
19
10
1
-
I+N
23
14
5
-
B+O+W
40
13
4
7
RAINBOW
82
37
10
-
-
8+2
3+7
1+0
7
RAINBOW
10
10
1
-
-
1+0
1+0
-
7
RAINBOW
1
1
1

 

I

ME

THEE

GODONEGOD

HOLY HOLY HOLY

666666999999999666666

DIVINELAW 99 LAWDIVINE

LOVE DIVINE 99 DIVINE LOVE

NAMES OF GOD 99 99 GOD OF NAMES

LETTERANDLANGUAGELANGUAGEANDLETTER

NUMBERSSIGNSSYMBOLSSYMBOLSSIGNSSNUMBERS

HOW GREAT THOU ART MY GOD HOW GREAT THOU ART

MAGICAL ALPHABET 999 MAGICAL 999 ALPHABET MAGICAL

AUM MANI PADME HUM 333 MUH EMDAP INAM MUA

LOVEEVOLVELOVELOVEEVOLVE 7777777 EVOLVELOVELOVEEVOLVELOVE

GODISISISISISISGODGODISISISISISIS 999999999 ISISISISISISGODGODISISISISISISGOD

 

 

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 351 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 126 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = 9 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

 

 

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 351 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 126 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 9 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

 

 

www.merriam-webster.com/dictionary/algorithm

a procedure for solving a mathematical problem (as of finding the greatest common divisor) in a finite number of steps that frequently involves repetition of an ...

algorithm [ˈælgəˌrɪðəm]
n
1. (Mathematics) a logical arithmetical or computational procedure that if correctly applied ensures the solution of a problem Compare heuristic
2. (Mathematics) Logic Maths a recursive procedure whereby an infinite sequence of terms can be generated Also called algorism
[changed from algorism, through influence of Greek arithmos number]
algorithmic adj
aal·go·rithm (lg-rm)
n.
A step-by-step problem-solving procedure, especially an established, recursive computational procedure for solving a problem in a finite number of steps.
algorithmically adv

algorithm (lg-rthm)
A finite set of unambiguous instructions performed in a prescribed sequence to achieve a goal, especially a mathematical rule or procedure used to compute a desired result. Algorithms are the basis for most computer programming.

Noun 1. algorithm - a precise rule (or set of rules) specifying how to solve some problem
algorithmic program, algorithmic rule
formula, rule - (mathematics) a standard procedure for solving a class of mathematical problems; "he determined the upper bound with Descartes' rule of signs"; "he gave us a general formula for attacking polynomials"
sorting algorithm - an algorithm for sorting a list
stemming algorithm, stemmer - an algorithm for removing inflectional and derivational endings in order to reduce word forms to a common stem algorithm
any methodology for solving a certain kind of problem.
See also: Mathematics

 

 

Algorithm
From Wikipedia, the free encyclopedia

Flow chart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" (or true) (more accurately the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).
In mathematics and computer science, an algorithm (i/ˈælɡərɪðəm/) is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning.

More precisely, an algorithm is an effective method expressed as a finite list[1] of well-defined instructions[2] for calculating a function.[3] Starting from an initial state and initial input (perhaps empty),[4] the instructions describe a computation that, when executed, will proceed through a finite [5] number of well-defined successive states, eventually producing "output"[6] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.[7]

Though al-Khwārizmī's algorism referred to the rules of performing arithmetic using Hindu-Arabic numerals and the systematic solution of linear and quadratic equations, a partial formalization of what would become the modern algorithm began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. Subsequent formalizations were framed as attempts to define "effective calculability"[8] or "effective method";[9] those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation 1" of 1936, and Alan Turing's Turing machines of 1936–7 and 1939. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem.[10]

Informal definition
For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.
While there is no generally accepted formal definition of "algorithm," an informal definition could be "a set of rules that precisely defines a sequence of operations."[11] For some people, a program is only an algorithm if it stops eventually; for others, a program is only an algorithm if it stops before a given number of calculation steps.[12]

A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers; an example (there are others) is described by the flow chart above and as an example in a later section.

Boolos & Jeffrey (1974, 1999) offer an informal meaning of the word in the following quotation:

No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ...you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.[13]

The term "enumerably infinite" means "countable using integers perhaps extending to infinity." Thus, Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be chosen from 0 to infinity. Thus an algorithm can be an algebraic equation such as y = m + n—two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):
Precise instructions (in language understood by "the computer")[14] for a fast, efficient, "good"[15] process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities)[16] to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively"[17] produce, in a "reasonable" time,[18] output-integer y at a specified place and in a specified format.
The concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term.

[edit] Formalization

Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform (in a specific order) to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):

Minsky: "But we will also maintain, with Turing . . . that any procedure which could "naturally" be called effective, can in fact be realized by a (simple) machine. Although this may seem extreme, the arguments . . . in its favor are hard to refute".[19]

Gurevich: "...Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage [1987], an algorithm is a computational process defined by a Turing machine".[20]

Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and/or stored for further processing. Stored data is regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures.

For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).

Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom", an idea that is described more formally by flow of control.

So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, setting the value of a variable. It derives from the intuition of "memory" as a scratchpad. There is an example below of such an assignment.

For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming.

[edit] Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see more at finite state machine, state transition table and control table), as flowcharts (see more at state diagram), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see more at Turing machine).

Representations of algorithms can be classed into three accepted levels of Turing machine description:[21]
1 High-level description:
"...prose to describe an algorithm, ignoring the implementation details. At this level we do not need to mention how the machine manages its tape or head." 2 Implementation description:
"...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level we do not give details of states or transition function." 3 Formal description:
Most detailed, "lowest level", gives the Turing machine's "state table". For an example of the simple algorithm "Add m+n" described in all three levels see Algorithm examples.
[edit] Implementation

Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.

[edit] Computer algorithms

Flowchart examples of the canonical Böhm-Jacopini structures: the SEQUENCE (rectangles descending the page), the WHILE-DO and the IF-THEN-ELSE. The three structures are made of the primitive conditional GOTO (IF test=true THEN GOTO step xxx) (a diamond), the unconditional GOTO (rectangle), various assignment operators (rectangle), and HALT (rectangle). Nesting of these structures inside assignment-blocks result in complex diagrams (cf Tausworthe 1977:100,114).
In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended "target" computer(s), in order for the target machines to produce output from given input (perhaps null).

"Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
Knuth: ". . .we want good algorithms in some loosely defined aesthetic sense. One criterion . . . is the length of time taken to perform the algorithm . . .. Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc"[22] Chaitin: " . . . a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"[23]
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant'"—such a proof would solve the Halting problem (ibid).

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This will be true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is . . . important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".[24]

Unfortunately there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example of using Euclid's algorithm will be shown below.

Computers (and computors), models of computation: A computer (or human "computor"[25]) is a restricted type of machine, a "discrete deterministic mechanical device"[26] that blindly follows its instructions.[27] Melzak's and Lambek's primitive models[28] reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters[29] (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.[30]

Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability".[31] Minsky's machine proceeds sequentially through its five (or six depending on how one counts) instructions unless either a conditional IF–THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution)[32] operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1).[33] Rarely will a programmer have to write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT.[34]

Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example".[35] But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.[36]

This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).

But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters".[37] When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" (division) instruction available rather than just subtraction (or worse: just Minsky's "decrement").

Structured programming, canonical structures: Per the Church-Turing thesis any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code" a programmer can write structured programs using these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language".[38] Tausworthe augments the three Böhm-Jacopini canonical structures:[39] SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.[40] An additional benefit of a structured program will be one that lends itself to proofs of correctness using mathematical induction.[41]

Canonical flowchart symbols[42]: The graphical aide called a flowchart offers a way to describe and document an algorithm (and a computer program of one). Like program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only 4: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm-Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles but only if a single exit occurs from the superstructure. The symbols and their use to build the canonical structures are shown in the diagram.

 

 

EVOLVE LOVE EVOLVE

LOVES SOLVE LOVES

EVOLVE LOVE EVOLVE

 

LOGARITHM


Logarithm - Wikipedia
en.wikipedia.org › wiki › Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed ...
Logarithmic scale · ?Natural logarithm · ?Common logarithm · ?Binary logarithm

 

Introduction to Logarithms - Math is Fun
www.mathsisfun.com › algebra › logarithms
Introduction to Logarithms. In its simplest form, a logarithm answers the question: How many of one number do we multiply to get another number?

?Working with Exponents · ?Logarithms Can Have Decimals · ?Exponents, Roots

 

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9
ALGORITHM
103
49
49
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1
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9
-
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9
LOGARITHM
-
-
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1
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9
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
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5
6
-
-
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G
=
7
3
1
G
7
7
7
-
-
-
-
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5
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7
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
M
=
4
9
1
M
13
4
4
-
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4
5
-
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49
-
9
LOGARITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
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-
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-
-
-
1+8
-
-
13
-
9
LOGARITHM
4
13
13
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2
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8
9
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1+3
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1+3
1+3
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-
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-
-
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-
4
-
9
LOGARITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

 

-
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10
ALGORITHM
103
49
49
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1
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10
LOGARITHM
-
-
-
-
1
2
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8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
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5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
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5
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8
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R
=
9
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1
R
18
9
9
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9
I
=
9
6
1
I
9
9
9
-
-
-
-
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5
-
-
-
9
-
-
49
-
10
LOGARITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
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4+9
-
1+0
-
1+0+3
4+9
4+9
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-
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1+8
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13
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1
LOGARITHM
4
13
13
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2
2
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1+3
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1+3
1+3
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4
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1
LOGARITHM
4
4
4
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9

 

LOGARITHM

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL MORDER

 

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ALGORITHM
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10
LOGARITHM
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1
2
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9
A
=
1
1
1
A
1
1
1
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1
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T
=
2
7
1
T
20
2
2
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L
=
3
2
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L
12
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-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
49
-
10
LOGARITHM
103
49
49
-
2
2
3
4
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
LOGARITHM
4
13
13
-
2
2
3
4
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LOGARITHM
4
4
4
-
2
2
3
4
6
7
8
9

 

MINUS 5

 

LOGARITHM

ALGORITHM

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
4
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
4
4
4

 

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

ALGORITHM

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL MORDER

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
1+0
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

MINUS THE 5

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
6
7
8
9
-
-
1+3
-
1+0
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
6
7
8
9

 

Algorithm - Wikipedia, the free e

Algorithm - Wikipedia, the free encyclopedia en.wikipedia.org/wiki/Algorithm

In mathematics and computer science, an algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and ...

 

A
=
1
-
9
ALGORITHM
103
49
4
A
=
1
-
10
ALGORITHMS
122
59
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
-
-
ALGORITHMS
-
-
-
-
-
-
-
1
A
1
1
1
-
-
-
-
1
L
12
3
3
-
-
-
-
1
G
7
7
7
-
-
-
-
1
O
15
6
6
-
-
-
-
1
R
18
9
9
-
-
-
-
1
I
9
9
9
-
-
-
-
1
T
20
2
2
-
-
-
-
1
H
8
8
8
-
-
-
-
1
M+S
32
14
5
A
=
1
-
10
ALGORITHMS
122
59
50
-
-
-
-
1+0
-
1+2+2
5+9
5+0
A
=
1
-
1
ALGORITHMS
5
14
5
-
-
-
-
-
-
-
1+4
-
A
=
1
-
1
ALGORITHMS
5
5
5

 

 

-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
9
-
8
-
1
+
=
24
2+4
=
6
=
6
-
-
-
-
-
15
-
9
-
8
-
19
+
=
51
5+1
=
6
=
6
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
3
7
-
9
-
2
-
4
-
+
=
26
2+6
=
8
=
8
-
-
1
12
7
-
18
-
20
-
13
-
+
=
71
7+1
=
8
=
8
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
12
7
15
18
9
20
8
13
19
+
=
122
1+2+2
=
5
1+0
5
-
-
1
3
7
6
9
9
2
8
4
1
+
=
50
5+0
=
5
1+0
5
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
9
-
-
-
-
-
-
9
occurs
x
1
=
9
5
10
A
L
G
O
R
I
T
H
M
S
-
-
27
-
-
10
-
41
-
1+0
-
-
-
-
9
9
-
-
-
-
-
-
2+7
-
-
1+0
-
4+1
5
1
A
L
G
O
R
I
T
H
M
S
-
-
9
-
-
1
-
5
-
-
1
3
7
6
9
9
2
8
4
1
-
-
-
-
-
-
-
-
5
1
A
L
G
O
R
I
T
H
M
S
-
-
9
-
-
1
-
5

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
T
=
2
-
1
T
20
2
2
L
=
3
-
1
L
12
3
3
M
=
4
-
1
M
13
4
4
5
-
5
-
-
5
-
-
5
O
=
6
-
1
O
15
6
6
G
=
7
-
1
G
7
7
7
H
=
8
-
1
H
8
8
8
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
6
7
8
9

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Preface to the New Edition

Page 5
Twelve years ago a little boy entered my imagination as he hopped across the centuries and played with numbers. I began to see how the simple architecture of our decimal system could be constructed in secret ways — not a building project this time but an abstract one. On the surface of our arithmetic countless combinations of numbers take part in tedious and exacting calculations but underneath it all there is pattern, governed by a repeating code of integers. The Sigma Code reduces numbers to a single digit and the illusion of the many is seen to be but the reflection of a few. This is not a book on maths: this is a book for anyone who can carry out simple sums in their heads, and who won't be short-changed knowingly.
When Number 9 first came out I received mail from many who played with numbers. They chased patterns; some had special numbers and even mystical systems. I was tempted to write about numerology but resisted. I wanted to write about the intricacy of what the.. numbers actually do and leave the reader to wonder about the larger irrational that seems to hover around such constructions.
If I were writing this book today the numbers would have featured in a wider context of structuring nature's patterns, and also playing the role of animator in algorithms that create unique architectural forms and shapes. I would also include my previous research into other base systems. But this book was a first step which came from a child-like urge, like playing with building blocks, to build out of our numbers — just the simple 1, 2, 3, up to number 9.

 

RESEARCH R E SEARCH ER RESEARCH

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

GODS NUMBER

7641-534259

GODS NUMBER

GOD NUMBERS

764-5342591

GOD NUMBERS

 

 

GOD NUMBER

 

-
-
-
-
3
GOD NUMBER
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
-
3
GOD
26
17
8
-
-
-
-
-
-
-
-
-
-
N
=
5
-
6
NUMBER
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
12
-
9
Add to Reduce
99
45
9
-
1
2
3
4
5
6
7
8
9
-
-
1+2
-
-
Reduce to Deduce
9+9
4+5
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
Essence of Number
18
9
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
-
3
GOD
26
17
8
-
-
-
-
-
-
-
-
-
-
N
=
5
-
7
NUMBER
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
12
-
10
Add to Reduce
99
45
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
1
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
D
=
4
3
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
17
-
3
-
26
17
17
-
-
-
-
-
-
-
-
-
-
N
=
5
4
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
5
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
6
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
B
=
2
7
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
29
-
6
-
73
28
28
-
-
-
-
-
-
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
1
2
3
8
10
6
7
8
9
N
=
5
-
6
NUMBER
73
28
1
-
-
-
-
-
1+0
-
-
-
-
-
-
12
-
9
Add to Reduce
99
45
9
-
1
2
3
8
1
6
7
8
9
-
-
1+2
-
-
Reduce to Deduce
9+9
4+5
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
Essence of Number
18
9
9
-
1
2
3
8
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
Essence of Number
9
9
9
-
1
2
3
8
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
-
3
GOD
26
17
8
-
-
-
-
-
-
-
-
-
-
N
=
5
-
7
NUMBER
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
12
-
10
Add to Reduce
99
45
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
1
1
G
7
7
7
-
1
-
-
-
-
-
7
8
-
O
=
6
2
1
O
15
6
6
-
1
-
-
-
-
6
-
8
-
D
=
4
3
1
D
4
4
4
-
1
-
-
4
-
-
-
8
-
N
=
5
4
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
U
=
3
5
1
U
21
3
3
-
1
-
3
-
-
-
-
8
-
M
=
4
6
1
M
13
4
4
-
1
-
-
4
-
-
-
8
-
B
=
2
7
1
B
2
2
2
-
1
2
-
-
-
-
-
8
-
E
=
5
8
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
R
=
9
9
1
R
18
9
9
-
1
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
1
2
3
8
10
6
7
8
9
N
=
5
-
6
NUMBER
73
28
1
-
-
-
-
-
1+0
-
-
-
-
-
-
12
-
9
Add to Reduce
99
45
9
-
1
2
3
8
1
6
7
8
9
-
-
1+2
-
-
Reduce to Deduce
9+9
4+5
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
Essence of Number
18
9
9
-
1
2
3
8
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
Essence of Number
9
9
9
-
1
2
3
8
1
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
-
3
GOD
26
17
8
-
-
-
-
-
-
-
-
-
-
N
=
5
-
7
NUMBER
73
28
1
-
-
-
-
-
-
-
-
-
-
-
-
12
-
10
Add to Reduce
99
45
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
B
=
2
7
1
B
2
2
2
-
1
2
-
-
-
-
-
8
-
U
=
3
5
1
U
21
3
3
-
1
-
3
-
-
-
-
8
-
D
=
4
3
1
D
4
4
4
-
1
-
-
4
-
-
-
8
-
M
=
4
6
1
M
13
4
4
-
1
-
-
4
-
-
-
8
-
N
=
5
4
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
8
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
O
=
6
2
1
O
15
6
6
-
1
-
-
-
-
6
-
8
-
G
=
7
1
1
G
7
7
7
-
1
-
-
-
-
-
7
8
-
R
=
9
9
1
R
18
9
9
-
1
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
1
2
3
8
10
6
7
8
9
N
=
5
-
6
NUMBER
73
28
1
-
-
-
-
-
1+0
-
-
-
-
-
-
12
-
9
Add to Reduce
99
45
9
-
1
2
3
8
1
6
7
8
9
-
-
1+2
-
-
Reduce to Deduce
9+9
4+5
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
Essence of Number
18
9
9
-
1
2
3
8
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
Essence of Number
9
9
9
-
1
2
3
8
1
6
7
8
9

 

 

THE GOD NUMBERS

 

-
-
-
-
13
THE GOD NUMBERS
-
-
-
T
=
2
-
3
THE
33
15
6
G
=
7
-
3
GOD
26
17
8
N
=
5
-
7
NUMBERS
92
38
2
-
-
14
-
13
THE GOD NUMBERS
151
70
7
-
-
1+4
-
1+3
-
1+5+1
7+0
-
-
-
5
-
4
THE GOD NUMBERS
7
7
7

 

 

-
-
-
-
13
THE GOD NUMBERS
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
-
-
-
-
-
-
-
-
-
N
=
5
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
14
-
13
THE GOD NUMBERS
151
70
7
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
15
-
3
-
33
15
15
-
-
-
-
-
-
-
-
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
17
-
3
-
26
17
17
-
-
-
-
-
-
-
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
B
=
2
10
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
12
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
13
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
29
-
6
-
92
38
29
-
-
-
-
-
-
-
-
-
-
-
-
-
-
13
THE GOD NUMBERS
-
-
-
-
1
4
3
8
15
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
1+5
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
1
4
3
8
6
6
7
8
9
N
=
5
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
14
-
13
THE GOD NUMBERS
151
70
16
-
1
4
3
8
6
6
7
8
9
-
-
1+4
-
1+3
-
1+5+1
7+0
1+6
-
-
-
-
-
-
-
-
-
-
-
-
5
-
4
THE GOD NUMBERS
7
7
7
-
1
4
3
8
6
6
7
8
9

 

 

-
-
-
-
13
THE GOD NUMBERS
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
-
-
-
-
-
-
-
-
-
N
=
5
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
14
-
13
THE GOD NUMBERS
151
70
7
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
B
=
2
10
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
12
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
13
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
13
THE GOD NUMBERS
-
-
-
-
1
4
3
8
15
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
1+5
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
1
4
3
8
6
6
7
8
9
N
=
5
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
14
-
13
THE GOD NUMBERS
151
70
16
-
1
4
3
8
6
6
7
8
9
-
-
1+4
-
1+3
-
1+5+1
7+0
1+6
-
-
-
-
-
-
-
-
-
-
-
-
5
-
4
THE GOD NUMBERS
7
7
7
-
1
4
3
8
6
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
13
THE GOD NUMBERS
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
-
-
-
-
-
-
-
-
-
N
=
5
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
14
-
13
THE GOD NUMBERS
151
70
7
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
13
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
B
=
2
10
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
U
=
3
8
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
R
=
9
12
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
13
THE GOD NUMBERS
-
-
-
-
1
4
3
8
15
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
1+5
-
-
-
-
G
=
7
-
3
GOD
26
17
8
-
1
4
3
8
6
6
7
8
9
N
=
5
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
14
-
13
THE GOD NUMBERS
151
70
16
-
1
4
3
8
6
6
7
8
9
-
-
1+4
-
1+3
-
1+5+1
7+0
1+6
-
-
-
-
-
-
-
-
-
-
-
-
5
-
4
THE GOD NUMBERS
7
7
7
-
1
4
3
8
6
6
7
8
9

 

 

NUMBER = 534259 = 1 = 534259 NUMBER

NUMBER = 234559 NUMBER

NUMBER = 534259 = 1 = 534259 NUMBER

 

 

NUMBERS = 5342591 = 2 = 5342591 NUMBERS

SBUMNER = 1234559 = SBUMNER

NUMBERS = 5342591 = 2 = 5342591 NUMBERS

 

-
-
-
-
Q
NUMBERS
-
Q
Q
-
1
2
3
4
5
6
7
8
9
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
6
7
8
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
6
7
8
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
6
7
8
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
6
7
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
6
7
8
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
6
7
8
-
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
6
7
8
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
-
-
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
6
7
8
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
-
-
-
-
Q
NUMBERS
-
Q
Q
-
-
-
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
-
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
9

 

 

-
-
-
-
Q
NUMBERS
-
Q
Q
-
1
2
3
4
5
6
7
8
9
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
6
7
8
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
6
7
8
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
6
7
8
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
6
7
8
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
6
7
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
6
7
8
9
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
6
7
8
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
-
-
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
6
7
8
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
-
-
-
-
Q
NUMBERS
-
Q
Q
-
-
-
-
-
-
-
S
=
1
-
1
S
19
1
1
-
1
-
-
-
-
-
B
=
2
-
1
B
2
2
2
-
-
2
-
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
-
-
29
4
7
NUMBERS
92
29
29
-
1
2
3
4
10
9
-
-
2+9
Q
-
Q
9+2
2+9
2+9
-
-
-
-
-
1+0
-
Q
-
11
-
7
NUMBERS
11
11
11
-
1
2
3
4
1
9
-
-
1+1
Q
-
Q
1+1
1+1
1+1
-
-
-
-
-
-
-
Q
-
2
-
7
NUMBERS
2
2
2
-
1
2
3
4
1
9

 

 

THE

BALANCING

ONE TWO THREE FOUR

FIVE

NINE EIGHT SEVEN SIX

4 FIVE 42 24 6

1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1

15 ONE TWO THREE FOUR 208 82 1
4 FIVE 42 24 6
17 NINE EIGHT SEVEN SIX 208 91 1

1234 5 6789

 

 

prime number - Whatis Techtarget
https://whatis.techtarget.com/definition/prime-number
A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole numbers that can be divided evenly into another number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

 


Prime number - Wikipedia
https://en.wikipedia.org/wiki/Prime_number

The first 25 prime numbers (all the prime numbers less than 100) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (sequence A000040 in the OEIS).
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers (2 × 3) that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.
Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several proofs of the theorem.

 

Prime numbers
www-groups.dcs.st-and.ac.uk/history/HistTopics/Prime_numbers.html
Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to ...
Prime numbers

Number theory index History Topics Index

Version for printing

Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
You can see more about these numbers in the History topics article Perfect numbers.

By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers.

In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes.

There is then a long gap in the history of prime numbers during what is usually called the Dark Ages.

 

PRIME NUMBERS

 

P
=
2
-
5
PRIME
61
34
7
N
=
2
-
7
NUMBERS
92
38
2
-
-
17
-
12
Add to Reduce
153
72
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
8
-
3
Essence of Number
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
34
-
11
-
61
34
34
-
-
-
-
-
-
-
-
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
29
-
10
-
92
38
29
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
6
-
8
9
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
6
-
8
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
6
-
8
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
6
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
6
-
8
-
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
6
-
8
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
6
-
8
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
6
-
8
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
6
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
6
-
8
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
6
-
8
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
6
7
8
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
6
-
8
9
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
6
-
8
9
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
6
7
8
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
6
7
8
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
6
7
8
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
6
7
8
9

 

6 +8 = 14 1+4 = 5

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
7
9
P
=
2
-
5
PRIME
61
34
7
-
-
-
-
-
-
-
-
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
7
9
S
=
1
12
1
S
19
10
1
-
1
-
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
-
-
U
=
3
7
1
U
21
3
3
-
-
-
3
-
-
-
-
M
=
4
8
1
M
13
4
4
-
-
-
-
4
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
7
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
9
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
9
R
=
9
11
1
R
18
9
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
1
2
3
8
15
7
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
-
2+7
P
=
2
-
5
PRIME
61
34
7
-
1
2
3
8
6
7
9
N
=
2
-
7
NUMBERS
92
38
2
-
-
-
-
-
-
-
-
-
-
17
-
12
Add to Reduce
153
72
9
-
1
2
3
8
6
7
9
-
-
1+7
-
1+2
Reduce to Deduce
1+5+3
7+2
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
9
9
9
-
1
2
3
8
6
7
9

 

 

I = 9 9 = I
ME = 9 9 = ME
BRAIN + BODY = 9 9 = BODY + BRAIN
LIGHT + DARK = 9 9 = DARK + LIGHT
ENERGY + MASS = 9 9 = MASS +ENERGY
MIND + MATTER = 9 9 = MATTER + MIND
MAGNETIC + FIELD = 9 9 = FIELD + MAGNETIC
POSITIVE + NEGATIVE = 9 9 = NEGATIVE + POSITIVE
973 OM AZAZAZAZAZAZAZAZAZZAZAZAZAZAZAZAZAZAOM 973

 

 
 
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