A NEW LAW REVEALING HIDDEN EQUILIBRIUM IN ALL NUMBER SEQUENCES

Σ ∆n = 0

JOYCE P. BOWEN, Ph.D.

1
 ©2003 Joyce P. Bowen
revised 2004
revised 2008

All Rights Reserved

ISBN 0-9615454-2-9

Cover Art by Yaounde Olu

The author may be contacted at

astropoint@aol.com

Astropoint Research Associates

Chicago, Illinois

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 ACKNOWLEDGEMENTS

I am grateful for constructive criticism, advice and/or philosophical support from

Emanuel Gugwor, Dr. Sow Aboubacar Sidy, Christine Bowen, Pamela Dominguez,
Charmaine Anderson, Jeanne Ricks, Lerone Bennett, Jr., and Jeffrey Shallit. Thank
you all.

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 TABLE OF CONTENTS

Page
Introduction……………………………………………………………………….7

Chapter One ………………………………………………………………….…..8

Rules of Analysis

Chapter Two……………………………………………………………………. 12
Circular and Periodic Numbers

Chapter Three………………………………………………………………..…14
Other Numbers

Chapter Four………………………………………………………………….... 22
Pi Patterns

Chapter Five…………………………………………………………………..…31
Whole Number Increments

Chapter Six………………………………………………………………….…..36
Irrational and Transcendental Numbers

Chapter Seven……………………………………………………………….....41
Base Two

Chapter Eight……………………………………………………………….. …42

Fibonacci and Lucas Numbers

Chapter Nine………………………………………………………………...….46
Pascal’s Triangle

Chapter Ten………………………………………………………………….….48
Prime Numbers

Chapter Eleven………………………………………………………………....53
Square Business

Chapter Twelve………………………………………………………………....63
Numbers of Proton and Neutron Stability

Chapter Thirteen……………………………………………………..……..….66
Notions about Nothing and Everything

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Chapter Fourteen…………………………………………………………………….69
Additive Assessment

Chapter Fifteen……………………………………………………………………….74
Multiplicative Assessment

Chapter Sixteen…………………………………………………………………..….78
Factorials

Chapter Seventeen……………………………………………………………..……79
Conclusions and Implications/The end of randomness

Analysis Catalogue…………………………………………………………………..82

Natural Numbers – 1 – 100…………………………………………………..83

Reciprocals of 1 – 100…………………………………………………..……98

Mirrors……………………………………………………………………..….112

Other Number Sequences……………………………………….……..…..118

Bibliography……………………………………..……………………………….…125

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For every continuous symmetry of the
laws of physics, there must exist a
conservation law.

For every conservation law, there must

exist a continuous symmetry.
(Proved by Emmy Noether in 1905)

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INTRODUCTION

In today=s society numbers are merely seen as tools for calculation. The notion
that numbers are things in themselves@ and have deeper import is not always
considered. In this book I will examine numbers with the goal of uncovering
deeper meanings and evidence of an underlying order to seemingly random
numerical systems. Increment Analysis, assessing number relationships between
adjacent numbers in numerical sequences, is a technique used to identify and
synthesize number relationships.

Increment Analysis differs from the calculus of sequence differences in

that the result of analysis is a charged number line. Increment Analysis
reveals previously unobserved patterns in Fibonacci numbers, Lucas
numbers and prime numbers. In a nutshell, Increment Analysis reveals
obvious, arithmetical characteristics of number sequences that were
heretofore hidden, and where the increments of all whole numbers, or
fragments of irrational or transcendental numbers sum to zero while
exhibiting beautiful symmetries.

There are mirror image groupings that cancel each other with a resultant of zero
in circular numbers. In some cases, 58 and 60 element mirror image
sequences are revealed. In addition, there are oscillating digits denoting a wave
pattern whose sum tends to converge to zero in non-circular number sequences.
When zeros are added before and after any sequence, the result is digits which
cancel to zero. Increment Analysis, therefore, may uncover a type of numerical
homeostasis with zero as a focal point. In other words, there seems to be a
Law of Digit Balance wherein the digits of all numbers in all number sequences
have increments that sum to zero in beautifully arranged symmetries.

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 CHAPTER ONE

RULES OF ANALYSIS
1. Select a number sequence and evaluate the increments between
adjacent numbers so that they become positive or negative in
relation to each other. Ignore decimal points and treat the
number as a continuous sequence. For example, the reciprocal of
CONJECTURE
seven is a series of repeating numbers 0.01428571…:

1 4 2 8 5 7 1…:

The increment between 1 and 4 is +3

“ “ “ 4 and 2 is –2
“ “ “ 2 and 8 is +6
“ “ “ 8 and 5 is –3
“ “ “ 5 and 7 is +2
“ “ “ 7 and 1 is -6

The resulting number string, therefore, is

+3 -2 +6 -3 +2 -6…
2. Isolate the resulting binary patterns and cancel them to zero,
leaving one or more remainders. In the above case, +3 -2 and –3
+2 cancel each other, and the +6 and –6 also cancel each other,
resulting in 0.

3. To see the underlying patterns of any sequence or sequence

segment that is not a circular number, you add zeros before and
after the sequence and proceed with the methodology outlined in
number 1 above. The digits will always cancel to zero.

4. Though it is important to keep in mind that each digit is an

increment that is ten times more precise than the last in the
decimal expansion of numbers, the numbers are taken at face
value.

5. For further details on analysis and extracting 2nd, 3rd, or N order

sequences from non-circular number segments without adding
zeros, see the chapter on Pi Patterns.

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CONJECTURE:
The increments between adjacent numbers in circular,1 or revolving
number sequences generate oscillating mirror image number
groupings that always cancel to 0.

EXPERIMENTAL DATA

Let nC = digits in a circular number sequence

Let ∆ = the increments between adjacent numbers in the sequence

If nc is extended indefinitely,

Then: Σ∆n C =0

The Reciprocal of Number Seven

The sum of the increments between the digits of the circular number, the
reciprocal of seven, yields oscillating digits that cancel to 0.
The reciprocal of the number seven, 1/7, = 0.01428571428571…

Analysis: the increment between 1 and 4 is +3; between 4 and 2 is –2; between
2 and 8 is +6; between 8 and 5, -3; between 5 and 7 is +2, etc. The resultant
number string is +3 –2 +6 –3 +2 –6 +3 –2 +6 –3 +2 –6 …∞
∞=0
Hence, the outcome yields 6-element mirror image groupings that cancel to zero.

1 Circular numbers produce the same arithmetic sequence of numbers, but starting in different positions, when
multiplied or divided by given numbers.
9
The Reciprocal of Number Thirteen

The sum of the increments between the digits of the reciprocal of thirteen yields
oscillating mirror-image digits that cancel to 0.
The reciprocal of the number thirteen, 1/13, is

0.0769230769230…

The resultant number string after increment analysis is:

+7 -1 +3 -7 +1 –3 +7 -1 +3 -7 +1 - 3…∞
∞=0

Again, the outcome yields 6-element mirror image groupings that cancel to zero.
The number thirteen, when divided by any number between one and nine, yields
two cyclic variations based on the numbers 0.076923 and 0.1538461. The
0.076923 pattern was analyzed above. Let’s look at 0.15384615…
15384615… after analysis yields

+4 - 2 +5 -4 +2 - 5 +4 -2 +5 - 4 +2 - 5…∞
∞=0

Once again, we see 6-element mirror image groupings that cancel to zero.

The Reciprocal of Number 17

The sum of the increments between the digits of the cyclic number, the reciprocal
of seventeen, 1/17, yield oscillating digits that cancel to 0.

The reciprocal of the number seventeen = 1/17=0.05882352941176470…

After synthesis:
+5 +3 0 - 6 +1 +2 - 3 +7 - 5 - 3 0 +6 -1 -2 +3 -7…∞
∞=0

This is a 16-element mirror image grouping that cancels to zero.

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The Reciprocal of Seven divided by its mirror image and then divided

by the result:

Dividing the reciprocal of seven (142857) by its mirror image (758241) and then
dividing the result (5.30769203…) by seven results in oscillating digits that cancel
to zero.
5.30769203 ÷ 7 = 0.7582417…

7 5 8 2 4 1 7…

The resultant number string, after synthesis, is

-2 +3 -6 +2 -3 +6…∞
∞=0

…a 6-element mirror image grouping that cancels to zero.

The number seven divided by thirteen results in oscillating digits that

cancel to zero.

7/13 = 0.538461538461538461…:

53 8 4 6 1 5…

After synthesis yields

-2 +5 -4 +2 -5 +4… ∞ = 0

6-element mirror images that cancel to zero.

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 CHAPTER TWO

CIRCULAR NUMBERS AND THE NUMBER NINE

All circular numbers are related to the number nine, in that they produce series of nines
when they are divided in half. It may be surmised, therefore, that the number nine is related
to zero, since the increments between the digits of circular numbers cancel to zero.
Consider the following results when the digits of circular numbers are added together to
produce a final number:

1/7=0.142857; 1 + 4 + 2 + 8 + 5 + 7 = 27; 2 + 7 = 9

1/13 = 0.076923; 7 + 6 + 9 + 2 + 3 = 27; 2 + 7 = 9

1/17 = 0.0588235294117647; 5+8 + 8 +3 + 3 + 5 + 2 + 9 + 4 + 1 + 1 + 7 + 6 + 4 + 7 = 72;

7+2=9

2/13 = 153846; 1 + 5 + 3 + 8 + 4 + 6 = 27; 2 + 7 = 9

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Number Pattern after synthesis

1/7=0.142857; +3 -2 +6 -3 +2 -6…. ∞ = 0

1/13 = 0.076923; +7 -1 +3 -7 +1 -3… ∞ = 0

1/17 = 0.0588235294117647; +5 +3 0 -6 +1 +2 -3 +7 -5 -3 0 +6 -1 -2 +3 -7…∞

∞
=0
2/13 = 153846; +4 -2 +5 -4 +2 -5…∞
∞=0

The mirror image symmetry produced by increment analysis of these numbers is sheer
beauty. Consider the reciprocal of 17 where a 16-element mirror image grouping is
generated!
THE NUMBER NINE CYCLES TO ZERO
In a related matter, whenever nine divides any number between 1 and 8 the result is
repeating digits (repdigits) whose increments equal zero. For example:

1/9 = 0.111111111 = 00000000........∞

∞=0

2/9 = 0.222222222 = 00000000........∞

∞=0

3/9 = 0.333333333 = 00000000........∞

∞=0

4/9 = 0.444444444 = 00000000........∞

∞=0

5/9 = 0.555555555 = 00000000........∞

∞=0

6/9 = 0.666666666 = 00000000........∞

∞=0

7/9 = 0.777777777 = 00000000........∞

∞=0

8/9 = 0.888888888 = 00000000........∞

∞=0

All of the foregoing implies a “zero-ness” of cycles (circles).

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 CHAPTER THREE

NON-CIRCULAR NUMBERS

The previous chapters illustrated the tendency of circular numbers to generate

mirror image oscillating digits that cancel to zero. The case will be made in this
chapter that non-monotonic increasing or decreasing number sequences
generate number patterns with adjacent binary pairs and mirror image
symmetries. Sometimes there is a remainder, a Point of Synthesis, and a mode
of 0. Increment Analysis reveals hidden patterns in all number sequences.

CONJECTURE

All number sequences that are non-monotonic increasing or decreasing generate

non-random number patterns with oscillating digits that result in a residual Point
of Synthesis and a mode of 0.
Let n = digits in a number sequence.

Let ∆ = increments between adjacent digits.

Let Ps = Point of Synthesis

If n is a non- monotonic increasing or decreasing number sequence,

Then: Σ∆n=P s ≥ 0

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EXPERIMENTAL DATA

The reciprocal of a rough approximation of the fine structure constant

(1/137) produces digits that cancel to zero.

1/137 = 0.007299270…

=0 +7 -5 +7 0 -7 +5 -7 0 ∞=0

The result is 6-element mirrors that cancel to zero, the Point of Synthesis.

(1/137 is a natural palindrome, i.e., 729 927, which forms a palindrome mirror)

RECIPROCAL OF # 22

The reciprocal of the number 22 is 0.0454545454545…

454545454545

The resultant number string after analysis is:

+1 –1 + 1 –1 +1 –1 +1 –1 +1 –1…

=0

After canceling the binaries, the remainder is 0. Zero, therefore, is the Ps.

THE NUMBER 4 ÷ 22

= 0.181818181818181818…

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 1 8 1 8 1 8 1 8 1…

After synthesis yields

+7 –7 +7 –7 +7 –7 +7 –7…∞
∞= 0

The Ps is 0.

The reciprocal of thirteen subtracted from the reciprocal of seven is

the same as the reciprocal of seven subtracted from the reciprocal of
thirteen, only negative.
0.142857142 (carried out to nine digits)
-0.076923076
0.065934066

and interestingly,

0.076923076
-0.142857142
- 0.065934066

The resulting number sequence in both cases is

0 6 5 9 3 4 0 6 6

After analysis we get

+6 -1 +4 -6 +1 -4 +6 0

After canceling the 6-element mirror-image grouping of +6 -1 +4, -6 +1 -4, we are left
with a Ps of +6, or simply 6. It must be kept in mind that this is carried out to nine digits; the

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remainder of +6 will be replaced by another digit if more decimal places are included. If a
final zero is added to the original sequence, it cancels to zero.

THE RECIPROCAL OF SEVEN DIVIDED BY ITS MIRROR IMAGE

The reciprocal of the number seven divided by its mirror image yields a seeming
variant of the reciprocal of thirteen:
758241 ÷ 142857 = 5.30769203769203769203…;

5 3 0 7 6 9 2 0 3 7 6 9 2 0 3 7 6 9 2 0 3…

The result after synthesis yields:

-2 –3 +7 –1 +3 –7 – 2 +3 +4 -1 +3 – 7 –2 +3 +4 –1 +3 –7 –2 +1…;
↓ ↓
+7 +7

results in the following second order sequence:

-2 –3 +7 –1 +3 - 7 –2 +7 -1 +3 –7 –2 +7 –1 +3… ;

A third order sequence reveals:

-2 -3 +7 -1 +3 -7 -2 +7 -1 +3 -7 -2 +7 +2
↓ ↓ ↓
-5 +7 +2 -7 -2 +7 +2 -7 -2 +7 +2;

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After canceling the 4-element oscillating mirrors (+7 +2, -7 -2) the
resulting Ps is –5.

RECIPROCAL OF 82.9

1 ÷ 82.9 =

0.012062726176115…

After synthesis, this yields:

+1 +1 -2 +6 -4 +5 -5 +4 -5 +6 -1 -5 0 +4

second order sequence:

+1 +1 -2 +6 -4 +5 -5 +4 -5 +6 -1 -5 0 +4

↓ ↓ ↓

+2 -2 +2 +5 -5 +4 -5 +5 -5 0 +4

third order sequence:

+2 -2 +2 +5 -5 +4 -5 +5 -5 0 +4

fourth order sequence:

+4 +4

Ps = +8

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SQUARE ROOT OF THE RECIPROCAL OF THIRTEEN

In some number sequences such as those in the last two examples, the mirrors
and oscillating digits are not obvious, though they exist nevertheless. Consider
the following example.
1 ÷ √13 =

1 ÷ √769230

= 877.05758

8 7 7 0 5 7 5 8:

After analysis we get:

-1 0 -7 +5 +2 -2 +3

-1 0 –7 = --8; +5 +3= +8;

--8 +8 = 0

The Ps, therefore, is 0.

FIVE DIVIDED BY TWENTY-ONE:

5 ÷21 =0.2380952;

2 3 8 0 9 5 2

Yields

+ 1 +5 -8 +9 -4 -3;

+1 +5 = +6; -8 +9 =+1; -4 -3=-7;

+6 +1 = +7; -4 -3 = -7

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 +7 -7 = 0

The Ps = 0.

SIX DIVIDED BY TWENTY-ONE

6 /21 = 0.2857142;

2 8 5 7 1 4 2

After analysis we get:

+6 -3 +2 -6 +3 -2

6-element mirror with the Ps = 0.

EIGHT DIVIDED BY 21 ( 8/21)

= 0.3809523

=3 8 0 9 5 2 3

5 -8 +9 -4 -3 +1
↓ ↓ ↓
-3 +9 -7 +1 second order sequence
↓ ↓
+6 -6;
+6 - 6 + third order sequence
↓
0

The Ps of 8/21 is, therefore, 0.

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NINE DIVIDED BY TWENTY-ONE ( 9/21):

= 0.4285714…

4 2 8 5 7 1 4

-2 +6 -3 +2 -6 +3

The Ps = 0

THE NUMBER 13492 --

=1819801
+7 –7 +8 -1 -8 +1

+7 -7 +8 -1 -8 +1
↓
0 0
Ps = 0
Note: The Ps of 1349 squared is, therefore, 0. The number 1349, by the way, was
derived in the following manner: The reciprocal of the number thirteen is a circular number,
as is the reciprocal of the number seven. Unlike seven, however, thirteen cycles two
groups of numbers when a number between one and nine is divided by thirteen: 0.076923
and 0.153846. The numbers 1, 3, 4, and 9 cycle on the digits 076923, hence the #1349.
The remaining numbers cycle the digits 153846.

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 CHAPTER FOUR

PI PATTERNS

ORGANIZED PATTERNS IN THE DIGITS OF PI

Pi, the number that expresses the relationship between the diameter and
circumference of a circle, has fascinated people for centuries. Around 2000 B.C.
the Babylonians used pi as 3 1/8. During that same period the ancient Egyptians
identified the relationship as (16/9)2 =3.1605. In the 12th century B.C. the
Chinese used pi = 3, and in the Bible in 550 B.C., I Kings vii, 23 implies pi = 3. In
the 3rd century B.C. Archimedes establishes pi using a geometric method as pi =
3.14163. These earlier researchers did not use the Greek letter as a symbol for
pi, however. It has only been used for the past 250 years.

In 1766 Johann Heinrich Lambert proves the irrationality of pi and in 1882 F.

Lindemann proves the transcendence of pi.

Digit hunters throughout the centuries have tried to determine the number of
digits in pi. To date, more than 51.5 billion digits have been identified, and no
identifiable pattern has emerged. Until now, that is. David Blatner in his
delightful book entitled The "Joy of Pi" (Walker Publishing Co., 1997) states
"There's little doubt that if we understood this number better--if we could find a
pattern in its digits or a deeper awareness of why it appears in so many
seemingly unrelated equations--we'd have a deeper understanding of
mathematics and the physics of our universe." He goes on to say "The digits of
pi appear so random that if there were a rule to the sequence, it may require
billions--or trillions--of digits to begin to see it." Perhaps not.

22
The following number experiment uses a quaternary, reiterative skip sequence
on the first 218 digits of pi to create new digits for increment analysis.

THE FIRST 218 DIGITS OF PI

3.14159265358979323846264338327950288419716939937510582097494459230781

640628620899862803482534211706798214808651328230664709384460955058223

172535940812848111745028410270193852110555964462294895493038196442881

097566593344

QUATERNARY REITERATIVE SKIP SEQUENCE

A quaternary reiterative skip sequence was used. This is done in the following manner:

1 - 2 - 3-4
1 -2-3 -4
1 -2-3 - 4
1 - 2 - etc. etc.

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In this sequence 1 reiterates 4, i.e., the one in the next sequence is the
same as the 4 in the last sequence. The following digits are the result of
using this quaternary, reiterative skip sequence on the first 218 digits of pi:

312393823398113312442848888831684632694952734281547
3259428439487634

Analysis:

First Order Sequence

Once these skip sequence digits were identified, a relationship between

them was assessed. The following first order sequence resulted from
determining the positive or negative increments between digits of the
resultant skip sequence. (It is important to keep in mind that though each
digit is an increment that is ten times more precise than the last in this
decimal expansion of pi, the numbers are taken at face value.

For example, the first numbers in the resulting sequence are 3 1 2 3 9

3....therefore, from 3 to 1 is -2; from 1 to 2 is +1; from 2 to 3 is +1; from 3 to
9 is +6; from 9 to 3 is -6; etc. etc.
Resultant Patterns—First Order Sequence

-2 +1 +1 +6 -6 +5 -6 +1 0 +6 -1 -7 0 +2 0 -2 +1

+2 0 -2 +6 -4 +4 0 0 0 0 -5 -2 +5 +2 -4 +2 -3 -1

+4 +3 -5 +5 -4 -3 +5 -4 +1 -2 +6 -7 +4 -1 +3

-4 -1 +3 +4 -5 -2 +6 -4 -1 +6 -5 +4 -1 -1 -3 +1

24
The following mirrors and oscillating digits were found in the previous

number string.

-2 +1 +1 +6 -6 +5 -6 +1 0 +6 -1 -7 0 +2 0 -2 +1

+2 0 -2 +6 -4 +4 0 0 0 0 -5 -2 +5 +2 -4 +2 -3 -1

+4 +3 -5 +5 -4 -3 +5 -4 +1 -2 +6 -7 +4 -1 +3

-4 -1 +3 +4 -5 -2 +6 -4 -1 +6 -5 +4 -1 -1 -3 +1

Binary patterns emerge in the first 218 digits of pi. A remarkable symmetry

seems evident where positive and negative numbers in reciprocal positions occur

in a seeming non-random manner. In addition, without exception, numbers or

number groupings on the opposite sides of zeroes are have opposite or

balancing polarities in the first order sequence. Observe the following while

keeping in mind the fact that this is the exact order in which they appeared in the

number string:

+6 -6

-6 +1 0 +6 -1

+2 0 -2 + 1 +2 0 -2

-4 +4

0000

-5 -2, +5 +2

25
+4 +3, -5 +5, -4 -3

+4 -1 +3, -4 -1 +3

In addition, note the fact that +5 -4 or +4 -5 repeat five times in various places

in the string. Also note that there are no 8s or 9s in the string.

The numbers not included in the number groupings identified in the first order

sequence noted above were found to also have binary relationships when a

synthesis methodology was employed.

Synthesis of the First Order Sequence to Create a Second Order Sequence

A synthesis methodology was devised to test the apparent binary nature of the

digits. Basically, synthesis is the resolution of binaries. The following "rules" were

utilized:

1. only adjacent numbers or number groupings were used, or those that are

separated by one or more zeroes; 2. these are subsequently added until a final

result, i.e., resolution of binaries, is achieved. For the first 218 digits, it took four

sequence generations to reach neutralization. The italicized numbers below

result from the number groupings beneath them and represent a subsequent

order.

-2 +2 0 +5 -5 0 +5 -7 (second order)

| / \ / \ | / \ | / \ |

-2 [+1 +1] [+6 -6] +5 [-6 +1] 0 [+6 -1] -7

26
0 0 +1 0 +6 0 0 (second order)

| / \ | / \ | / \ / \

0 [+2 0 -2] +1 [+2 0 -2] +6 [-4 +4] [0 0 0 0]

-7 +7 -6 (second order)

/ \ / \ / \

[-5 -2] [+5 +2] [-4 +2 -3 -1]

+7 0 -7 +1 -1 -1 (second order.)

/ \ / \ / \ / \ / \ / \

[+4 +3] [-5 +5] [-4 -3] [+5 -4] [+1 -2] [+6 -7]

+6 (second order)

/ \

[+4 -1 +3]

-2 -1 +4 -4 -1 +1 +4 (second order)

/ \ / \ / \ | | / \ |

[-4 -1 +3] [+4 -5] [-2 +6] -4 -1 [+6 -5] +4

27
-1 -4 +1 (second order.)

| / \ |

-1 [ -1 -3] +1

Third Order Sequence

0 0 0 0 -7 0 0 +7 (third order)

/ \ | / \/ \ | | | / \

[-2 +2] 0 [+5 -5 0 +5] -7 0 0 +1 0 +6

0 0 0 -6 0 0 -1 +6 -2 (third order)

| | / \ | / \ / \ | | |

0 0 -7 +7 -6 +7 0 -7 +1 -1 -1 +6 -2

-1 0 0 +3 -3 (third order)

| / \ / \ / \ / \

-1 +4 -4 -1 +1 +4 -1 -4 +1

Fourth Order Sequence

0 0 0 0 0 0 +6 (fourth order)

| | | | / \ / \ |

0 0 0 0 -7 0 0 +7 [-6 0 0 -1] +6

28
 (-7 0 0 +7 0 0 0 -7 )

0 0 (fourth order)

/ \ / \

[-2 -1] 0 0 +3 -3

(-3 0 0 +3 -3)

A resolution of increments in the first 218 digits of pi, therefore, results in

the following sequence:

0 0 0 0 0 0 +6 0 0 ; where the mode is zero and the Point of Synthesis

is +6.

SUMMARY OF 2ND THROUGH 4TH ORDER SEQUENCES

SECOND ORDER:

-2 +2 0 +5 - 5 0 +5 -7 0 0 +1 0 +6 0 0 -7 +7

-6 +7 0 -7 +1 -1 -1 +6 -2 -1 +4 -4 -1 +1

[+4 -1] [ -4 +1]

THIRD ORDER:

0 0 0 0 -7 0 0 +7 0 0 0 -7 +6

-3 0 0 +3 -3

FOURTH ORDER:

0 0 0 0 0 0 +6 0 0

29
Based on the foregoing number experiment there are mirrors and binary

patterns underlying the first 218 digits of pi where the increments tend

toward zero. More research would have to be done, of course, to determine

whether or not this pattern is ongoing up to and beyond the more than 51.5 billion

digits of pi that have been identified thus far. What is definite, however, is that

any sequence extracted from the digits of pi have increments that sum to zero,

as will be demonstrated later.

30
 CHAPTER FIVE

WHOLE NUMBER INCREMENTS

The major focus thus far has been on reciprocals of numbers and their

increments that sum to zero. Now I will examine whole number increments. It

will be found that they always sum to zero.

With whole numbers, to assess the increments, you must add a zero to the

beginning of the number and a zero to the end (unless the number ends in zero.

In that case, you don’t add an extra zero). The rationale for this is that there is

an understood zero, or absence of a number at the beginning and ending of a

whole number.

CONJECTURE;

The increments of any whole number sum to zero.

EXPERIMENTAL DATA:

Let n = a whole (natural) number

Let ∈ = increment

Then, ∞
Σ ∆=0
0…n

31
EXPERIMENTAL DATA
Number sample Synthesis

1 0 1 0 =

+1 -1 = 0

2 0 2 0=

+2 -2 = 0

13 0 1 3 0=

+ 1 +2 –3 =

+3 -3 = 0

477 0 4 7 7 0=

+4 +3 0 -7 =

+7 -7 = 0

4579 045790 =

+4 +1 +2 +2 –9 =

+9 -9 = 0

36920 036920=

+3 +3 +3 –7 –2=

+9 -9 = 0

32
7787537 077875370

+7 0 +1 –1 –2 –2 +4 –

7=

+8 –5 +4 -7

+8 -1 -7 =

+8 -8 = 0

3123 031230 =

+3 – 2 +1 +1 –3 =

+3 -2 +2 -3 = 0

2138 021380 =

+2 -1 +2 +5 –8 =

+1 +7 -8 =

+8 -8 = 0

146 01460 =

+1 +3 +2 -6 =

+6 -6 = 0

175 01750 =

+1 +6 -2 -5 =

+7 -7 = 0

33
2903 029030

+2 +7 –9 +3 –3 =

+9 -9 +3 -3 = 0

3167 031670 =

+3 –2 +5 +1 -7 =

+1 +6 -7 =

+7 -7 = 0

1699 016990 =

+1 +5 +3 0 -9 =

+9 0 -9 = 0

3851 038510 =

+3 +5 -3 -4 -1 =

+8 -8 = 0

797 0790 =

+7 +2 –2 –7 = 0

4049 040490 =

+4 -4 +4 +5 -9 =

0 +9 -9

34
877 08770 =

+8 –1 0 -7 =

+8 -8 =0

953 09530 =

+9 -4 -2 -3 =

+9 -9 =0

2459 024590 =

+2 +2 +1 +4 -9 =

+9 -9 = 0

382 03820 =

+3 +5 -6 -2 =

+8 -8 = 0

267 02670 =

+2 +4 +1 -7 =

+7 -7 = 0

35
 CHAPTER SIX

IRRATIONAL AND TRANSCENDENTAL NUMBERS

An irrational number is a number that is not capable of being expressed as an

integer or as a quotient of an integer. A transcendental number is a number,

such as pi, which is not the root of any polynomial equation with integer

coefficients; i.e., it is not an algebraic number. Pi is also irrational. When any

series of consecutive digits is lifted from any irrational or transcendental

number and zero is placed in front of and behind the fragment, the sum of

the increments is always 0!

Conjecture:

The sum of increments in number sequences extracted from irrational and

transcendental numbers always equals 0.

Pi, to 24 digits = 3.14159265358979323846264….

Extract the sequence: 6535897;

36
 Add zeros before and after the fragment:

065358970 =

+6 -1 -2 +2 +3 +1 -2 -7 =

+6 -3 +5 -1 -7=

+6 +2 -8 =

+8 -8 = 0

Extracting another sequence from the pi expansion:

8462=

084620 =

+8 -4 +2 -4 -2 =

+8 -2 -6

+6 -6 = 0

37
Feigenbaum numbers have not proven to be transcendental, but are generally

believed to be. They are related to properties of dynamical systems with period

doubling. The ratio of successive differences between period doubling bifurcation

parameters approaches the number 4.669…..

Feigenbaum constant to the 30th digit:

4.66920160910299067185320388204…

Extracting sequence :

6718532:

Adding zeros:

067185320 =

+6 +1 -6 +7 -3 -2 -1 -2 =

+8 -8 = 0

38
 The irrational square root of 2 =

√2 = 1.4142135623730950488016887242097…;

Extract sequence:

73095:

add zeros:

0730950 =

+7 -4 -3 +9 -4 -5 =

+7 -7 +9 -9 = 0

39
A “PROOF” THAT ALL OF THE INCREMENTS BETWEEN DIGITS
IN TRANCENDENTAL NUMBERS SUM TO ZERO

If it can be proven that the increments between all digits of transcendental

numbers sum to zero, it may lend credence to the position that the digits in
transcendental numbers are not random. To demonstrate this point, I will use a
variation of the proof that the late Middle Ages French scholar Nicole d’Oresme
(ca.1323 – 1382) used to prove the divergence of the harmonic series.
D’Oresme pointed out that 1/3 + ¼ is greater than ½; so is 1/5 + 1/6+ 1/7 + 1/8;
so is 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/ 16 ; and so on. In other
words, by taking 2 terms, then 4 terms, then 8, then 16 terms, and so on, you can
group the series into an infinite number of blocks, every one of which is bigger
than one-half. The entire sum must, therefore, be infinite. No matter what, there
is always another one-half to be added; and that means that the total increases
without limit.2

In the following proof, larger and larger blocks of the decimal expansion of a
transcendental number are considered utilizing a process called additive
assessment (see chapter 14). If the digits in transcendental numbers are taken
in larger and larger blocks with zeroes added before and after the blocks, the
sum of increments is always zero, therefore, the sum of increments in the digit
expansion will always be zero through infinity.
Let n = integer

Let nts = any number segment in a transcendental number

Let 02 = zeroes before and after any segment

Let ∆ = increment segments through infinity

If ∆ nts +n + 02 = 0;

Then Σ ∆ nts + n + 02 ----- ∞ = 0 Q.E.D.

2 John Derbyshire. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in
Mathematics. Joseph Henry Press. Washington, D.C. 2003

40
 CHAPTER SEVEN

BASE TWO
When assessing numbers in base two, we get a similar result when assessing

the face value of the numbers. For example, #7 in Base 10 equals 111 in base

2.

111 =

01110 =

+1 0 0 -1 = 0

#13 in base 2 equals 1101:

1101 =

011010 =

+1 0 -1 +1 -1 = 0

#17 in base 2 equals 10001:

10001 =

0100010 =

+1 -1 00 +1 -1 = 0

41
 CHAPTER EIGHT
FIBONACCI AND LUCAS NUMBERS

The Fibonacci Numbers

Fibonacci numbers were first demonstrated by Leonardo of Pisa, better known

as Fibonacci, in his book titled Liber Abaci published in 1202. In it he discussed
the number of rabbits that would be born from a pair of rabbits at the beginning of
every month. The resulting number of rabbits revealed the following pattern:
1,1,2,3,5,8,13,21,34,55, etc. This series converges to a number called the
“golden ratio,” which is rounded off here to 1.6180339887….. Basically, you take
a term, divide it by the previous term, and the number converges to the golden
ratio. This number is found in many places in nature. The formula that
generates this number is:

( 1 + √5) = 1.6180339887…..
2

In an examination of the first 500 Fibonacci numbers, Increment Analysis has

revealed an interesting pattern– when the digits are summed modulo nine, a
recurring sequence with a mirror pattern emerges. The sequence is:

112358437189887641562819…1…….
This series of numbers repeats continually beginning again at every 24th digit.
The 24th digit always sums to nine. The increment pattern that emerges is:
0 +1 +1 +2 +3 -4 -1 +4 -6 +7 +1 -1

0 –1 -1 -2 -3 +4 +1 -4 +6 -7 +8 -8

The series sums to a perfect mirror with an anomaly, +1 +8, -1 –8, which sums to
+9 and –9, which sums to zero.

42
The Lucas Numbers

The Lucas Numbers are similar to the Fibonacci numbers, but they start with 2
and 1 instead of the Fibonacci’s 0 and 1. The series is named after Edouard
Lucas (1842 – 1891), professor of mathematics at the Lycee Charlemagne in
Paris who developed a well-known formula for the Fibonacci numbers as well as
the numbers that bear his name. The first few Lucas numbers are:
2,1,3,4,7,11,18,29,47,76,123,199,322, etc.

When the first 200 Lucas numbers are summed modulo nine, as with the
Fibonacci numbers, a recurring series emerges which also repeats at every 24th
number. In every case, the pivot number sums to eight. The series is:

213472922461786527977538…2…
The resulting mirror becomes evident:

-1 +2 +1 +3 -5 +7 -7 0 +2 +2 -5 +6

+1 -2 -1 -3 +5 +2 -2 0 -2 -2 +5 -6
This is a perfect mirror with an anomaly, +7 +2, -7 -2, that sums to +9 -9, which

= 0.

43
 The first 25 Fibonacci numbers
Number ∑ mod 9
1 1
1 1
2 2
3 3
5 5
8 8
13 4
21 3
34 7
55 1
89 8
144 9
233 8
377 8
610 7
987 6
1597 4
2584 1
4181 5
6765 6
10946 2
17711 8
28657 1
46368 (F#24) 9
75025 1

44
 The first twenty-five Lucas numbers
Number ∑ mod 9

2 2
1 1
3 3
4 4
7 7
11 2
18 9
29 2
47 2
76 4
123 6
199 1
322 7
521 8
843 6
1364 5
2207 2
3571 7
5778 9
9349 7
15127 7
24476 5
39603 3
64079 (L#24) 8
103682 2

45
 CHAPTER NINE

PASCAL’S TRIANGLE

Pascal’s Triangle was designed by Blaise Pascal in the 17th century. The
Chinese, however, knew of this triangle centuries before Pascal. It has been
found that the permutations relate directly to terms of the Binomial Theorem. It
can also be used in the analysis of probabilities. From an Increment Analysis
standpoint, Pascal’s Triangle creates a series of palindrome mirrors. (A
palindrome is a number that reads the same way forward or backward).

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

46
 The previous triangle, completed through the seventh row, demonstrates
the following pattern:

+1

+1 -1

+2 0 -2

+3 +2 -2 -3

+4 +5 0 -5 -4

+5 +9 +5 -5 -9 -5

The sum of the increments in this triangle, and to one of any power, is
equal to one, i.e., unity. An intriguing question: would the sum of
increments be 1 in a Pascal Pyramid?.

47
 CHAPTER TEN

PRIME NUMBERS

Prime numbers are numbers that have only factor of themselves and one. They
are enigmatic, in that no discernable pattern of organization is evident. When
Einstein said God doesn’t play dice when expressing exasperation about
quantum physics, he could have said the same thing about prime numbers.

There have been some advances in understanding them. Researchers have

been able to make progress regarding the conjecture that there are an infinite
number of pairs of prime numbers that differ only by two. It has also been found
that the frequency of twin primes decreases as one gets to larger numbers.
Brun’s constant, another revealing aspect of their nature, is found by adding the
reciprocals of successive twin primes. When this is done, the sum converges to
a specific numerical value which is 1.902160582310… The reciprocals of all of
the primes, on the other hand, diverge.

The Goldbach Conjecture, that every even number is the sum of two primes has
not yet been proven. It is known, however, that all twin primes are of the multiple
6k +/- 1.
Other mathematicians who have thrown their hats into the prime ring are
Mersenne (Mersenne Primes), Sophie Germaine ( Sophie Germaine Primes) and
Fermat (Fermat Primes) just to name a few. One of the most intriguing of prime
conjectures, however, is the Riemann Hypothesis. Briefly stated, when studying
the distribution of prime numbers, Riemann extended Euler’s zeta function,
defined for real part greater than one, to the entire complex plane. When he did,
he noted that his zeta function had trivial zeros at –2, -4, -6,… and that all

48
nontrivial zeros were symmetric about the line Re(s) + ½. The hypothesis is that
all nontrivial zeros are on this line. The Clay Mathematics Institute has offered to
pay $1 million dollars to anyone that can prove this hypothesis to be true.

Increment Analysis has uncovered some peculiarities about primes. When the
primes are assessed as sums modulo nine, they all resolve to digits of the
reciprocal of the number seven. Those digits are 0.142857 142857…..The
primes, however, do not follow in a particular order that has yet been noted.
What has been revealed by Increment Analysis is:
• No prime number with two or more digits sums to 6, 3, or 9.

• No prime with two or more digits ends with the number 5.

• The final digit of any prime number of two or more digits can only be 1, 3,
7, or 9.

• There is a yin yang configuration of the primes. The only even prime,
number 2, is separated from every other prime by odd numbered

increments. Every odd prime is separated from every other prime by an

even numbered increment.

• Though it has been shown that the frequency of twin primes decreases as

one gets to larger numbers, Increment Analysis demonstrates the fact that

the frequency of larger increments increases as one gets to larger

numbers.

49
 FREQUENCY OF ALL ADJACENT NUMBER INCREMENTS IN THE FIRST 101 PRIMES:

25

20

15 3-D Column 1
3-D Column 2
10
3-D Column 3
5

#10

#14
#2

#6

♦ # 2 appears 25 x

♦ # 4 appears 25 x

♦ # 6 appears 25 x

♦ # 8 appears 7 x

♦ # 10 appears 8 x

♦ # 12 appears 4 x

♦ # 14 appears 3 x

♦ # 18 appears once

In addition, the increment between the 20,861st and 20,862nd prime (235,397

and 235,439) is 42.

THE FIRST 101 PRIMES SUMMED MODUL0 9 DEMONSTRATING DIGITS

FROM THE RECIPROCAL OF THE NUMBER SEVEN

(1)23 = The Supernal Triad

5724815241572857481287248151524574152812481472485785281724515727

4572874152154578172872482154858117

50
PRIME EXCLUSION PRINCIPLES

Based on the foregoing, it is safe to assume that, when assessing a number with
two or more digits, you know it is not prime if the final digit is a 5. It is also not
prime if the sum modulo nine is not one of the digits of the reciprocal of seven.

PRIMAL ELEMENTS

When considering the Periodic Table of the Elements, the prime number
elements are arranged in interesting patterns. The periodic table is divided into
classes numbered from 1 through 18. The 18th division is comprised of the “noble
gases.”
I II IIIb IVb Vb Vib VIIb VIIIb Ib IIb III IV V VI VII 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Li none Ac none V none Tc noneMt none Cu none B none N none Cl He

Na Nb Bh Ag Al Bi I

K Ta Pm Au Ga Md Lu

Rb Pr Bk Ho Lr

Totals

4 0 1 0 4 0 3 0 1 0 4 0 4 0 3 0 4 1

The “prime” elements, i.e., the elements with the same number of protons as the
prime numbers, tend to alternate, and they tend to fall in prime numbered
columns, with a few exceptions. When the total number of prime elements in
each group is reduced to a number, the following sequence results:
4 0 1 0 4 0 3 0 1 0 4 0 4 0 3 0 4 1;

401040301040403041 =

04010403010404030410 =

51
 +4 -4 +1 -1 +4 -4 +3 -3 +1 -1 +4 -4 +4 -4 +3 -3 +4 -3 -1 =

+28 -28;

Sum modulo 9 =

+10 -10 =

+1 -1

This last section, the analysis of the resulting number sequence reveals a
pattern of primes in the natural elements that resolves to ±1.

52
 CHAPTER ELEVEN

SQUARE BUSINESS
In this chapter we will examine the increments in “squares”-- zero sum squares,
magic squares, and Latin squares.
2 x 2 ZERO SUM SQUARES

I have discovered a new class of 2 x 2 squares that utilizes zero and a negative
number to form Zero Sum squares. Zero Sum squares generate series of
positive and negative numbers that sum to zero. The following is a Zero Sum 2
x 2 square comprised of the first three primes and zero wherein each row,
column or diagonal sums to positive and negative prime numbers that together
sum to zero.

3 0

2 -5

53
The previous Zero Sum 2 x 2 prime square generates the following

patterns:

Horizontals : +3 -3

Verticals: +5 -5

Diagonals: +2 -2

Another Zero Sum Prime Number square:

5 0

2 -7

This Zero Sum square generates:

Horizontals: +5 -5

Verticals: +7 -7

Diagonals: +2 -2

54
Zero Sum Prime Number Square

17 0

2 -19

This square generates:

Horizontals: +17 -17

Verticals: +19 -19

Diagonals: +2 -2

55
Zero Sum Prime Number Square:

29 0

2 -31

This generates:

Horizontals: +29 -29

Verticals: +31 -31

Diagonals: +2 -2

56
Zero Sum 2 x 2 non-prime Square

27 0

2 -29

This generates:

Horizontals: +27 -27

Verticals: +29 -29

Diagonals: +2 -2

57
Another Zero Sum Square:

79 0

2 -81

This generates:

Horizontals: +79 -79

Verticals: +81 -81

Diagonals: +2 -2

58
MAGIC SQUARES

At this point a magic square will be examined using increment analysis. A magic
square is one wherein the positive integers of the verticals, horizontals and
diagonals all sum to the same number, the magic constant. There are no 2 x 2
magic squares. The formula for generating such a square is:
n2
M2(n) = 1/n Σ =1/2n (n2 +1)
k=1

If every number in a magic square is subtracted from n2 +1, another magic

square is obtained called the complementary magic square.3
The following third order magic square has integers that all sum to the number
15.
Third Order Magic Square

8 1 6

3 5 7

4 9 2

3 http://mathworld.wolfram.com/MagicSquare.html

59
This generates:

Horizontals: 816 = -7 +5 Verticals: 834 = -5 +1

159 = +4 +4 357 = +2 +2

492 = +5 -7 672 = +1 -5

Diagonals: 852 = -3 -3

456 = +1 +1

When added together, the sum of all increments in this magic square is negative
four. The horizontals and verticals sum to zero, but the diagonals, always
homogenous, tilt the square toward a remainder.

Fourth Order Magic Square

4 14 15 1

9 7 6 12

5 11 10 8

16 3 13
2

60
The above Fourth Order Magic square generates:

Horizontals Verticals Diagonals

+10 +1 -14 +5 -4 +11 +3 +3 +3

-2 -1 +6 -7 +4 -9 +5 +5 +5

+6 -1 -2 -9 +4 -7

-14 +1 +10 +11 -4 +5

As in the third order magic square, this one has horizontals and verticals that
sum to zero, and homogenous diagonals. The resulting +8 +8 +8 = +24, the
remainder.
LATIN SQUARES

A Latin square is an array of symbols wherein every symbol occurs

exactly once in each row and column of the array. Leonhard Euler,
the great mathematician, introduced them in 1783 as a “nouveau
espece de carres magiques”, a new kind of magic square.4
The following is a Second Order Latin square

1 2

2 1

This Second Order Latin square generates:

Horizontals Verticals Diagonals

+1 +1 0

-1 -1 0

This second order Latin Square is a zero sum square.

4 http://www.cut-the-knot.org/arithmetic/latin.shtml

61
The following is a Fourth Order Latin Square

1 2 3 4

2 3 4 1

3 4 1 2

4 1 2 3

This Fourth Order Latin Square

Horizontals Verticals Diagonals

+1 +1 +1 +1 +1 +1 +2 -2 +2

+1 +1 -3 +1 +1 -3 0 0 0

+1 -3 +1 +1 -3 +1

-3 +1 +1 -3 +1 +1

This Latin square exhibits the same type of pattern seen in the magic squares –

the horizontals and vertical increments sum to zero, and homogeneous diagonals

result in a remainder of +2.

62
 CHAPTER TWELVE

“MAGIC” NUMBERS (of proton and neutron stability)

20

28

50

82

126; the sum of these numbers =

316

Adding zeros to this number yields:

03160 =

+3 -2 +5 -6 =

+8 -8

There is, therefore, and “eightness” connected with these numbers.

When 316 is divided by eight, the result is the repeating sequence
0.0253164556962 0253164556962 0253164… =
+15 -15

63
Other interesting features:

√316 = 17.776388834631177700182883350817

The reciprocal of 316 = 0.003164556962025 3164556962025

3164556….seems to say its own name.

Extracting any sequence from this number string sums to zero when zeroes are
added before and after the sequence.
An interesting thing happens when you flip these numbers and line them up next
to each other sideways creating the following number string:
2820285082126

without adding zeroes, the first generation yields:

+6 -6 -2 +2 +6 -3 -5 +8 -6 -1 +1 +4

-8 +8 -8

second generation:

+6 -8 +8 -8 +8 -6 -1 +1 +4

64
Now, when adding zeros to the original number sequence we get the
following result.

2820285082126 =

028202850821260 =

+2 +6 -6 -2 +2 +6 -3 -5 +8 -6 -1 +1 +4 -6 =

+29 -29

65
 CHAPTER THIRTEEN

NOTIONS ABOUT NOTHING AND EVERYTHING : A LOOK AT

ZERO AND INFINITY
Based on the evidence in this paper that all number sequence increments sum to
zero, zero becomes quite important and deserves closer scrutiny. The number
zero is enigmatic. Not always a member of the pantheon of numbers, the
concept of zero was introduced by the Babylonians around 300 BC. They used
two slanted wedges to represent an empty space. Indian mathematicians later
transformed zero from placeholder to a number, and later, when the Arabs
adopted Hindu-Arabic numerals, they also adopted zero. 5 The Mayans also had
a zero in their counting system. Christianity opposed it, and, in fact, zero and
infinity (nothing and everything) destroyed the Aristotelian philosophy. The void
and infinity, two sides of a coin, threatened the existence of God. For these
reasons, it took some time before zero was accepted by the West. Today, we
take zero for granted, even though it poses paradoxes. It is far more than a
placeholder, and I think the jury is still out regarding its true nature.
Rules regarding zero today are:

Zero divided by any number is zero

Any number divided by zero is undefined

Any number multiplied by zero is zero

Add zero to any number and it remains unchanged

One divided by zero equals infinity (non-traditional)

Dividing a number by infinity yields zero (non-traditional)

Dividing a number by zero yields infinity (non-traditional)

5 Zero, The Biography of a Dangerous Idea. Seife, Charles. Penguin Books. NY. 2000.

66
At this point I will take a departure from traditional mathematics theory and let
zero speak for itself to get a clearer understanding of its true nature. For one,
zero seems to exhibit three different modes: Positive, Negative and Neutral.
Positive zero would be the concept of “everything”, Negative zero would be the
concept of nothing (no-thing), and Neutral zero would be the concept of
“balance” or ‘‘fullness.” Zero is not just an “emptiness,” but a “fullness” that
tends to displace itself in greater and greater increments in the expansion of
numbers. In this sense, it brings to mind Cantor’s Continuum Hypothesis with its
‫א‬0, ‫א‬1 ‫א‬2…‫א‬n. A question can be asked about what happens when the positive
and negative increments between numbers are squared, cubed, quadrupled, etc.
Would we witness an expansion of the complex plane? Are we witnessing larger
and larger zeros, and hence, larger and larger infinities?

ZERO DOMAIN DYNAMICS (ZDD)

Tradition tells us that we can’t divide by zero because we consider the result
“undefined.” When zero speaks for itself, however, an interesting picture
emerges. For instance, If it were feasible to perform the following operations, the
results would be :

(2x0) = 2 (3x0) = 3 (4 x 0) = 4
0 0 0

or

1/0 x 0 = 1; 5/0 x 0 = 5; 6/0 x 0 = 6

The foregoing is considered illogical based on current mathematical philosophy.

It is not possible, the thinking goes, for zero to equal all numbers. This is
precisely what zero is saying, however, and since this is the case, zero is

67
demonstrating equality with infinity. Indeed, on a Riemann Sphere, zero and
infinity are at opposite poles. If we could imagine a stacking of Riemann Spheres,
zero and infinity would be indistinguishable.

When adding, subtracting, multiplying or dividing numbers in the zero realm,

unexpected and “illogical” results prevail. In the zero domain, almost anything
goes. This is true when considering the “phantom values” of numbers, i.e., the
balanced binaries that result when zeroes are added in front of, and behind,
numbers. For example:

31 x 45 = 1395;

0310 x 0450 = 013950

(Phantom Values) (+3 -3) ( +5 -5) = +9 -9

In the zero domain, therefore, ( ∀3 )( ∀5) = ∀9

Technically, though, the result of the above operation is 0, since we are

multiplying 0 x 0.

Another example:

1849673 /14 = 132119.5;

018496730 / 0140 = 013211950;

+14 -14 / +4 - 4 = +11 -11,

therefore, ±14 divided by ±4 = ±11,

but again, we are dividing zero by zero to get zero.

68
 CHAPTER FOURTEEN

ADDITIVE ASSESSMENT

The additive assessment of a number sequence or series is the process of

sampling bits of information in bigger and bigger pieces and assessing the
samples. For example, take a number sequence and analyze the first two
numbers of the sequence, and then the first three, the first four, etc. until the
entire sequence has been assessed. In increment analysis, additive assessment
is used to assess the increments of larger and larger blocks of a number
sequence (or series). Zeroes are added before and after the segment. When this
is done, the segment always sums to zero, with a graduated pattern of positive
and negative numbers. Following, the additive assessment technique is
demonstrated for the number e (the basis of natural logarithms) up to the 25th
integer, pi, primes (up to the 13th prime) the first 10 Fibonacci numbers, the
reciprocal of the number seven, the reciprocal of number thirteen and phi.

e = 2.718281828449045235360287…

020 = + 2 -2
0270 = + 7 -7
02710 = + 7 -7
027180 = +14 -14
0271820 = +14 -14
02718280 = +20 -20
027182810 = +20 -20
0271828180 = +27 -27
02718281820 = +27 -27
027182818280 = +33 -33
0271828182840 = +34 -34
02718281828450 = +38 -38
027182818284590 = +38 -38
0271828182845900 = +38 -38
02718281828459040 = +42 -42
027182818284590450 = +43 -43
0271828182845904520 = +43 -43
02718281828459045230 = +44 -44
027182818284590452350 = +46 -46

69
0271828182845904523530 = +46 -46
02718281828459045235360 = +49 -49
027182818284590452353600 = +49 -49
0271828182845904523536020 = +51 -51
02718281828459045235360280 = +57 -57
027182818284590452353602870 = +57 -57

Π (Pi)
(to 31 digits)
3.141592653589793238462643327950…

030 = + 3 - 3
0310 = + 3 - 3
03140 = + 6 - 6
031410 = + 6 - 6
0314150 = +10 -10
03141590 = +14 -14
031415920 = +14 -14
0314159260 = +18 -18
03141592650 = +18 -18
031415926530 = +18 -18
0314159265350 = +20 -20
03131592653580 = +23 -23
031415926535890 = +24 -24
0314159265358970 = +24 -24
03141592653589790 = +26 -26
031415926535897930 = +26 -26
0314159265358979320 = +26 -26
03141592653589793230 = +27 -27
031415926535897932380 = +32 -32
0314159265358979323840 = +32 -32
03141592653589793238460 = +34 -34
031415926535897932384620 = +34 -34
0314159265358979323846260 = +38 -38
03141592653589793238462640 = +38 -38
031415926535897932384626430 = +38 -38
0314159265358979323846264330 = +38 -38
03141592653589793238462643320 = +38 -38
031415926535897932384626433270 = +43 -43
0314159265358979323846264332790 = +45 -45
03141592653589793238462643327950 = +45 -45
031415926535897932384626433279500 = +45 -45

70
 PRIMES

(The first thirteen)

2 3 5 7 11 13 17 19 23 29 31 37 41

020 = +2 - 2
0230 = +3 - 3
02350 = +5 - 5
023570 = +7 - 7
02357 110 = +11 -11
02357 11 130 = +13 -13
02357 11 13 170 = +17 -17
02357 11 13 17 190 = +19 -19
02357 11 13 17 19 230 = +23 - 23
02357 11 13 17 19 23 290 = +29 -29
02357 11 13 17 19 23 29 310 = +31 -31
02357 11 13 17 19 23 29 31 370 = +37 -37
02357 11 13 17 19 23 29 31 37 410 = +41 -41

FIBONACCI NUMBERS

(The first ten)

1 1 2 3 5 8 13 21 34 55
010 = +1 -1
0110 = +1 -1
01120 = +2 -2
011230 = +3 -3
0112350 = +5 -5
01123580 = +8 -8
0112358 13 0 = +13 -13
0112358 13 21 0 = +21 -21
01123358 13 21 34 0 = +34 -34
0112358 13 21 34 55 0 = +55 -55

71
THE RECIPROCAL OF THE NUMBER SEVEN –0.142857142….

010 +1 -1
0140 +4 -4
01420 +4 -4
014280 +10 -10
0142850 +10 -10
01428570 +12 -12
014285710 +12 -12
0142857140 +15 -15
01428571420 +15 -15
014285714280 +21 -21
0142857142850 +21 -21
01428571428570 +23 -23
014285714285710 +23 -23
0142857142857140 +26 -26
01428571428571420 +26 -26
014285714285714280 +32 -32
0142857142857142850 +34 -34
01428571428571428570 +34 -34
014285714285714285710 +34 -34

THE RECIPROCAL OF NUMBER THIRTEEN…0.076923076….

O70 +7 -7
0760 +7 -7
07690 +10 -10
076920 +10 -10
0769230 +11 -11
07692300 +11 -11
076923070 +18 -18
0769230760 +18 -18
07692307690 +21 -21
076923076920 +21 -21
0769230769230 +22 -22
07692307692300 +22 -22
076923076923070 +29 -29

72
 Φ (PHI) (THE GOLDEN MEAN) – 1.6180339…

010 +1 -1
0160 +6 -6
01610 +6 -6
016180 +13 -13
0161800 +13 -13
01618030 +16 -16
016180330 +16 -16
0161803390 +22 -22

The prime and Fibonacci numbers “say their own names.” In other words, the
last prime or Fibonacci of each segment gives the name of the segment, which
follows the natural order of each sequence with no repetition characteristic of the
other numbers assessed thus far. This might be true for every uniformly
monotonic increasing number sequence.
ADDITIVE SIGNATURES

As demonstrated in the previous examples, every number sequence has an

additive signature that is characteristic of its assessment. The signatures of the
numbers thus far assessed are shown below.

e ±
2,7,7,14,14,20,20,27,27,33,34,38,38,38,42,43,43,44,46,46,49,49,51,57,57
Primes ±
2,3,5,7,11,13,17,19,23,29,31,37,41
±
π 3,3,6,6,10,14,14,18,18,18,20,23,24,24,26,26,26,27,32,32,34,34,38,38,38,38,38,43,45,45,45

Fibona ±
cci # 1,1,2,3,5,8,13,21,34,55

1/7 ±
1,4,4,10,10,12,12,15,15,21,21,23,23,26,26,32,34,34,34

1/13 ±
7,7,10,10,11,11,18,18,21,21,22,22,29

±
Phi 1,6,6,13,13,16,16,22

73
 CHAPTER FIFTEEN

MULTIPLICATIVE ASSESSMENT

THE CIRCULARITY OF “ONENESS”

In previous chapters we added zeroes to numbers to produce balanced binaries.

In actuality, any sequence that begins and ends with the same number creates a
perfectly balanced binary. This is most likely due to The Law of Conservation as
identified by the great mathematician Emmy Noether. In the foregoing, we have
started and ended with zeroes to produce perfectly balanced binaries. This is the
only number that can be added to a sequence to create balanced binaries
without changing the “value” of the sequence.

In this chapter, we will explore the binaries created using repunits (repeating
ones) and repdigits (repeating digits) multiplied by each other. It will be noted
that series of ones, when multiplied by series of ones, always result in the
number line from one through 9 and back again. Sometimes, there are even
cycles within the cycles.
1. 1 x 1 = 1
2. 11 x 11 = 121
3. 111 x 111 = 12321
4. 1111 x 1111 =1234321
5. 11111 x 11111 = 123454321
6. 111111 x 111111 = 12345654321
7. 1111111 x 1111111 = 1234567654321
8. 11111111 x 11111111 = 123456787654321
9. 111111111 x 111111111 = 12345678987654321
10. 1111111111 x 1111111111 = 1234567900987654321
11. 11111111111 x 11111111111 = 123456790120987654321
12. 111111111111 x 111111111111 = 12345679012320987654321
13. 1111111111111 x 1111111111111 = 1234567901234320987654321
14. 11111111111111 x 11111111111111 = 123456790123454320987654321
15. 111111111111111 x 111111111111111 = 12345679012345654320987654321
16. 1111111111111111 x 1111111111111111 = 12345679012345677654320987654321
17. 11111111111111111 x 11111111111111111 = 1.2345679012345678765432098765432e+32
18. 111111111111111111 x 111111111111111111 = 1.2345679012345678987654320987654e+34
19. 1111111111111111111 x 1111111111111111111 = 1.2345679012345679009876543209877e+36
20. 11111111111111111111 x 11111111111111111111 = 1.2345679012345679012098765432099e+38
21. 111111111111111111111 x 111111111111111111111 = 1.2345679012345679012320987654321e+40
22. 1111111111111111111111 x 111111111111111111111=1.2345679012345679012343209876543e+42

74
Natural patterns within the lines numbered 1 – 22, when multiplying ones by each
other, resulted in balanced binaries. Starting with # 10, there developed
segments within the segments, creating binaries within binaries.
Line Patterns
1. 1
2. +1 -1
3. +2 -2
4. +3 -3
5. +4 -4
6. +5 -5
7. +6 -6
8. +7 -7
9. +8 -8

10. Change point, beginning of segments in segments, starting with 00;

from this point on, only the segments within segments underlined will be
assessed. Zeroes did not need to be added – they were already there).

11. 0120 +2 -2
12. 012320 +3 -3
13. 01234320 +4 -4
14. 0123454320 +5 -5
15. 012345654320 +6 -6
16. 012345677654320 +7 -7
17. 0123456787654320 +8 -8
18. 012345678987654320 +9 -9

19. Change point, two segments within the line segment:

0123456790 0987654320
+9 -9 +9 -9

20. 0123456790120 +11 -11

21. 012345679012320 +12 -12
22. 01234567901234320 +13 -13

75
In the foregoing sections, even groupings of ones are multiplied by the same
number of ones. In the following examples, uneven groupings of ones are
multiplied. The results are still the same, however, balanced binaries.

1111 x 11111 = 12344321

1111111 x 11111 = 12345554321
111 x 111111111 = 12333333321
11 x 11111 = 122221

THE CIRCULARITY OF “TWONESS”

When groupings of twos are multiplied by other groupings of twos, the result
always begins and ends with the number four, creating balanced binaries in the
process.

2 x2=4 4
22 x 22 = 484 +4 -4
222 x 222 = 49284 +11 -11
2222 x 2222 = 4937284 +15 -15
22222 x 22222 = 493817284 +22 -22
222222 x 222222 = 49382617284 +26 -26
2222222 x 2222222 = 4938270617284 +33 -33
22222222 x 22222222 = 493827150617284 +37 -37
222222222 x 222222222 = 49382715950617284 +41 -41
2222222222 x 2222222222 = 4938271603950617284 +47 -47
22222222222 x 22222222222 = 493827160483950617284 +52 -52

THREENESS

Groupings of three multiplied by each other, unlike ones and twos, do not
produce balanced binaries. They always result in +8, and when zeroes are
added, +11 -11.

333 x 33333 = 11099889

76
“FOURNESS” THROUGH “EIGHTNESS:”

These numbers form predictable patterns when multiplied by themselves, but are
not naturally circular and do not form balanced binaries unless zeroes are added.

THE NUMBER NINE:

The number nine is the only number that generates a negative number,
which is always –8. (It is the converse of #3 which always generates +8). When
zeroes are added, it always generates +11 -11, just as # 3 does.

77
 CHAPTER SIXTEEN

FACTORIALS

A factorial is the product of a given series of whole numbers. The numbers 1! –

10! yield the following patterns:

Assessment (Phantom) Values

1! = 1 010 = +1 -1

2! = 2 020 = +2 -2

3! = 6 060 = +6 -6

4! = 24 0240 = +4 -4

5! = 120 0120 = +2 -2

6! = 720 0720 = +7 -7

7! = 5040 05040 = +9 -9

8! = 40320 040320 = +7 -7

9! = 362880 0362880 = +12 -12

10! = 3628800 03628800 = +12 -12

Note that the factorial values are always even numbers with the exception of #1,

yet the phantom (assessment) values alternate.

78
 CHAPTER SEVENTEEN

CONCLUSIONS AND IMPLICATIONS

The End of Randomness

Kurt Gödel (1906 – 1978) turned the world of mathematics upside-down with his
famous proof that showed that if a system is consistent, then you can show that it
is incomplete. Mathematician Gregory J. Chaitin of the IBM Thomas J. Watson
Research Center, author of The Limits of Mathematics suggests that the
structure of arithmetic is random. He states, “Although almost all numbers are
random, there is no formal axiomatic system that will allow us to prove this fact.”

Increment Analysis and The Law of Digit Balance point to an underlying order in
mathematics that defies randomness, and 50 + element mirror image increments
lends credence to this notion. Moreover, zero seems to be the background
“noise” of all numbers. Increment analysis opens up a whole new set of options
for analyzing number sequences. New unsuspected relationships and hidden
patterns become apparent that are beautifully symmetrical. As a result of the
evidence presented, the following implications become apparent:

• There is a Law of Digit Balance wherein the increments

between integers of any number sequence sum to zero. This
includes sequences comprised of positive and negative
numbers.

♦ The increments between digits of circular numbers manifest mirror image

patterns that sum to zero in all cases.

♦ The first 218 digits of pi have increments that converge to zero when a

quaternary skip sequence is utilized to isolate digits.

79
♦ The number seven is prominent in many numbers that produce mirror image

groupings that cancel to zero

♦ The prime numbers are organized around the digits of the reciprocal of seven

♦ Transcendental numbers have increments that sum to zero when extracting a

sequence of any size, hence, transcendental numbers sum to zero.

♦ Zero does not merely represent the absence of something, it also represents

a fullness and a balance.

♦ Fibonacci and Lucas numbers have increments that are perfect mirrors with

anomalies that add to nine, and demonstrate 24 – step repetition cycles.

♦ All numbers begin and end in zero.

Practical Applications?

When considering the practical applications of The Law of Digit Balance I am

reminded of a popular television commercial for an insurance company. In it, a
whimsical cartoon character is handed a black dot. The character, puzzling,
doesn’t know what to do with it until, suddenly, something falling from the sky
appears which threatens to squash him. He drops the dot, which becomes a hole
in the ground offering a means of escape. The point is that he had a tool that
was useless until more information was forthcoming. Further research may open
broader vistas in the world of numbers and yield surprising practical applications.
Chief among these might be information that can lead to a better understanding
of “trinary logic” and its application to trinary computers.

Actually, the idea of trinary computers is not a remote dream. One was
developed (SETUN) in 1958 by Nikolai Brusentsov and his team at the Moscow
State University and was based on ternary (trinary) logic (-1, 0, 1), which
distinguishes it completely from the usually binary operating computers of the

80
present and the past. While the binary logic just allows two states yes (1) | no (0),
the ternary logic has three different logical states: yes (1) | no (-1) | both or
maybe (0). In this book, Increment Analysis always reveals balanced positive
and negative numbers resulting in zero underlying all number sequences.

The sheer beauty of pattern revealed by The Law of Digit Balance, however,
provides esthetic satisfaction in itself.

81
82
 NATURAL NUMBERS 1 – 100
Number Sum of increments

1 010 = +1 -1 = 0

2 020 = +2 –2 = 0

3 030 = +3 –3 = 0

4 040 = +4 –4 + 0

5 050 = +5 –5 = 0

6 060 = +6 –6 = 0

7 070 = +7 –7 = 0

8 080 = +8 – 8 = 0

9 090 = +9 –9 = 0

10 010 = +1 -1 = 0

11 0110 = +1 0 –1 = 0

83
12 0120 = +1 +1 – 2 =

+2 -2 = 0

13 0130 = +1 +2 –3 =

+3 –3 = 0

14 0140 = +1 +3 –4 =

+4 -4 = 0

15 0150 = +1+4 -5 =

+5 -5 = 0

16 0160 = +1 +5 –6 =

+6 –6 = 0

17 0170 = +1+6 –7 =

+7 –7 = 0

84
18 0180 = +1 +7 –8 =

+8 - 8 = 0

19 0190 = +1 +8 –9 =

+9 – 9 = 0

20 020 = +2 – 2 = 0

21 0210 = +2 –1 –1 =

+2 -2 = 0

22 0220 = +2 0 -2 = 0

23 0230 = +2 +1 –3

+3 -3 = 0

24 0240 = +2 +2 –4 =

+4 –4 = 0

85
25 0250 = +2 +3 –5 =

+5 -5 = 0

26 0260 = +2 +4 –6 =

+6 –6 = 0

27 0270 = +2 +5 –7 =

+7 -7 = 0

28 0280 = +2 +6 –8=

+8 –8 = 0

29 0290 = +2 +7 –9 =

+9 -9 = 0

30 030 = +3 -3 = 0

86
31 0310 = +3 –2 –1 =

+3 -3 = 0

32 0320 = +3 –1 –2 =

+3 -3 = 0

33 0330 = +3 0 -3 = 0

34 0340 = +3 +1 –4 =

+4 -4 = 0

35 0350 = +3 +2 –5 =

+5 -5 = 0

36 0360 = +3 +3 -6 =

+6 -6 = 0

87
37 0370 = +3 +4 –7 =

+7 -7 = 0

38 0380 = +3 +5 –8 =

+8 –8 = 0

39 0390 = +3 +6 –9 =

+9 -9 = 0

40 040 = +4 –4 = 0

41 0410 = +4 –3 –1 =

+4 -4 = 0

42 0420 = +4 –2 –2 =

+4 -4 = 0

43 0430 = +4 –1 –3 =

+4 -4 = 0

88
44 0440 = +4 0 -4 = 0

45 0450 = +4+1 –5 =

+5 –5 = 0

46 0460 = +4+2 –6 =

+6 -6 = 0

47 0470 = +4+3 – 7 =

+7 -7 = 0

48 0480 = +4+4 –8 =

+8 –8 = 0

49 0490 = +4+5 –9 =

+9 -9 = 0

89
50 050 = +5 –5 = 0

51 0510 = +5 –4 –1 =

+5 -5 = 0

52 0520 = +5 –3 –2 =

+5 -5 = 0

53 0530 = +5 –2 –3 =

+5 -5 = 0

54 0540 = +5 –1 –4 =

+5 -5 = 0

55 0550 = +5 0 -5 = 0

90
56 0560 = +5 +1 -6 =

+6 -6 = 0

57 0570 = +5 +2 –7 =

+7 -7 = 0

58 0580 = +5+3 -8 =

+8 -8 = 0

59 0590 = +5+4 -9 =

+9 -9 = 0

60 060 = +6 –6 = 0

61 0610 = +6 -5 –1 =

+6 -6 = 0

91
62 0620 = +6 -4 –2 =

+6 -6 = 0

63 0630 = +6 -3 –3 =

+6 -6 = 0

64 0640 = +6 -2 –4 =

+6 -6 = 0

65 0650 = +6 -1 –5 =

+6 -6 = 0

66 0660 = +6 0 -6 = 0

67 0670 = +6 +1 –7 =

+7 -7 = 0

92
68 0680 = +6+2 – 8 =

+8 -8 = 0

69 0690 = +6+3 –9 =

+9 -9 = 0

70 070 = +7 –7 = 0

71 0710 = +7 –6 –1 =

+7 -7

72 0720 = +7 –5-2 =

+7 -7 = 0

73 0730 = +7 –4 –3 =

+7 -7 = 0

74 0740 = +7 –3 –4 =

+7 -7 = 0

93
75 0750 = +7 -2 –5 =

+7 -7 = 0

76 0760 = +7 -1 -6 =

+7 -7 = 0

77 0770 = =7 0 -7 = 0

78 0780 = +7 +1 –8 =

+8 -8 = 0

79 0790 = +7 +2 –9 =

+9 -9 = 0

80 080 = +8 -8 = 0

81 0810 = +8 –7 –1 =

+8 -8 = 0

94
82 0820 = +8 –6 –2 =

+8 -8 = 0

83 0830 = +8 -5 –3 =

+8 -8 = 0

84 0840 = +8 -4 –4 =

+8 -8 = 0

85 0850 = +8 -3 –5 =

+8 -8 = 0

86 0860 = +8 -2 -6 =

+8 -8 = 0

95
87 0870 = +8 -1 -7 =

+8 -8 = 0

88 0880 = +8 0 -8 = 0

89 0890 = +8 +1 -9 =

+9 -9 = 0

90 090 = +9 -9 = 0

91 0910 = +9 -8 -1 =

+9 -9 = 0

92 0920 = +9 -7 –2 =

+9 -9 = 0

93 0930 = +9 -6 –3 =

+9 -9 = 0

96
94 0940 = +9 –5 –4 =

+9 -9 = 0

95 0950 = +9 -4 -5 =

+9 -9 = 0

96 0960 = +9 -3 –6 =

+9 -9 = 0

97 0970 = +9 -2 -7 =

+9 -9 = 0

98 0980 = +9 -1 -8 =

+9 -9 = 0

99 0990 = +9 0 -9 = 0

100 0100 = +1 -1 0 = 0

97
 RECIPROCALS of #1 – 100
(If zeros are added before and after the number, it always sums to zero).

NUMBER PATTERN MODE

+1 -1 Oscillating 1s
1/1 = 1.0

+5 -5 Oscillating 5s
½ = 0.50

1/3 = 0.333333333 ( +3) 0000000000 0

¼ = 0.250 +2 +3 –5 Oscillating 5s
second order:
(+5 -5)

1/5 = 0.20 +2 -2 Oscillating 2s

1/6 = 0.166666666666… (+1 +5) 00000000 0

1/7 = 142857142857… +3 -2 +6 -3 +2 -6 Six-element mirror

1/8 = 0.125 +1 +1 +3 Mixed

second order: (Ps = +5)
+5

1/9 =0.1111111111… (+1) 000000000 0

1/10 = 0.10 +1 -1 Oscillating 1s

1/11 = +9 -9 +9 -9… Oscillating 9s

0.0909090909090…

98
1/12 = (+8 -5) 00000000000 0
0.0833333333333…

1/13 = +7 -1 +3 -7 +1 -3 Six-element mirror

0.0769230769230…

1/14 = (+7) –6 +3 -2 +6 -3 +2 Six-element mirror

0.07142857142857… …

1/15 = 0.06666666666…. (+6) 000000000000… 0

1/16 = 0.0625 +6 -4 +3 Mixed

(Ps = +5)

1/17 = +5 +3 0 -6 +1 +2 -3 16-element mirror

0.05882352941176470 +7
-5 -3 0 +6 -1 -2 +3 -7

1/18 = 0.0555555555… (+5) 00000000000… 0

1/19 = +5 -3 +4 -3 -2 +4 +2 18-element mirror

0.052631578947368421 +1 +1
0… -5 +3 -4 +3 +2 -4 -2
-1 -1

1/20 = 0.05 +5 -5 Oscillating 5s

1/21 = +4 +3 -1 -5 +8 Mixed (Ps = 9)

0.047619047619…

1/22 = 0.04545454545… +1 -1 +1 -1… Oscillating 1s

99
 +4 -1 +1 +3 +1 -6 +4 –
1/23 = 6 +8 22-element mirror
0.043478260869565217 -2 +3
39130… -4 +1 -1 -3 -1 +6 -4
+6 -8 +2 -3

1/24 = (+4 -3 +5)

0.041666666666666… 0000000000000 0

1/25 = 0.04 +4 -4 Oscillating 4s

1/26 = ( +3 ) +5 -4 +2 -5 +4 6-element mirror

0.038461538461538461 -2 +3 –4 +2
53846…

1/27 = +3 +4 -7 +3 +4 -7 Oscillating 7s
0.037037037037… Second order:
+7 -7

1/28 = (+3 +2) +2 -6 +3 -2 6-element mirror

0.0357142857142857… +6 -3

+3 +1 0 +4 -6 +5 -2
1/29 = +3 -2 28-element mirror
0.034482758620689655 -4 -2 +6 +2 +1
17241379310…
-3 -1 0 -4 +6 -5 +2 -
3 +2 +4 +2 -6 -2 -1
1/30 = (+3)
0.033333333333333333 00000000000000000 0
…

1/31 =
0.0322580645161290…

100
1/32 = 0.03125 +3 -2 +1 +3 Mixed
(Ps = +5)

1/33 = +3 -3 +3 -3 +3 -3… Oscillating 3s

0.030303030303…

1/34 = (+2 +7) -5 -3 16-element mirror;

0.02941176470588235… 0 Oscillating –5 -3 0, +5
+6 -1 -2 +3 -7 +5 +3 +3 0
0
-6 +1 +2 -3 +7 -5 -3
0

1/35 = (+2) +6 -3 + 2 -6 +3 6-element mirror

0.02857142857142857… -2 +6 -3…

1/36 = (+2 +5) 000000000000 0

0.027777777777…
(Ps = +7)

1/37 = +2 +5 -7 +2 +5 -7 +2 Oscillating 7s
0.027027027027… +5 -7…
Second order:
+7 -7 +7 -7 +7 -7…
(+2)
1/38 = +4 -3 -2 +4 +2 +1 +1 18-element mirror
0.026315789473684210 -5 +3
5263… -4 +3 +2 -4 -2 -1 -1
+5 -3
(+4 -3)

1/39 = +2 +3 +1 -2 -3 -1… 6-element mirror

0.025641025641…
Ps = 0

101
 Mixed

1/40 = 0.025 +2 +3 (Ps = +5)

1/41 = 0.02439 02439… +2 +2 -1 +6… Mixed

(Ps = +9)

1/42 = 0.0238095 (+2) Mixed

238095… +1 +5 -8 +9 -4 -3…
(Ps = 0)

1/43 = +2 +1 -1 +3 0 +3 -7 Mixed;
0.023255813953488372 +2 +6
0930 232558139535 -4 -2 +1 +4 0 -5 +4 (Ps = 0)
-5 -2 +9 -6 -3…

1/44 =
0.02272727272727…

1/45 = (+2) 000000000000 0

0.02222222222222…

1/46 = -1 +6 -4 +6 -8 +2 -3 22-element mirror

0.021739130434782608 +4 -1 +1 +3
695652… Ps = 0
+1 -6 +4 -6 +8 -2 +3
-4 +1
-1 -3

102
 +2 -1 +1 +5 -1 -1 +4
1/47 = -4 +2 46-element mirror
0.021276595744680851 –3 0 +2 +2 -8 +8 -3 -
0638297872340425… 4 -1 +6 -3 +5 -6 +7 Ps = 0

-2 +1 -1 -5 +1 +1 -4
+4 -2 +3 0 -2 -2 +8 -
8 +3 +4 +1
-6 +3 –5 +6 -7***

1/48 = (+2 -2 +8 -5 ) 0
0.020833333333333333 000000000000000000
33…

+2 -2 +4 -4 +8 -7+5 -
1/49 = 3 -1 +4 –1 -2 -3 +6 -5 42-element mirror
0.020408163265306122 +1 0 +2 0 +4 +1
4489795918367346… -2 +2 -4 +4 -8 +7 -5 Ps = 0
+3 +1 –4
+1 +2 +3 -6 +5 -1 0 -
2 0 -4 -1 ***

1/50 = 0.02 +2 (-2) +2 ( -2)

1/51 = +1 +8 -3 -6 +7 +1 -4 Mixed
0.01960784313725490… -1 0 +4 -5 +3 -1 +5 -
9 (Ps = 0)

1/52 = ( +1 +8) -7 +1 -3 6-element mirror

0.019230769230769230 +7 -1 +3…
…

103
1/53 = +1 +7 0 -2 +1 +2 -7 Mixed
0.01886792452830 +2 +1
1886792452830 -3 +6 -5 -3 (Ps = 0)
1886792…

1/54 = (+1) +7 -3 -4 +7 -3 - Oscillating sevens

0.0185185185185185… 4…
(Ps = 0)***
Second order:
(+1) +7 -7 +7 -7

1/55 = (+1) +7 -7 +7 -7 +7 - Oscillating sevens

0.018181818181818… 7…

1/56 = 0.017857 (+1 +6 +1) -3 +2 -6 Six-element mirror

142857 142857… +3 -2 +6…

1/57 = +1 +6 -2 -1 -1 +5 -3 Mixed
0.017543859649122807 +4 -3 (Ps = 0)
0 175438596491228… -2 +5 -8 +1 0 +6 -8
+7 -7…

(+1)
1/58 = +6 -5 +2 -3 +2 +4 +2 28-element mirror
0.017241379310344827 -6 -2
58620689655 17241… -1 +3 +1 0 +4

-6 +5 -2 +3 -2 -4 -2
+6 +2 +1 -3 -1 0 -
4…

104
 (+1)
1/59 = +5 +3 -5 +5 -8 +4 -3 58-element mirror
0.016949152542372881 +3 -1
3559322033898305… -2 +1 +4 -5 +6 0 -7
+2 +2 0 +4 -6 -1 0 -2
+3 0 +5 +1 -1

-5 -3 +5 -5 +8 -4 +3 -
3 +1
+2 -1 -4 +5 -6 0 +7 -
2 -2 0
-4 +6 +1 0 +2 -3 0 -5
-1 +1

1/60 = (+1 +5)

0.016666666666666666 0000000000000000000 0
…

1/61 = +1 +5 -3 +6 -6 +1 0 - 60-element mirror?

0.016393442622950819 2 +4 -4 0 +7 -4 -5 +8
6721311475409836… -7 +8 -3 +1 -5 -1 +2 -
2 0 +3 +3 -2 -1 -4 +9

-1 -5 +3……

1/62 = (+1) Mixed

0.0161290322580645 +5 -5 +1 +7 -9 +3 -1
161290322580645 0 +3 +3 -8 +6 -2 +1 - (Ps = 0 )
161… 4 +5 -5…

105
1/63 = 0.015873 015873 +1 +4 +3 -1 - 4 -3 Six-element mirror
015873 015873 015873
0158…

1/64 = 0.015625
+1 +4 +1 -4 +3 Mixed

(Ps = 5)

1/65 = 0.0153846 (+1)

153846 153846153846 Six-element mirror
153846 153… +4 -2 +5 -4 +2 -5

(+1)
1/66 = +4 -4 +4 -4 +4 -4 +4 Oscillating 4s
0.0151515151515…. -4…
(Ps = 0)

1/67 = (+1) +3 +5 -7 +3 -2 +4
0.014925373134328358 -4 -2 +2 +1 -1 -1 +6 - 62-element mirror???
208955223880597… 5 +2 +3 -6 -2 +8 +1 -
4 0 -3 0 +1 +5 0
-8 +5 +4 -2 …

(+1 +3)
1/68 = 14-element mirror
0.014705882352941176 +3 -7 +5 +3 0 -6 +1
4705882352941176… (+2)
-3 +7 -5 -3 0 +6 -1

106
1/69 = +1 +3 0 +5 -7 +5 -2 - Mixed
0.014492753623188405 2 +3 -4 +1 -2 +7 0 -4
79710 14492753623… -4 +5 +2 +2 -2 -6 - (Ps = 0 )
1….

1/70 = 0.0142857 (+1) +3 -2 +6 -3 +2 - 6-element mirror

142857142857… 6

1/71 = +3-4+8-4+1-5+7-7+4-2 Mixed

0.014084507042253521 0+3-2+2-3-10+1+4+1-1-
1267605633802817… 6+5+1-3 0+5-8+2+6-7+6

1/72 = (+1) +2 +5 0
0.013888888888888888 000000000000
8…

1/73 = 0.013698630 +1 +2 +3 +3 8-element mirror

13698630 13698630 -1 -2 -3 -3
136986301…

1/74 = (+1)
0.013513513513513513 Oscillating 4s
5… +2 +2 -4 +2 +2 -4
Second order:

+4 -4 +4 -4…..

1/75 = (+1) +2 0
0.01333333333333… 00000000000000

107
1/76 = (+1) +2
0.013157894736842105 18-element mirror
26 31578947368421… -2 +4 +2 +1 +1 -5 +3
-4 -3
+2 -4 -2 -1 -1 +5 -3
+4 -3

1/77 = 0.0129870
129870 129870 129870 +1 +1 +7 -1 -1 -7 6-element mirror
129870…

1/80 = 0.0125 +1 +1 +3 Mixed

(Ps = 5)

1/81 = 0.012345679 +1 +1 +1 +1 +1 +1 + +1
012345679 012345679… 1 +2

1/82 = 0.012195 12195 (+1) Mixed

12195 1219512195 +1 -1 +8 -4
12195… (Ps = +5 )

+1+1-2+4+4-7+8-7+5 0 – Mixed
1/83 = 6-1+8-4-1 0 +4-4+1+5-
0.012048192771084337 6+6-2-2+4-9+3+6-5+3 0
3493975903614458 +1+3…

1/84 = 0.01190476 (+1 0)

190476 190476 190476 Mixed
19047619 +8 -9 +4 +3 -1
(Ps = 6)

108
 (+1)
0 +6 -1 -2 +3 -7 +5
1/85 = +3 16-element mirror
0.01176470588235294 0 -6 +1 +2 -3 +7 -5
1176470588235294… -3

1/86 = +1 0 +5 –4+5+2-9+6+3- Mixed

0.011627906976744186 2-1+1-3 0 –3+7-2-6+4+2-
0465 1=4 0 +5-4+5+2-9+6+3-
116279069767… 2-1+1

+1 = +3 +5-5-2+3-3+6-1-
1/87 = 4+2+1-3-1-1+7-5+6-9+8- Mixed
0.011494252873563218 8+4+1+4-2 0 -– +1 0 +3
39080459770 11494… +5-5

1/88 = (+1) 0 -3 +3 -3 +3 -3 Oscillating 3s

0.011363636363636363 +3….
6…

109
 +1 0 +1 +1 +2 +4-4 0 –
1/89 = 5+5+1-5+6+2-2 0 –2 –3 44-element mirror
0.011235955056179775 +6 –8+8 +1-1 0 –1-1-2-
2808988764044944 4+4 0 +5 –5 0

1/90 =
0.011111111111111111 (+1) 000000000000000 0
1111111111111111…

1/91 = 0.010989 010989 +1 -1 +9 -1 +1 -9 6-element mirror

010989 010989
010989…

1/92 = (+1-1)+8-2+3-4 22-element mirror

0.010869565217391304 +1-1-3-1+6-4+6-8+2-3
3478260869565217 +4- 1+1+3+1-6+4-6 +8
-2+3-4+1-1-3-1+6

1/93 = +1 -1 +7 -2 -3 +4 +2 Mixed
0.010752688172043 0 +6 -5 -2 +4 -1 (-3)
010752688172043…

1/94 = (+1-1)+6-3+5-6+7-2+1-1- 14-element mirror

0.010638297872340425 5+1+1-4+4-2+3 0 –2-
5319148936170213 2+8-8+3+4+1
-6+3-5+6-7+2-1+2

110
1/95 = (+1-1)+5-3+4-3- +5-3+4-3-2+4+2+1+1
0.010526315789473684 2+4+2+1+1-5+3-4+3+2-
2105263157894737 4-2-1-1+5-3+4-3- +5-3+4-3-2+4+2 +1-1
2+4+2+1+1
-5+3-4+4…

1/96 = (+1-1)+4-3+5 0 0 0… Ps = 6
0.010416666666666666
666666666666666…

1/97 = (+1-1)+3-3+9-7+5+1-5+2- Mixed

0.010309278350515463 5+5-4+4-1+2-3+6-8+6-2
9175257731958763 -3+3+2 0 –4-2+8-4+3-1-
1-3

1/98 = (+1-1)+2-2+4-4+8-7+5-3 46-element mirror

0.010204081632653061 -1+4-1-2-3+6-5+1 0 +2
2244897959183673 0+4+1-2+2 -4+4-8+7
-5+3+1-4…

1/99 = +1-1+1-1…. Oscillating 1s

0.010101010101010101
0101010101010101

1/100 = 0.01 +1-1 0

111
 MIRROR SEQUENCES

(For reciprocals of numbers between 1 – 100)

1/7 = 142857142857… +3 -2 +6 -3 +2 -6 Six-element mirror

1/13 = +7 -1 +3 -7 +1 -3 Six-element mirror

0.0769230769230…

1/14 = (+7) –6 +3 -2 +6 -3 +2 Six-element mirror

0.07142857142857… …

1/17 = +5 +3 0 -6 +1 +2 -3 16-element mirror

0.05882352941176470 +7
-5 -3 0 +6 -1 -2 +3 -7

1/19 = +5 -3 +4 -3 -2 +4 +2 18-element mirror

0.052631578947368421 +1 +1
0… -5 +3 -4 +3 +2 -4 -2
-1 -1
+4 -1 +1 +3 +1 -6 +4 –
1/23 = 6 +8 22-element mirror
0.043478260869565217 -2 +3
39130… -4 +1 -1 -3 -1 +6 -4
+6 -8 +2 -3

1/26 = ( +3 ) +5 -4 +2 -5 +4 6-element mirror

0.038461538461538461 -2 +3 –4 +2
53846…

1/28 = (+3 +2) +2 -6 +3 -2 6-element mirror

0.0357142857142857… +6 -3

+3 +1 0 +4 -6 +5 -2
1/29 = +3 -2 28-element mirror
0.034482758620689655 -4 -2 +6 +2 +1
17241379310… -3 -1 0 -4 +6 -5 +2 -
3 +2 +4 +2 -6 -2 -1

112
1/34 = (+2 +7) -5 -3 16-element mirror;
0.02941176470588235… 0 Oscillating –5 -3 0, +5
+6 -1 -2 +3 -7 +5 +3 +3 0
0
-6 +1 +2 -3 +7 -5 -3
0

1/35 = (+2) +6 -3 + 2 -6 +3 6-element mirror

0.02857142857142857… -2 +6 -3…

(+2)
1/38 = +4 -3 -2 +4 +2 +1 +1 18-element mirror
0.026315789473684210 -5 +3
5263… -4 +3 +2 -4 -2 -1 -1
+5 -3 (+4 -3)

1/39 = +2 +3 +1 -2 -3 -1… 6-element mirror

0.025641025641…
Ps = 0

1/46 = -1 +6 -4 +6 -8 +2 -3 22-element mirror

0.021739130434782608 +4 -1 +1 +3
695652… Ps = 0
+1 -6 +4 -6 +8 -2 +3
-4 +1 -1 -3
+2 -1 +1 +5 -1 -1 +4
1/47 = -4 +2 46-element mirror
0.021276595744680851 –3 0 +2 +2 -8 +8 -3 -
0638297872340425… 4 -1 +6 -3 +5 -6 +7 Ps = 0

-2 +1 -1 -5 +1 +1 -4
+4 -2 +3 0 -2 -2 +8 -
8 +3 +4 +1
-6 +3 –5 +6 -7

1/49 = +2 -2 +4 -4 +8 -7+5 -
0.020408163265306122 3 -1 +4 –1 -2 -3 +6 -5 42-element mirror
4489795918367346… +1 0 +2 0 +4 +1
Ps = 0
-2 +2 -4 +4 -8 +7 -5
+3 +1 –4
+1 +2 +3 -6 +5 -1 0 -
2 0 -4 -1

113
1/52 = ( +1 +8) -7 +1 -3 6-element mirror
0.019230769230769230 +7 -1 +3…
…

1/56 = 0.017857 (+1 +6 +1) -3 +2 -6 Six-element mirror

142857 142857… +3 -2 +6…

(+1)
1/58 = +6 -5 +2 -3 +2 +4 +2 28-element mirror
0.017241379310344827 -6 -2
58620689655 17241… -1 +3 +1 0 +4

-6 +5 -2 +3 -2 -4 -2
+6 +2 +1 -3 -1 0 -
4…

1/59 = (+1)
0.016949152542372881 +5 +3 -5 +5 -8 +4 -3
3559322033898305… +3 -1 58-element mirror
-2 +1 +4 -5 +6 0 -7
+2 +2 0 +4 -6 -1 0 -2
+3 0 +5 +1 -1

-5 -3 +5 -5 +8 -4 +3 -
3 +1
+2 -1 -4 +5 -6 0 +7 -
2 -2 0
-4 +6 +1 0 +2 -3 0 -5
-1 +1

1/61 = +1 +5 -3 +6 -6 +1 0 - 60-element mirror?

0.016393442622950819 2 +4 -4 0 +7 -4 -5 +8
6721311475409836… -7 +8 -3 +1 -5 -1 +2 -
2 0 +3 +3 -2 -1 -4 +9

-1 -5 +3……

114
1/63 = 0.015873 015873 +1 +4 +3 -1 - 4 -3 Six-element mirror
015873 015873 015873
0158…

1/65 = 0.0153846 (+1)

153846 153846153846 Six-element mirror
153846 153… +4 -2 +5 -4 +2 -5

1/67 = (+1) +3 +5 -7 +3 -2 +4 ?
0.014925373134328358 -4 -2 +2 +1 -1 -1 +6 - 66-element mirror???
208955223880597… 5 +2 +3 -6 -2 +8 +1 -
4 0 -3 0 +1 +5 0
-8 +5 +4 -2 …

(+1 +3)
1/68 = 14-element mirror
0.014705882352941176 +3 -7 +5 +3 0 -6 +1
4705882352941176… (+2)
-3 +7 -5 -3 0 +6 -1

1/70 = 0.0142857 (+1) +3 -2 +6 -3 +2 - 6-element mirror

142857142857… 6

1/73 = 0.013698630 +1 +2 +3 +3 8-element mirror

13698630 13698630 -1 -2 -3 -3
136986301…

(+1) +2
1/76 = 18-element mirror
0.013157894736842105 -2 +4 +2 +1 +1 -5 +3
26 31578947368421… -4 -3

+2 -4 -2 -1 -1 +5 -3
+4 -3

115
1/77 = 0.0129870 6-element mirror
129870 129870 129870 +1 +1 +7 -1 -1 -7
129870…

(+1)
1/85 = 0 +6 -1 -2 +3 -7 +5 16-element mirror
0.01176470588235294 +3
1176470588235294… 0 -6 +1 +2 -3 +7 -5
-3

+1 0 +1 +1 +2 +4 -4 0 44-element mirror
1/89 = -5 +5 +1 -5 +6 +2 -2
0.011235955056179775 0 -2 –3 +6 -8 +8 +1
2808988764044944… -1 0 -1 -1 -2 -4 +4 0
+5
-5 -1 +5 -6 -2 +2 0
+2 +3 -6 +8 -8 -1**

6-element mirror
1/91 = 0.010989 010989 +1 -1 +9 -1 +1 -9
010989 010989
010989…

(+1 -1) +8 -2 +3 -4 +1
1/92 = -1 -3 -1 +6 -4 +6 22-element mirror
0.010869565217391304 -8 +2 -3 +4 -1 +1 +3
3478260869565217 +1 -6 +4 -6

(+1 -1) +5 -3 +4 -3 -2
1/95 = +4 +2 +1 +1 18-element mirror
0.010526315789473684
2105263157894737 -5 +3 -4 +3 +2 -4 -2 -
1 -1

116
1/98 = (+1 -1) +2 -2 +4 -4 +8 42-element mirror
0.010204081632653061 -7 +5 -3 -1 +4 -1 -2 -
2244897959183673… 3 +6 -5 +1 0 +2 0 +4
+1
-2 +2 -4 +4 -8 +7 -5
+3 +1 -4…

117
 OTHER NUMBER SEQUENCES
(When zeros are added before and after segment, the mode is always zero)

NUMBER
INCREMENT MODE
ANALYSIS PATTERN

Mass, Gravity, +5 +3 -5 0 0 repeaters

Communications harmonic
(Bruce Cathie)

16944

23 = +6 -6 +6 -6 +6 Oscillating 6s
2.828428
(2nd order sequence)
Synergetics conversion -1 +6 -6 +6 0 Oscillating 6s
constant (R. Buckminster
Fuller)

1.06066

Scheherazade Number (first 8 -5 +5 -5 Oscillating 5s

primes factorial) R. B. Fuller

30, 030 x 17 = 510,510!

Scheherazade Number (first 7 -3 0 +3 -3 Oscillating 3s

primes factorial) R. B. Fuller
30, 030

Primes through 29 (R.B. Fuller) -2 +2 +3 -3 +3 -3 -1 Oscillating binary pairs

+1 -3
6,469,693,230

Seven-illion Scheherazade +1 -1 +1 Oscillating

Number (R.B. Fuller) 1s
(5th order sequence)
24,421,743,243,121,524,300,0
00

118
Matter plus anti-matter Earth -3 0 +3 +4 -4 +4 -8 +1 Oscillating 3s and 4s
Field Harmonic (Bruce Cathie)
(2nd order sequence)
4114846801

e= 4-element mirrors (-6 +7,

2.718281828459045… +5 -6 +7 -6 +6 -7 +7 +6 –7)
-6 +6 -4 +1 +4 binary pairs

Planck’s Constant 0 -4 +4 -6 +6 +3 Binary pairs

h = 6.626069 x 10-34

Speed of light in a vacuum +7 0 -2 +2 -7 +2 +1 +3 Binary pairs

2.99792458 x 108 m/s

Atomic mass unit +5 0 -6 +6 -1 0 Binary pair (-6 +6)

1.6605655
(2nd order sequence)

C (Euler-Mascheroni Constant) +2 0 -5 -1 +4 +1 Pseudo-mirrors

0 -2 +5
0.5772156649

√2= +3 -3 +3 -3 Oscillating 3s
1.414213…
(2nd order sequence)

Schrodinger Constant +1 -1 +1 +9 Oscillating 1s

1.644934067
(2nd order sequence)

119
Electric permittivity constant -7 +7 -1 +1 -7 Binary pairs, oscillating
7s
8.85418781...x10 (2nd order sequence)

Euler’s constant reciprocal = -3 +3 -6 +6 0 -6 +6 +2 Oscillating 3s, 6s

-4 -3
1.73245471460063605660684
1… ( 2nd order sequence)

Reciprocal of #137 (a rough 0 +7 -5 +7 0 6-element mirrors

approximation of the fine -7 +5 -7 0 +7
structure constant)

0.007299270072...

Reciprocal of #49 +2 -2 +4 -4 +8 -7 +5 Binary pairs and 4-

0.020408163265306… element mirrors
-4 +4 -6 +6

(2nd order sequence)

7/9 + 9/7 = +2 +6 -3 +2 -6 +3 6-element mirrors
1.396825396825… -2 +6 -3 +2 -6 +3
up to 13 digits

7/9 + 9/7 reciprocal = -6 +4 +4 -9 +9 -9 +9 -9 Oscillating 9s

0.715909090909… +9 -9 +9
up to 12 digits

137/13 reciprocal = 6-element mirrors

+9 -5 +5 -9 +5 -5
0.094890510…

120
137/13 = -1 +5 -2 +5 -4 6-element mirrors
10.538461538461… +2 -5 +4 -2 +5 -4
+2 -5 +4

137/7 reciprocal = +5 -4 -1 +9 -5 -4 4-element mirrors (-1 +9,

+1 -9 +1 –9)
0.051094890510…

137/7 = +8 -4 +2 -6 +3 -2 +6 6-element mirrors

19.5714285714… -3
+2 -6 +3

Coupling constant = +8 -3 -3 +3 0 Oscillating 3s

0.08542455

(2nd order sequence)

13/81 = -4 +1 -3 +7 -1 +3 -7 6-element mirrors

6.2307692307 +1 -3 +7

13/81 reciprocal = +5 -6 +4 +5 -6 +5 -6 Repeaters (+5 –6)

0.160493827… +5

13/49 = +4 -6 +6 -4 0 4-element mirrors

0.265306122…

(2nd order sequence)

121
√26/81 = +1 0 -1 0 +2 0 -5 4-element mirrors
0.566557723 +1

+7 -6 +6 -7 +7 -6 Binary pairs
9/123 = 0.073170731…
(2nd order sequence)

-1 +2 -1 +7 +1 -2 +1 -7 8-element mirrors
28/274 =0.102189781

Connected with electron spin -1 0 +1 0 +8 -8 +8 - Oscillating 8s, 1s

(Harold Aspden) 6
1.001159652193

Absolute zero; average period +3 +3 0 -3 0 0 +3

of human gestation: 273: “threeness”
reciprocal =
1/273 = 0.003663003

(Peter Plichta: God’s Secret

Code)
+5 -5 Binary pair
Sidereal month: 27.32 days

(2nd order sequence)

+3 0 +5 -5 +5 -1 Oscillating 5s;
Synodic month reciprocal,
1/29.53 days = 0.033863867

122
 Mirrors; repeaters
Euler’s Constant =
+2 0 -5 -2 +5 0 -2 +5
0.577215664901532
-9 +5 -2 -1

(2nd order sequence)

+5 -5 +7 -8 +3 0 +6 -1 Oscillating fives
Golden ratio = 0 -1 +3 +5 -1 +1 -5
1.618033988749894…

Magnetic permeability +4 -4 +4 -2 -4 0 -1
constant = Oscillating 4s
1.2566370614…x1? (3rd order sequence)

Nuclear magneton = -5 +5 -5 +7 +1 -5 Oscillating 5s

5.050783 x 10 –22 J/T

6-element mirrors
#0.094890510
+9 -5 +5 -9 +5 -5

Binary pairs
#7787537 +1 -1 -4 +4

(2nd order sequence)

-7 +7 +1 -1 -7 Oscillating 7s, binary

#7078760 pairs

123
 -1 +2 +1 +3 -6 +7
0.21347189763921347189763 12-element mirrors
+1 -2 -1 -3 +6 -7…
9…
(Cyclic divisibility,
mathpages.com)

#134718976392 10-element mirrors

+2 +1 +3 -6 +7 +1
-2 -1 -3 +6 -7
(Cyclic divisibility,
mathpages.com)

-7 +6 -3 -1 -2 +1
#706320182568 10-element mirrors
+7 -6 +3 +1 +2

#865281023607 -2 -1 -3 +6 -7 -1 10-element mirrors

+2 +1 +3 -6 +7
(cyclic integer)

+6 -3 +2 -6 +3 -2 6-element mirrors
Auric Key reciprocal; 1/2720
= 3.968253968253…
(Syndex Synergetics Synopsis,
Iona Miller)

124
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 “I have examined Dr. Bowen’s abstract and found it an
innovative contribution to mathematics.”

Emmanuel Gugwor, Mathematics Instructor

Prologue H.S. and Truman College, Chicago

“Looking at the patterns of these numbers you might discover the deeper secrets of many lives.”

Sow Aboubacar Sidy, Ph.D.

Adjunct Professor of Mathematics
Loyola University of Chicago

Professor of Mathematics,
Columbia College, Chicago

ISBN 0-9615454-2-9

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