Characteristics of Numbers Download Site

An Approach to the Characteristics of Numbers

by Frank C. Fung - 1st published in November, 2003. ( 810 pages )

This paper is dedicated to the late Charles H.L. Fung

TABLE OF CONTENT

SECTION 1 - Introduction & Summary ( 30 pages )
-------------------- Title of the Chapter --------------------filename :pages
1-01Introductionconint22.doc1
1-02The concept of { OH-WON-TOOLE (0-1-2) }
- the main connecting theme throughout		conowt22.doc5
1-03Approach & Summary to 'Section 2'
- Examining the Hyperbola		conaas22.doc2
1-04Approach & Summary to 'Section 3'
- A General Study of Numbers		conaas33.doc3
1-05Approach & Summary to 'Section 4'
- Fermat's 'Sum-Of-Two-Squares' Equation		conaas44.doc6
1-06Approach & Summary to 'Section 5'
- Fermat Numbers		conaas55.doc5
1-07Approach & Summary to 'Section 6'
- The Fermat Little Theorem		conaas66.doc1
1-08Approach & Summary to 'Section 7'
- Squares of Squares		conaas77.doc3
1-09Approach & Summary to 'Section 8'
- Logarithm of '3' [ base '2' ]		conaas88.doc1
1-10Approach & Summary to 'Section 9'
- Global Positioning System		conaas99.doc1
1-11Approach & Summary to 'Section10'
- The Logarithmic Process		conaas00.doc2

SECTION 2 - Examining the Hyperbola ( 75 pages )
-------------------- Title of the Chapter --------------------filename :pages
2-01The Multiplicative Product expressed as
a Difference of Squares		condif22.doc1
2-02The Hyperbola - { X*Y = Constant }conhyp22.doc2
2-03The 'Integer X-Y Grid' & the Hyperbolaconhyp23.doc5
2-04The Hyperbola - { U^2 - V^2 = Constant }conhyp33.doc3
2-05The Hyperbola - Cylindriacal Co-ordinatesconhyp44.doc11
2-06The Characteristic Rhombus for the Hyperbolaconrho22.doc17
2-07Constructing the hyperbola - using the concept of
[ The 'X' & the 'cross' ]		conxac22.doc18
2-08Constructing the hyperbola - using the concept of
[ The 'Circle' & the 'Chord' ]		concac22.doc10
2-09The Hyperbola - Area under the Curve
( Cartesian co-ordinates )		conhyp55.doc5
2-10The Hyperbola - Area under the Curve - ( in
preparation for Cylindrical co-ordinates )		conhyp56.doc9
2-11The Hyperbola - Area under the Curve
Cylindrical Co-ordinates considerations		conhyp66.doc4

SECTION 3 - A General Study of Numbers ( 27 pages )
-------------------- Title of the Chapter --------------------filename :pages
3-01'Even' Numbers Vs. 'Odd' Numbersconevo22.doc3
3-02{ Power-Of Two } - [ POT-X ] vs. [ PAROT-X ]
& [ POF-X ]		conpar22.doc2
3-03'Odd' Numbers - 'P-ONE' vs. 'M-ONE' 'odd' integers
( '4*K + 1' vs. '4*K - 1' formats )		conpon22.doc4
3-04'P-M-M-P' Mappingconpmm22.doc5
3-05The Pythagorus Equation - Finding Integers
Solutions		conpyt22.doc13

SECTION 4 - Fermat's 'Sum-Of-Two-Squares' Equation ( 276 pages )
-------------------- Title of the Chapter --------------------filename :pages
4-01The Fermat 'Sum-Of-Two-Squares' equation
- A First Glance		confer22.doc5
4-02The Fermat 'Sum-Of-Two-Squares' equation
- A Second Look		confer23.doc4
4-03Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Setting up the Equations		condua22.doc3
4-04Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Setting up 'Q'		condua23.doc6
4-05Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - A First Analysis of the 'right-hand-side'		condua24.doc13
4-06Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'Case Two' on the 'left-hand-side'		condua25.doc12
4-07Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'LGPE' & 'LGME' are both 'even'		condua26.doc20
4-08Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'right-hand-side' when 'IQ-ODD' is 'P-ONE'		condua33.doc26
4-09Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'right-hand-side' when 'IQ-ODD' is 'M-ONE'		condua34.doc27
4-10Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'left-hand-side' with 'Case Three' operative		condua44.doc2
4-11Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'left-hand-side' with 'Case One' operative		condua45.doc2
4-12Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'left-hand-side' with 'Case Two' ( 1 of 4 )		condua55.doc22
4-13Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'left-hand-side' with 'Case Two' ( 2 of 4 )		condua56.doc26
4-14Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'left-hand-side' with 'Case Two' ( 3 of 4 )		condua57.doc24
4-15Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - 'left-hand-side' with 'Case Two' ( 4 of 4 )		condua58.doc27
4-16Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Summary of Findings		condua66.doc6
4-17Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Worked Example [ Q = 16 * 105 ]		condua77.doc3
4-18Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Worked Example [ Q = 32 * 105 ]		condua78.doc4
4-19Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Worked Example [ Q = 64 * 105 ]		condua79.doc5
4-20Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Worked Example [ Q = 128 * 105 ]		condua80.doc6
4-21Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Worked Example [ Q = 256 * 105 ]		condua81.doc6
4-22Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Worked Example [ Q = 512 * 105 ]		condua82.doc7
4-23Dual Solution to the Fermat 'Sum-Of-Two-Squares'
Equation - Combinatorial Counting for the examples		condua83.doc3
4-24{ Paired Factors } - Squares vs. Non-Squarescondua88.doc16
4-25The 'Prime Number Theorem'conpnt22.doc1

SECTION 5 - Fermat Numbers ( 127 pages )
-------------------- Title of the Chapter --------------------filename :pages
5-01'Fermat Numbers' - A First Glanceconfnu22.doc2
5-02The 'Augmented Binary Format' - { Bundle of ( 1's) }
& { Bundle of ( 0's) }		confnu33.doc8
5-03Factors of [ 2^32 + 1 ] in 'Augmented Binary
Format' and the 'Multiplication Box' explained		confn522.doc7
5-04Factors of [ 2^64 + 1 ] in 'Augmented Binary
Format'		confn622.doc2
5-05Factors of [ 2^128 + 1 ] in 'Augmented Binary
Format'		confn722.doc3
5-06'Fermat Numbers' - Setting up the 'BOX'confnu44.doc10
5-07'Ricochet-In-The-Box' - [ 'Lower-left-hand'
Corner ]		confnu45.doc6
5-08'Ricochet-In-The-Box' - [ 'Upper-right-hand'
Corner ]		confnu46.doc6
5-09'Meshing' in the Middle and Anywhere Elseconmes22.doc10
5-10The Art of 'Ricocheting' - 'Strictly Horizontal'
vs. 'Strictly Vertical'		conric22.doc6
5-11The Art of 'Ricocheting' - Switching Once Onlyconric33.doc26
5-12The Art of 'Ricocheting' - Switching a Second Time
on a 'Completed Pair'		conric34.doc13
5-13The Art of 'Ricocheting' - Switching a Second Time
on a 'Broken-Pair'		conric35.doc3
5-14The Art of 'Ricocheting' - Special Case for the
[ 'Upper-right-hand' Corner ]		conric55.doc8
5-15The shape of the 'BOX'consob22.doc15
5-16STOP! - But 'Fermat Numbers' are not about
'Fermat Numbers'		consto22.doc2

SECTION 6 - The Fermat Little Theorem ( 86 pages )
-------------------- Title of the Chapter --------------------filename :pages
6-01The 'Fermat Little Theorem' - 'FIRON' : the 1st
Instance of 'One' for [ a^X modulo P ]		confli22.doc25
6-02Multiplication Table for { modulo 13 } &
for { modulo 11 }		conmtm22.doc4
6-03'Swing from the Diagonal' - explained via the
'Multiplication Table' for [ modulo 13 ]		conmtm33.doc15
6-04'Swing from the Diagonal' for the
'Multiplication Table' for [ modulo 11 ]		conmtm55.doc7
6-05Constructing the 'Multiplication Table' { modulo P }
- with { P = 2*Lambda + 1 }		conmtm77.doc35

SECTION 7 - Squares of Squares ( 116 pages )
-------------------- Title of the Chapter --------------------filename :pages
7-01'Squares-Of-Squares' and the
[ 'Square-Of-Square' Exponents ] (SOSE-i)		consos22.doc3
7-02[ 2^29 - 1 ] = 233 * 1103 * 2089consos33.doc2
7-03'Squares-Of-Squares' Analysis for
{ modulo 233 }		consos44.doc21
7-04The 'Disk-of-Power-Indicators' { modulo 233 }consos45.doc7
7-05'Squares-Of-Squares' Analysis for
{ modulo 1103 }		consos46.doc6
7-06'Squares-Of-Squares' Analysis for
{ modulo 2089 }		consos47.doc8
7-07{ [ 2^(SOSE-28) ] = 2 modulo 233 / 1103/ 2089 }consos55.doc4
7-08Why '28' - because { FIRON-2/29 } = 28consos56.doc4
7-09'FIRON' Groups for { modulo 31 / 61 / 163 }consos57.doc4
7-10The 'Chinese Remainder Theorem'concrt22.doc7
7-11{ 'Square-of-Square' Diagrams } for { modulo 105 }consos66.doc8
7-12{ 'Square-Of-Square' Diagrams } for
{ modulo 3 / 5 / 7 } & { modulo 15 / 21 / 35 }		consos67.doc5
7-13Revisiting { modulo 105 }consos68.doc10
7-14Other than [ 2^29 - 1 ]conmer22.doc7
7-15Mersenne & Non-Mersenne Numbersconmnm22.doc20

SECTION 8 - Logarithm of '3' [ base '2' ] ( 10 pages )
-------------------- Title of the Chapter --------------------filename :pages
8-01Calculating Logarithms - An Elementary Viewconlog22.doc2
8-02The { Logarithm of '3' [ Base '2' ] }conlog33.doc3
8-03Logarithms using 'Square-Of-Square'
& the 'I-CHING'		conlog44.doc5

SECTION 9 - Global Positioning System ( 1 page )
-------------------- Title of the Chapter --------------------filename :pages
9-01The Global Positioning System ( GPS) in numberscongps22.doc1

SECTION 10 - The Logarithmic Process ( 43 pages )
-------------------- Title of the Chapter --------------------filename :pages
10-01The Search for Logarithm and the Logarithmic
process		conlfn22.doc7
10-02The Logarithmic Curve and the numner 'e'
- (2,718281..........)		conlfn33.doc6
10-03[ 2^4 = 4^2 ] and its implication on
{ [ Log X ] / X }		conlfn44.doc14
10-04{ [ Log X ] / X } for { 1 < X < 2 }conlfn55.doc4
10-05In Search of the 'missing' [ 3 ] from [ 0 ] ,
[ 1 ] , [ 2 ] & [ 4 ]		conlfn66.doc5
10-06Approximation Procedure for { Logarithms [ base 'e' ] }conlfn88.doc7

Section 11 - Findings & Concluding Remarks ( 2 pages )
-------------------- Title of the Chapter --------------------filename :pages
11-01Summary Of Findingsconsof22.doc1
11-02Concluding Remarksconcon22.doc1

Appendices ( 18 pages )
-------------- Content of the Appendix --------------filename :pages
Appendix AValues of [ Powers-Of-Two ] up to [ 2^128 ]conaaaaa.doc7
Appendix B'Sum-of-Two-Squares' for { positive 'odd'
integers } of the format '4*K + 1' up to '257'		conbbbbb.doc4
Appendix CThe Cosine function as defined by
the Triangle		conccccc.doc7

View the EPILOG to the paper dated mid-November, 2003.

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