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On Unique-Factorization Combinatorics via the Pascal Triangle

by Frank C. Fung ( 1st published in October, 2012. )

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Summary for the paper :

In this paper , we shall attempt to answer the question :
- What is the Full Count of { Category [ T ] Composite Numbers } that can be formulated
  - with an initial set of [ H ] distinct-and-different prime-numbers ?
This is an important question because :
- we should first find out how many composite-numbers there are ,
  
  before we can start the [ cut-off procedure ] :
  - to throw-out the unwanted composite-numbers above the Upper-Limit of our investigation.

Table of Content

Table of Content :

Table Of Content
	Part I --- Introduction and Preliminaries
Section I	Introduction
	Part II --- Re-visiting the Pascal Triangle
Section II	The Pascal Triangle via Binomial Expansions
Section III	A Special Property of the Pascal Triangle
	Part III --- Calculations for { Category X Composite Numbers }
Section IV	Setting up for the Calculations in the next 10 Sections
Section V	Formulating { Category X Numbers } using 1 prime-number
Section VI	Formulating { Category X Numbers } using 2 prime-numbers
Section VII	Formulating { Category X Numbers } using 3 prime-numbers
Section VIII	Formulating { Category X Numbers } using 4 prime-numbers
Section IX	Formulating { Category X Numbers } using 5 prime-numbers
Section X	Formulating { Category X Numbers } using 6 prime-numbers
Section XI	Formulating { Category X Numbers } using 7 prime-numbers
Section XII	Formulating { Category X Numbers } using 8 prime-numbers
Section XIII	Formulating { Category X Numbers } using 9 prime-numbers
Section XIV	Formulating { Category X Numbers } using 10 prime-numbers
Section XV	Summarizing on the Formulating Process
	Part IV --- Formalizing the Calculation Formula
Section XVI	The formal Combinatorics Equation for { Category [ T ] }
	Part V --- The Combinatorics Function via Matrices
Section XVII	Creating the 11 x 11 [ PASCAL Matrix ]
Section XVIII	Creating the 10 x 10 [ ALPHA Matrix ]
Section XIX	Creating the 10 x 10 [ ETA Matrix ]
Section XX	Creating the 10 x 10 [ Z Matrix ]
Section XXI	Symmetry Considerations for the [ Z Matrix ]
	Part VI --- Characteristics of the Function { z = f( H, T ) }
Section XXII	Expressing [ z ] as a function of [ H] and [ T ]
	Part VII --- Concluding Remarks
Section XXIII	Concluding Remarks

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return to Title Paper : An Approach to the Structural Integrity of Prime Number Systems

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Original dated 2012-10-08