On Prime Number Systems and Core Structural Pattern Propagations

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

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Section VI - Selecting a Core Structural Pattern for our Demonstrations

Summary for the Section :

• In this Section , we shall select the { Logarithmic Distribution Pattern } for the prime-numbers [ 7 / 17 / 37 / 47 ]
  • as the { Core Structural Pattern } for our Demonstrations on Structural Pattern Propagations ,
    • to follow in the next 5 Sections .

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Reasons for selecting [ 7 / 17 / 37 / 47 ] :

• Let us quickly state the reasons for selecting the 4 prime-numbers [ 7 / 17 / 37 / 47 ] here :

  Reason One :

  • For our demonstrations to follow , we shall be looking at :
    • { Category II Composite Numbers } that are the products of 2 prime-numbers ,

    • { Category III Composite Numbers } that are the products of 3 prime-numbers ,

    • { Category IV Composite Numbers } that are the products of 4 prime-numbers ,

      and so-on-and-so-forth .

    Thus ,

    • if we were to select a high-number of prime-numbers for our { Core Structural Pattern } ,
      • say 10 prime-numbers ,

      then , when we present our Demonstration Diagrams for the { Category II / III / IV / V Composite Numbers } ,

      • the Demonstration Diagrams will be quite congested and hard-to-read ,

        thereby defeating our original purpose of showing how [ Core Structural Patterns ] may propagate .

    We shall therefore limit our selection to 4 prime-numbers or less initially .

  Reason Two :

  • We do not wish to place any undue restrictions on our demonstration of the { propagation mechanism } .
    • If we were to use the 4 prime-numbers [ 2 / 3 / 5 / 7 ] ,
      • this might give a wrong impression that the { propagation mechanism } will only work for the [ first N prime-numbers ] .

      If we were to use the 4 prime-numbers [ 7 / 11 / 13 / 17 ] ,

      • this might give a wrong impression that the { propagation mechanism } will only work for [ consecutive prime-numbers ] .

    Thus , in order to avoid both , we have selected 4 prime-numbers that are sufficiently separate from one-another ,

    • so as to give the { propagation mechanism } a more proper general feel .

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Setting up the X-Axis for [ X = LOG(N) ] :

• Let us now set up the [ X-Axis ] as shown in the diagram below :

  

  • for the purpose of recording [ X = LOG(N) ] ,
    • where [ N ] is the prime-number or the composite-number under investigation .

• Let us now quickly bring-in the Natural Logarithms for the prime-numbers [ 7 / 17 / 37 / 47 ] :
  • as per this table below :

    Value of [ N ]	7	17	37	47
    LOG ( N )	1.9459	2.8332	3.6109	3.8501
    ************	************	************	************	************

  We can then mark-off the values of [ X = LOG(N) ] for the prime-numbers [ 7 / 17 / 37 / 47 ] :

  • as per this diagram below .

  

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Setting up the { Core Structural Pattern } :

• We then set up accordingly the { Core Structural Pattern } for the prime-numbers [ 7 / 17 / 37 / 47 ]
  • as per this diagram below .

  

  And we take special note here that :

  • the Geometric Mean of [ 7 / 17 / 37 / 47 ] is [ 21.3286 ] ,

    and

  • the Natural Logarithm of [ 21.3286 ] is roughly [ 3.0600 ] ,
    • as marked-off in the diagram above .

• Let us now also set up , at this point , the corresponding Structural Frame [ FRAME 1-A ] ,
  • for our comparison purposes later in Section XI of this paper .

  

• We are now ready to take a closer look at the { Category II / III / IV / V Composite Numbers }
  • arising from the prime-numbers [ 7 / 17 / 37 / 47 ] .

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go to the next section : Section VII - Derived Structural Patterns for [ Category II Composite Numbers ]

go to the last section : Section V - The Reasons for the different modulo's the [ Prime Numbers Systems ]

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08