On Prime Number Systems and Core Structural Pattern Propagations

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

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Section XII - The Range for each Category of Composite Numbers

Summary for the Section :

• In this Section , we shall look at the range
  • for each Category of Composite Numbers derived from the 4 prime-numbers [ 7 / 17 / 37 / 47 ] .

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The Range for each Category of Composite Numbers :

• We note here that :
  • for the the 4 prime-numbers [ 7 / 17 / 37 / 47 ] :
    • [ 7 ] is the smallest prime-number

      and

    • [ 47 ] is the largest prime-number .

  It then follows that :

  • For the derived [ Category II Composite Numbers ] ,
    • the range would run from [ 49 = 7 x 7 ] to [ 2209 = 47 x 47 ] .

  • For the derived [ Category III Composite Numbers ] ,
    • the range would run from [ 343 = 7 x 7 x 7 ] to [ 10 3823 = 47 x 47 x 47 ] .

  • For the derived [ Category IV Composite Numbers ] ,
    • the range would run from [ 2401 = 7 x 7 x 7 x 7 ] to [ 487 9681 = 47 x 47 x 47 x 47 ] .

  • and so-on-and-so-forth .

  We then have this schematic presentation of the above ranges , on a [ logarithmic scale basis ] , as per this diagram below .

  

• And we see here that :
  • The range for each Category is then :
    • dependent on a specific { power of the [ smallest prime-number ] } on the low-side ,

      and

    • dependent on the same specific { power of the [ largest prime-number ] } on the high-side .

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Comparing the Geometric Mean for each Category of Composite Numbers :

• We note here , first-of-all , that :
  • the Geometric Mean of [ 7 / 17 / 37 / 47 ] is roughly [ 21.3286 ] .

  Let us now denote this partciular Geometric Mean as the value [ G ] , i.e. :

  • 

• Let us now bring-in this table below :
  • showing the Geometric Mean for each successive Category of Composite Numbers .

    Category	Geometric Mean	Comment
    **************	**************	************************
    Category II	454.91	the square of [ 21.3286 ]
    Category III	9702.51	the cube of [ 21.3286 ]
    Category IV	20 6941.00	the 4th-power of [ 21.3286 ]
    Category V	441 3761.81	the 5th-power of [ 21.3286 ]

  We then have this schematic presentation of the above Geometric Means

  • as per this diagram below .

  

• A quick explanation of the phenomenon here :
  • For the 10 derived [ Category II Composite Numbers ] :
    • each of 4 prime-numbers are necessarily ranked on-par with one-another in the formation of these { Category II Composite Numbers } .

    And that-is-to-say :

    • each of the 4 prime-numbers are always of-equal-strengths in the propagation process

      and

      any one prime-number cannot be favored over another here when the [ derived category ] is viewed on an overall basis .

    Thus , for the 10 derived [ Category II Compositive Numbers ] here involving exactly 20 [ prime factors ] ,

    • the split for the 20 prime-factors here must necessarily be [ 5 : 5 : 5 : 5 ] for the 4 prime-numbers [ 7 / 17 / 37 / 47 ] .

    Consequently , the Geometric Mean of all derived { Category II Composite Numbers } must necessarily be :

    • 

      yielding :

    • 

    Therefore :

    • 

  • For the 20 derived [ Category III Composite Numbers ] :
    • again each of 4 prime-numbers are necessarily ranked on-par with one-another in the formation of these { Category III Composite Numbers } .

    Thus , for the 20 derived [ Category III Compositive Numbers ] here involving exactly 60 [ prime factors ] ,

    • the split of the 60 prime-factors here must necessarily be [ 15 : 15 : 15 : 15 ] for the 4 prime-numbers [ 7 / 17 / 37 / 47 ] .

    Consequently , the Geometric Mean of all derived { Category III Composite Numbers } must necessarily be :

    • 

      yielding :

    • 

    Therefore :

    • 

  • For the 35 derived [ Category IV Composite Numbers ] :
    • each of 4 prime-numbers are necessarily ranked on-par with one-another in the formation of these { Category IV Composite Numbers } .

    Thus , for the 35 derived [ Category IV Compositive Numbers ] here involving exactly 140 [ prime factors ] ,

    • the split of the 140 prime-factors here must necessarily be [ 35 : 35 : 35 : 35 ] for the 4 prime-numbers [ 7 / 17 / 37 / 47 ] .

    Consequently , the Geometric Mean of all derived { Category IV Composite Numbers } must necessarily be :

    • 

      yielding :

    • 

    Therefore :

    • 

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Re-looking at the Powers-of-Primes :

• Let us take a quick look at the powers of [ 17 ] and [ 37 ] ,
  • with [ 17 ] and [ 37 ] here being the intermediate prime-numbers in-between :
    • [ 7 ] on the low-side and [ 47 ] on the high-side .

  

• And we simply take note here that :
  • the powers of the [ intermediate prime-numbers ] may very well serve as [ dividers / segregators ]
    • within each [ category of compsite numbers ] .

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Where to go from here :

• Admitted , the propagation of Structural Patterns into the different [ categories of Composite Numbers ]
  • is not an easy subject that can be readily understood at the first glance .

  We shall therefore do another round of [ Core Structural Pattern ] propagations using the six (6) prime numbers ,

  • [ 7 / 17 / 37 / 47 / 67 / 97 ] ;

    in an attempt to bring out some of the { Symmetry and Counter-Flow features }

    • inherent therein within the propagation process itself .

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go to the next section : Section XIII - Identifying the Core Structural Pattern for [ 7 / 17 / 37 / 47 / 67 / 97 ]

go to the last section : Section XI - A Quick Comparison of the Structural Frames

return to the Prime Number System and Core Pattern Propagation Homepage

Original dated 2012-10-08