An Approach to Prime Numbers , the L.C.M. , and the Zeta Functions

by Frank C. Fung - 1st published in May, 2009.

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Section VIII - Area Under the Hyperbola

Summary for the Section :

• In this Section , we shall look at the possibility of using the { area under the hyperbola } ,
  • as a first estimate on how many numbers are in the knock-off area in the { 2-D prime multiplication table } .

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Putting the [ 42 x 42 Multiplication Table ] on a 2-Dimensional plane :

• Let us bring-back , at this point , the [ 42 x 42 Multiplication Table ] from Appendix B , first presented in Section VII .

  We shall now transform the said-[ multiplication table ] :

  • into a corresponding 2-dimensional grid of [ 42 vertical-lines and 42 horizontal-lines ] ,

    so that :

  • each [ cell ] in the original [ multiplication table ] is now represented by a corresponding [ nodal-point ] on the new grid in the 2-Dimensional { X-Y plane } .

  The new grid has therefore [ 42 x 42 ] , or 1,764 { nodal points } .

• We then have these two (2) diagrams below showing the new [ 42 x 42 grid ] ,
  • and we have marked-off , on both the { X-Axis } and the { Y-Axis } , the original [ 42 prime numbers ] in the [ multiplication table ] ,
    • via [ short-black-color lines ] .

• And the { nodal-points } responsible for knocking-off { non-primes } in-between [ 210 ] and [ 2310 ] are then as shown in { purple-color } ,
  • in both diagrams above .

  The 3 [ hyperbola's ] serving as boudary-lines dividing the regions , are then as shown in the diagram on-the-right via the { red-color curves } ;

  • and these 3 { hyperbola's } are :
    • the hyperbola : { X*Y = 210 } ,
    • the hyperbola : { X*Y = 2310 } , and
    • the hyperbola : { X*Y = 30030 } ;

      respectively .

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The Irregular Distribution of { Prime Numbers } :

• We take note , at this point , that the { prime numbers } in the range from [ 11 ] to [ 210 ] do not seem to exhibit a regular pattern :
  • there are quite-a-few-number of { 2-digit gaps } for the { twin primes } :
    • [ 11 / 13 ] , [ 17 / 19 ] , [ 29 / 31 ] , [ 41 / 43 ] , [ 59 / 61 ] , [ 71 / 73 ] , [ 101 / 103 ] , [ 107 / 109 ] ,

      [ 137 / 139 ] , [ 149 / 151 ] , [ 179 / 181 ] , [ 191 / 193 ] and [ 197 / 199 ] ;

  • there is a [ 14-digit gap } in-between the prime numbers [ 113 ] and [ 127 ] ;
  • there is a [ 10-digit gap } in-between the prime numbers [ 139 ] and [ 149 ] ;
  • there is a [ 10-digit gap } in-between the prime numbers [ 181 ] and [ 191 ] ;
  • there is a [ 12-digit gap } in-between the prime numbers [ 199 ] and [ 211 ] .

  The question then arises :

  • Can we still use a comparison of the { areas in-bewteen the hyperbola's } :
    • as a first estimate on how many { nodal-points } there are in the { knock-off region } , marked-off in { purple-color } .

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The Area Under the Curve :

• Let us now take a quick look at the { area under the curve } for the hyperbola's .

• A comparison here then reveals that the ratio of :
  • the { purple-color area } on-the-left : the { yellow-color area } on-the-right ,
    • is roughly [ 0.120 : 1.000 ] .

• Let us now give a count on the number of { nodal points } in the { purple-color area } ,
  • and the full count here is [ 349 ] , made-up of :
    • 7 { nodal points } for the 7 squares : 289 / 361 / 529 /841 / 961 / 1369 / 1849 / 2209 ;

    • 342 { nodal points } for the 171 other knock-off numbers in-bewteen [ 211 ] and [ 2310 ] .

  Since we have [ 42 x 42 ] or 1,764 { nodal points } in the { yellow-color area } ,

  • the ration here on the { nodal points } for [ purple-color : yellow-color ] ,
    • is then [ 349 : 1,764 ] , or [ 0.197 : 1.000 ] .

• Quite clearly , the estimation procedure here needs further refinements ;
  • the prime numbers are , at this instance , more concentrated towards the lower-end of the spectrum .

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• We shall look more closely at the hyperbola's , next .

go to the next section : Section IX - Singularities and Symmetries in 2-D / 3-D / 4-D Hyperbola's

go to the last section : Section VII - A Fast-Track Approach to identify { prime numbers }

return to the Prime Numbers / L.C.M. / Zeta Functions HomePage

original dated 2009-5-08