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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Epilog III - The gap in-between { consecutive prime numbers } to reach Infinity

Summary for this Epilog :

• This epilog is simply added as a safety measure to ensure that this point below hasn't been missed-out , namely that :
  • The gap in-between any pair of { consecutive prime numbers } may reach [ infinity ] .

  And this is based on our Fast-Track Procedure to identify { prime numbers } in Section VII .

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A First Explanation :

• Let us first cite a numerical example here using the Fast-Track Procedure as outlined in Section VII .

  And we shall use [ 31 ] as the { pivotal prime number } here ; so that :

  • the next lower-ordered { prime number } is then [ 29 ] .

  And the { Product-of-Primes } we shall use at this instance , denoted by [ N ] here , is then :

  • [ N ] = 2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29

• We then have this table below for the { Candidates for Prime Numbers } :

  Range	Candidates for Prime Numbers							Comments
  ******************	**********	**********	**********	**********	**********	**********	******************	**************
  1 thru [ N ]	1	31	37	41	43	47	{ Set C }	base row
  [ N + 1 ] thru [ 2*N ]	N + 1	N + 31	N + 37	N + 41	N + 43	N + 47	N + { Set C }	candidate row
  [ 2*N + 1 ] thru [ 3*N ]	2*N + 1	2*N + 31	2*N + 37	2*N + 41	2*N + 43	2*N + 47	2*N + { Set C }	candidate row
  [ 3*N + 1 ] thru [ 4*N ]	3*N + 1	3*N + 31	3*N + 37	3*N + 41	3*N + 43	3*N + 47	3*N + { Set C }	candidate row
  ..........	And so-on-and-so-forth							candidate rows
  [ 29*N + 1 ] thru [ 30*N ]	29*N + 1	29*N + 31	29*N + 37	29*N + 41	29*N + 43	29*N + 47	29*N + { Set C }	candidate row
  [ 30*N + 1 ] thru [ 31*N ]	30*N + 1	30*N + 31	30*N + 37	30*N + 41	30*N + 43	30*N + 47	30*N + { Set C }	candidate row

  • where the set { Set C } here is comprised of :
    • { prime numbers from [ 53 ] up to [ N ] } , and

    • { natural numbers less than [ N ] that are not divisible by [ 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , or 29 ] } .

• Let us now concentrate on the { candidates in the first 2 columns } , and :
  • each pair of candidates in-the-same-row are of the formats :
    • { x*N + 1 } and { x*N + 31 } respectively ,
      • with the value [ x ] running from { [ 1 ] thru [ 30 ] } .

  There are then three ( 3 ) possible scenario's here :

  • Scenario I :
    • Both candidates are { prime numbers } ,
      • in which case the [ gap ] is simply [ 30 ] .

  • Scenario II :
    • One candidate is { prime } and the other is not ,
      • in which case the [ gap ] is greater than [ 30 ] .

  • Scenario III :
    • Both candidates are not { primes } ,
      • in which case the [ gap ] is also greater than [ 30 ] ,

        ( in-between the { next prime number lower than [ x*N + 1 ] } and the { next prime number higher than [ x*N + 31 ] } ) .

• Thus , regardless of which of the 3 scenario's is the valid one :
  • the gap in-between 2 consecutive { prime numbers } here is always greater than or equal to [ 30 ] .

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The General Statement :

• Suppose we were able to identify a very very large { prime number } [ P ] ,
  • we can then set up the value [ Q ] being given by :
    • [ Q ] = [ P - 1 ] .

  And once we have identified the value of [ P ] , we immediately know for sure that :

  • somewehere out-there-in-deep-outer-space ,

    there exists a gap in-between 2 consecutive { prime numbers } that is greater than or equal to [ Q ] .

• And this arises directly from the Fast-Track Procedure demonstration we have immediately above .

  ( replace [ 31 ] with any { prime number greater than [ 5 ] } and follow the exact-same procedure ) .

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There are infinitely many { prime numbers } :

• Since it has been established , since Euler's time , that there are infinitely many { prime numbers } ,
  • it then follows that the gap in-between prime numbers may also reach [ infinity ] ,
    • and will reach [ infinity ] , eventually .

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An issue for further considerations :

• An Issue - on empirical proofs :
  • Suppose we were able to build a very powerful computer and identified a { prime number } in the { 10-billion-billion-billion region } ,
    • immediately we know that , somewehere out-there-in-deep-outer-space , there is a [ prime gap ] of { 10-billion-billion-billion or thereabouts } .

    And that is to say :

    • once we hit this region , wherever it may be , we will go thru { 10-billion-billion-billion natural numbers } without finding a { prime number } .

  • The interesting question is :
    • Will the existence of a very very large [ prime gap ] be an obstacle to any empirical proof of theories on { prime numbers } ?

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go to the last epilog : Epilog II - The Separability of Riemann's f(x) function into its component parts

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

original dated 2009-6-02