On the Unique-Factorization Equation within Prime Number Systems

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

Top Of Page

Section XI - Short-Cut #2 - [ 1543 over 7 ]

Summary for the Section :

• In this Section , we shall take a look at the [ 2nd ] of 3 short-cuts to arrive at a solution to the equation .

go to Top Of Page

The basis for the Short-Cut :

• Let us recall again that :
  • [ 7 ] is the smallest prime-number under our consideration in the [ Modulo 30 Prime Number System ] .

  And since our { Output String of 412-elements } has a cut-off Upper-Limit at [ N = 1543 ] ,

  • it is no long absolutely necessary for us to determine :
    • which of the { candidates-for-primes above [ N = 1543 ] }

      gets knocked-off as [ non-prime numbers ] .

• Let us now quickly write-out this arithmetic relation involving [ 7 ] and [ N = 1543 ] :
  • [ 1543 ] divided by [ 7 ] = [ 220.43 ]

    yielding :

  • [ 1543 ] = [ 7 ] x [ 220.43 ]

  And what this equation immediately above does tell us here is that :

  • { Assumed Prime-Numbers } greater than [ 220.43 ] cannot be involved :

    in the knocking-off of { candidates-for-primes below [ N = 1543 ] } .

  And that-is-to-say :

  • For the next { candidate-for-prime } greater than [ 220.43 ] at [ 221 ] ,
    • poised as always to be a possible { Assumed Prime-Number } ;

    we have :

    • [ 1547 ] = [ 7 ] x [ 221 ]

      and this [ 1547 ] here is already greater than the cut-off Upper-Limit of [ N = 1543 ] .

• As such , we can now temporarily suspend the { candidates-for-primes above [ 220.43 ] }
  • from being included into our initial list of { Assumed Prime-Numbers } .

go to Top Of Page

The new Procedure for arriving at { Output String of 412-elements } :

• Based on the above , our new proposal for the Procedure to arrive at the { Output String of 412-elements } is then as follows :
  • Step One :
    • We suspend all { candidates-for-primes above [ 220.43 ] } from becoming { Assumed Prime-Numbers } .

  • Step Two :
    • We proceed to asssume a set of { Assumed Prime-Numbers } for the { candidates-for-primes below [ 220.43 ] } .

  • Step Three :
    • We proceed to knock-off { candidates-for-primes } in the usual maner ,
      • based now on the reduced set of { Assumed Prime-Numbers } in [ Step Two ] above .

  • Step Four :
    • Once we are done with the Knocking-Off process in [ Step Three ] above ,
      • we can now mark-off :
        • the { candidates-for-primes in-between [ 220.43 ] and [ 1543 ] }

          that didn't get knock-off in [ Step Three ] above

          • as the additional { Assumed Prime-Numbers } .

      And we can now place a [ red-color ball ] into the each of the bins associated with these additional { Assumed Prime-Numbers }

      • to include these new additions into the full-list of { Assumed Prime-Numbers } .

    This then completes the formulation process for the { Output String of 412-elements } .

go to Top Of Page

A Special Note here on the Correct Solution :

• We simply note here that :
  • IF :
    • we did correctly identified all the { [ prime-numbers ] up to [ 220.43 ] } ,

      and

      put these into the initial { List of [ Assumed Prime-Numbers ] } but added nothing else ,

    THEN :

    • after we are done with { Step Three } and { Step Four } in the Procedure above ,
      • we would end up with an { Output String of 412-elements }

        with each of the [ 412 bins ] spotting a single [ color-ball ] .

  As such , our proposed Procedure above is indeed a valid procedure

  • for closing-in on the solution .

go to Top Of Page

The advantages of the Short-Cuts here :

• Let us now quickly look the advantages offered here :

  

  And we can see from this diagram here that :

  • [ 7 / 11 / 13 / 17 / 19 / 23 / 29 / 31 / 37 / 41 / 43 / 47 ] are now marked-off as [ prime numbers ] ,
    • based on [ Short-Cut #1 ] ;

  • [ candidates-for-primes ] above [ 220.43 ] have been suspended ,
    • based on our [ Short-Cut #2 ] here .

  Since we have established that [ 49 = 7 x 7 ] is not a prime-number ,

  • this then leaves us only 44 { candidates-for-primes } , from [ 53 ] thru to [ 217 ] ,
    • for our consideration to mark-off as an [ Assumed Prime-Number ] .

• Rather neat , I would say .

go to Top Of Page

go to the next section : Section XII - Short-Cut #3 - Using [ Modulo 210 ] in lieu

go to the last section : Section X - Short-Cut #1 - [ 49 = 7 x 7 ]

return to the Unique-Factorization Equation Homepage

Original dated 2012-10-08