On the Unique-Factorization Equation within Prime Number Systems

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

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Section II - A Statement of Objectives

Summary for the Section :

• In this Section , we shall set out the directions for the rest of the paper .

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A Statement of Objectives :

• For any [ Modulo M Prime Number System ] up to given Upper Limit ,
  • we can always :
    • assume a particular set of { candidates-for-primes } to be the prime-numbers in question ,

      and

    • proceed to knock-off [ composite numbers ] based on that particular set of [ assumed prime-numbers ] .

  And if we did pick prime-numbers correctly , then :

  • everything would fall-into-place , so-to-speak , i.e. :
    • all prime-numbers will be marked-off and none missed-out , and

    • all composite-numbers will have a unqiue set of [ prime factors ] ,
      • dictated by unique-factorization .

• But :
  • if we did miss-out on prime-numbers and/or wrongly identified a composite-number as an [ assumed prime-number ] ,

    then :

    • several things can happen :
      • one or more { candidates-for-primes } is not marked-off :
        • either as a prime-number and/or composite-number/s ;

      • doubling-up or worse on a { composite-number } under certain circumstances and :
        • the { composite-number } will have more than one set of { prime-factors of assumed-prime-numbers }

          and

          Unique-Factorization is abandoned .

• Our objective here is then :
  • to go thru the process of zeroing-in on the solution set ,

    where :

    • all of the genuine [ prime numbers ] and nothing-else

      are marked-off as members of the set of [ Assumed Prime Numbers ] .

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Our Approach here :

• In the remainder of this paper , we shall use :
  • the [ Modulo 30 Prime Number System ] up to [ N = 1543 ] ,

    for our demonstration of the mechanics .

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go to the next section : Section III - Setting up the [ 412 Empty Bins ]

go to the last section : Section I - Introduction

return to the Unique-Factorization Equation HomePage

Original dated 2012-10-08