Structure of the Multiplication Table for [ modulo P ]

In our construction scheme for the { Multiplication Table for [ modulo P ] } , there are always two (2) classes of { core elements } and they are :

• the { Characteristic Pyramid } , &
• the { Square-Of-Square RING's } .

And these are explained in details in this Section .

For the 'odd' { prime number } , [ P ] , greater than [ 3 ] , we again set-up the value [ Q ] such that :

• [ Q ] = [ P - 1 ]

The { Characteristic Pyramid } is then a diagram based on the { 'even' prime factors } of [ Q ] , detailing a particular set of { 'squarimg relations' [ modulo P ] } .

But let us first look at a few examples , before a more formal explanation below :

• EXAMPLE #1 --- the { Characteristic Pyramids } for the { prime numbers } [ 47 ] & [ 83 ] , respectively , i.e. :
  • 47 = 2 * 23 + 1
  • 83 = 2 * 41 + 1

• We note here that the { 2 numbers } on the bottom row , i.e. { Row 1 } , are the solutions to the equation :
  • 

  And :

  • squaring a [ number in { Row 1 } ] will always result in a [ number in { Row 0 } ] , as indicated by the links .
• EXAMPLE #2 --- the { Characteristic Pyramids } for the { prime numbers } [ 53 ] & [ 149 ] , respectively , i.e. :
  • 53 = 4 * 13 + 1
  • 149 = 4 * 37 + 1

• We note here that the { 4 numbers } on the bottom row , i.e. { Row 2 } , are the solutions to the equation :
  • 

  And :

  • squaring a [ number in { Row 2 } ] will always result in a [ number in { Row 1 } ] ,
  • squaring a [ number in { Row 1 } ] will always result in a [ number in { Row 0 } ] ,

    as indicated by the links .

• EXAMPLE # 3 --- the { Characteristic Pyramids } for the { prime numbers } [ 137 ] & [ 233 ] , respectively , i.e. :
  • 137 = 8 * 17 + 1
  • 233 = 8 * 29 + 1

• We note here that the { 8 numbers } on the bottom row , i.e. { Row 3 } , are the solutions to the equation :
  • 

  And :

  • squaring a [ number in { Row 3 } ] will always result in a [ number in { Row 2 } ] ,
  • squaring a [ number in { Row 2 } ] will always result in a [ number in { Row 1 } ] ,
  • squaring a [ number in { Row 1 } ] will always result in a [ number in { Row 0 } ] ,

    as indicated by the links .

We can then make these statements for the { Characteristic Pyramid } for [ modulo P ] , in general , as follows :

• the bottom row is always the { solution set } for the equation :
  • , if [ Q ] is divisible by [ 2 ] but not by [ 4 ] , or

  • , if [ Q ] is divisible by [ 4 ] but not by [ 8 ] , or

  • , if [ Q ] is divisible by [ 8 ] but not by [ 16 ] ;

  • and so-on-and-so-forth .

• squaring a [ number in { any row } ] will always result in a [ number in { the row immediately above } ] ,

  as indicated by the links .

• { Row 0 } always contains a single number , the number [ 1 ] .

The { Square-Of-Square RING } is then a diagram based on the { 'odd' prime factors } of [ Q ] , detailing a particular set of { 'squarimg relations' [ modulo P ] } .

Let us first look also at an example , for the { prime number } [ 233 ] , before going into more details below , i.e. :

• 233 = 8 x 29 + 1

We then have the { RING OF 28 } for [ modulo 233 ] , as marked by the { ring } of 28 { red-color balls } in the diagram below :

• And these { 28 numbers } are then the solutions to the equation :
  • 

    other than the number [ 1 ] , which is always the universal solution to all equations of the format :

    • , where 'N' is any { positive integer } .

  And these { 28 numbers } are arranged in a { circular ring } fashion , with each link indicating one of the 28 'squaring relations' as shown to-the-right of diagram , above .

We can then make these statements for the { Square-Of-Square RING's } , in general , as follows :

• members on the { Square-Of-Square RING } are always solution to the equation :
  • , where [ F ] is an { 'odd' prime factor } of [ Q ] .

• members on the { RING } are always arranged in { circular ring } fashion ,
  • and whenever possible , each link would indicate a 'squaring relation' [ modulo P ] ,

    but we shall see , later , why modifications to this statement are necessary , when we investigate :

    • the { prime number } [ 3,361 ] & the Mersenne Prime [ 2,147,483,647 ] , i.e. [ 2^31 - 1 ] .

      in Section V and Section VI , respectively .

Let us now take a look at 'outer-portion' of the { RING OF 28 } for [ modulo 233 ] , and we bring back this diagram from above :

Let us now focus on this set of 3 consecutive 'squaring relations' , as follows :

• ,
• ,
• .

And we bring-in again the { Characteristic Pyramid } for [ modulo 233 ] , as per the diagram on-the-left , below , and do the following :

• multiply { Row 3 } by [ 2 mod 233 ] ,
• multiply { Row 2 } by [ 4 mod 233 ] ,
• multiply { Row 1 } by [ 16 mod 233 ] , &
• multiply { Row 0 } by [ 23 mod 233 ] ;

• to arrive at the { derived pyramid } , as per the diagram on-the-right , above .

And we note here that all the 'squaring relations' for all the links here remain valid for this { derived pyramid } .

We can then attached { right-hand-side } of this { derived pyramid } , as marked-off 'to-the-right' of the { white-color dotted-line } , to the { RING OF 28 } :

• matching the { red-color ball } labelled [ 23 ] , { red-color ball } onto { red-color ball } .

And we also do the same for the other { 27 red-color balls } on the { RING OF 28 } ,

• using a corresponding / correct set of 3 consecutive 'squaring relations' each time ,

  to complete the process for the { outer-portion } of the { RING OF 28 } .

And we shall mention here , in passing , why the { Characteristic Pyramid } is named as-such ,

• because all the other { pyramids } are derived from it .

Let us now look again at the entire range from [ 1 ] to [ 232 ] for [ modulo 233 ] ,

• excluding , for the time being , [ 0 mod 233 ] :
  • which is usually omitted from a Multiplication Table , because { [ 0 ] x [ anything ] = [ 0 ] } .

The positions of these { 232 members } are then identified via :

• the { Characteristic Pyramid } accounting for { 8 members } ,

• 28 { derived pyramids } with { 8 members } each , each attached onto a unique position on the { RING OF 28 } ,
  • to account for [ 8 x 28 ] or { 224 members } .

  And all the 'squaring links' have remained valid throughout .

Thus , the 'squaring relations' among the full set of { 232 members } have now been recognized .

While some readers may venture , at this point , to comment on the positions of :

• each pair of { Additive Conjugates } [ modulo 233 ] , &

• each pair of { Multiplicative Conjugates } [ modulo 233 ] ;

we shall reserve our comments until later , after we have looked into the { prime numbers } :

• [ 1,103 ] = 2 x 19 x 29 + 1 ,
  • involving 2 { 'odd' prime numbers } for the { prime factors } of [ Q ] .

• [ 3,361 ] = 32 x 3 x 5 x 7 + 1 ,
  • involving 3 { 'odd' prime numbers } for the { prime factors } of [ Q ] .

So , let us move-on now to [ modulo 1103 ] in the next Section .