Structure of the Multiplication Table for [ Modulo P ]

In the Section , we shall construct a { Multiplication Table } for [ modulo 1103 ] ,

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

For the { prime number } [ P ] being equal to [ 1,103 ] , we set up the corresponding value [ Q ] such that :

• [ Q ] = [ P - 1 ] = [ 1,102 ]

And we can write :

• 1,103 = 2 x 19 x 29 + 1

We can then set-up the { core elements } for building the { Multiplication Table } for [ modulo 1103 ] , as follows :

• The { Characteristic Pyramid } , based on the { solution set } to the equation :
  • 

  

• the "Square-Of-Square" { RING OF 18 } , based on the { solution set } to the equation :
  • 

    but excluding the solution-value [ 1 ] .

  

• the "Square-Of-Square" { RING OF 28 } , based on the { solution set } to the equation :
  • 

    but excluding the solution-value [ 1 ] .

  

We can then construct the { Multiplication Table } for [ modulo 1103 ] as follows :

A quick explanation here on this [ 19 rows x 29 columns ] { Multiplication Table } :

• the { base / anchor values } for the { 19 rows } are made-up-of :
  • the number [ 1 ] & the { 18 members } of the { RING OF 18 } ;

• the { base / anchor values } for the { 29 columns } are made-up-of :
  • the number [ 1 ] & the { 28 members } of the { RING OF 28 } .

The [ 19 x 29 ] { Multiplication Table } is therefore split into four (4) separate sections :

• the number [ 1 ] , as marked-off in { red-color } ,
  • being the product of { [ 1 ] x [ 1 ] } [ modulo 1103 ] ;

• the other { 18 numbers } on the { 1st column } , counted as one separate section & marked-off in { pink-color } ,
  • arising from [ 1 ] x [ RING OF 18 ] ;

• the other { 28 numbers } on the { 1st row } , counted as another separate section & also marked-off in { pink-color } ,
  • arising from [ 1 ] x [ RING OF 28 ] ;

• the [ 18 x 28 ] area , as marked-off in { orange-color } ,
  • arsing from the product of the [ RING OF 18 ] & the [ RING OF 28 ] .

And we shall take note here of a very important feature for this [ 19 x 29 ] { Multiplication Table } :

• none of the { 551 values } here on this { Multiplication Table } repeats itself .

Let us now take a closer look at this [ 18 x 28 ] { orange-color } area .

And let us now arbitrarily pick a { number } here and do a { keep on 'squaring' } operation to see if it arrives back at the same { number } :

• and we pick the { number } [ 604 ] , as mark-off in { dark-grey-color } in the { Multiplication Table } below , for this purpose :
  • 

And we see here that when we 'square' the number [ 604 mod 1103 ] , we move diagonally 'forward' to the number [ 826 mod 1103 ] .

Let us now try to understand why this is so :

• We start-off again with the relation :
  • 

• and 'squaring' [ 613 mod 1103 ] will move us one-notch 'forward' on the { RING OF 18 } ,
  • and therefore one-notch { vertically downwards } on the { Multiplication Table } , as per this equation :
    • 

  and 'squaring' [ 512 mod 1103 ] will move us one-notch 'forward' on the { RING OF 28 } ,

  • and therefore one-notch { horizontally to-the-right } on the { Multiplication Table } , as per this equation :
    • 

• so that we end-up diagonally one-notch 'forward' at [ 826 mod 1103 ] , as per this equation :
  • 

Thus , 'squaring' any of the 504 { numbers } in this [ 18 x 28 ] { orange-color area } will move us 'forward' in a diagonal direction .

And { keep-on 'squaring' } 28 times will move us , from the original [ 604 mod 1103 ] , to [ 441 mod 1103 ] as marked-off in { white-color } in the { Multiplication Table } above ,

• following the { light-green color } path , as marked above ,

  which shall always remain unbroken because both the { RING OF 18 } and the { RING OF 28 } are 'circular' in nature .

Let us now repeat this [ { keep-on-squaring } 28 times ] operation another eight (8) rounds to arrive back at the number [ 604 mod 1103 ] ,

• as per this { Multiplication Table } below :

And a summary for the total { 9-rounds operation } is then as follows :

We can then conclude that :

• When we merge the { RING OF 18 } and the { RING Of 28 } together ,
  • we come up with 2 separate { RING's OF 252 } .

We then have this set of 2 diagrams , being schematic representations of the 2 { RING's OF 252 } :

And we note here that the { mutiplicative conjugate pair } [ 604 , 1061 ] are now on 2 separate { RING's } .

• Rather interesting , but we shall have more on this later when we study [ modulo 3361 ] .

But let us also understand why :

• When we merge the [ RING OF 18 ] and the [ RING OF 28 ] together ,
  • we get 2 x [ RING's OF 252 ] .

This is because :

• G.C.D. ( 18 , 28 ) = 2 , &
• L.C.M. ( 18 , 28 ) = 252 ;

  and these 2 { numbers } are always controlling on the merging of 2 { RING's } .

  • based on { combinatorics } considerations .

We have , in this Section , constructed a { Multiplication Table } of { 551 elements } for [ modulo 1103 ] , using the { core elements } :

• the { Characteristic Pyramid } , the { RING OF 18 } , & the { RING OF 28 } .

And we simply note here that the other { 551 elements } , which would then make-up the full field of { 1,102 elements } ,

• can always be derived by applying the { Characteristic Pyramid } to each of the { 551 elements } in the { Multiplication Table } .

Thus , the entire field of { 1,102 elements } have been characterized .

Let us now move-on to [ modulo 3361 ] .