Structure of the Multiplication Table for [ modulo P ]

by Frank Charles Fung ( 1st published in June , 2006 )

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Epilog I : A Question on Mersenne Prime

This question arose at 9:10 p.m. on Tuesday 2006-7-11 , while the author was walking thru the { Hong Kong Park } in Hong Kong .

• but re-phrasing the question properly took another 3-days .

Anyways , here it is .

The { Mersenne Prime } [ 2^31 - 1 ] :

• For the { Mersenne Prime } [ 2^31 - 1 ] , we set-up the value [ P ] , such that :
  • [ P ] = 2,147,483,647 = [ 2^31 - 1 ] .

  And we can write , alternately :

  • [ P ] = 2 x 9 x 7 x 11 x 31 x 151 x 331 + 1

  Therefore :

  • [ P ] = [ L.C.M. of ( 2 , 9 , 7 , 11 , 31 , 151 , 331 ) ] + 1

  And consequently ,

  • [ P ] is 'relatively prime' with respect to [ 2 ] , [ 9 ] , [ 7 ] , [ 11 ] , [ 31 ] , [ 151 ] , & [ 331 ] .

Setting-up the [ Upper-Limit ] & the [ Lower-Limit ] :

• We note here that the { Square-Root of [ P ] } is roughly [ 46,340.95 ] .

  We then set-up the [ Upper Limit ] and the [ Lower-Limit ] for the range of our investigations as follows :

  • Upper Limit --- [ 2,147,395,600 ] = 46,340 x 46,340
  • Lower Limit --- [ 2,147,488,281 ] = 46,341 x 46,341

  And we then set up the value [ N ] , to be investigated , such that :

  • [ 2,147,395,600 ] < [ N ] < [ 2,147,488,281 ]

The Quesestion :

• The QUESTION here is :
  • whether the fact that :
    • [ N ] is 'relatively prime' with respect to [ 2 ] , [ 9 ] , [ 7 ] , [ 11 ] , [ 31 ] , [ 151 ] & [ 331 ] ,

      is already a { sufficient condition } that [ N ] is 'prime' ;

      • And if so , why so ?

      • And if not , why not ?

Recalling the { 120-degree Triangle }:

• We then have this diagram for the [ 120-degrees Triangle ] :
  • for the Mersenne Prime [ 2,147,483,647 ] .

  

• And the two (2) sets of equations here are :
  • Set One :

    

    with :

    • [ X ] = 1,897,507,765
    • [ Y ] = 433,738,152
    • [ Z ] = 2,147,483,647
  • Set Two :

    

    with :

    • [ U ] = 43,813
    • [ V ] = 4,698
    • [ W ] = 2,147,483,647

• Any further thoughts here ?

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( Note : This file was forund to be corrupted when we try to reload on 2009-10-26 .

This version here is the revised version based on the what we can pick-up from the corrupted file .)

go to the next section : Epilog II --- The Fractions Equation

go to the last section : Section VII --- The { Chinese Remainder Theorem } revisited

return to the HomePage for { Structure of the Multiplication Table for [ Modulo P ] }

Original dated 2006-6-25 / Updated 2006-12-31 / 2009-10-26