Home #1 Mathematics A Theorem of Gauss ... by Giovanni A. Orlando
A Theorem of Gauss ... by Giovanni A. Orlando PDF Print E-mail
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Friday, 22 June 2012 10:04

***

Greetings in the day of Love ... the day of the Partners ...

    And if the Partners are many ... like the Soccer Teams ... they need to challenge each other ...

    And if the State Chief are few ... like the Euro meeting ... they need to work to blend their mind.

    In each case ... is a challenge of Consciousnesses ... the Winner ... if is the case ... represent the Stronger Mind ... or in the case of Necessity to Blend ... the Result is the G.C.M ... the Great Common Measure.

   ***

    And Unfortunately ... To-day ... people does not use their Mind ... or ... they have a Mechanical Mind.

    Students does not study ... people do not read ... and the Level of Consciousness ... remain Low.

    Are you looking for examples ... of the Mechanical Mind? ...

    They are everywhere:

  • People purchase Computers not like Helpers ... but like Mind-Substitute: iPad, Faster Computers with faster processors. This was my field of expertise for over than 25 years ... I know what I tell you.
  • Do you want to pray ... but have no time because need to watch the soap opera? ... Purchase a Bible and visit a Church and give some money to them.
  • More Energy? ... No problem. Let us go ... to Death Sea, Red Sea, Mediterranean Sea, Caribbean Sea ... Explode and Destroy everything ... Who cares?
  • Pasta ? ... Let take it from Germany ... They have a stronger euro ... It is available also in Italy.
  • Sex? ... Let go to Vegas ... and hire a ... Show-girl ... or a Show-boy ... No problem.
  • Everything have a price ... I say, 'No' ... No. Everything have a price in the Lower Mind, the Mind of Evil Ones ... Not in the Mind of God and the Light-Houses.

***

     This theorem is very simple and regards so-called Elementary Mathematics, that is not so elementary.

     I got this Theorem from an US Book on Algebra, titled: 'Algebra, An Elementary Text book' by G. Crystal.

     The Theorem is the following:

      Theorem. If a, b, c, d ... be a series of integers where the G.C.M (the Greatest Common Measure) is g. Show that integers (positive and negative) A, B, C, D, ... can be found ...

g = Aa + Bb+ Cc + Dd + ...

    (Gauss's Disquisitiones Arithmeticae, Th. 40)

     The theorem follows by the definition of G.C.M (check page 46 (71-pdf) of the book)

     You need to use your mind ...

     I will give two examples ...

     Example 1: Show that two integers A and B, can be found, such that,

5 A - 7 B = 1

     Solution. 7 = 1 x 5 + 2 ... 5 = 2 x 2 + 1. Now the solution can be find expressing 1, in function of 5 and 7.

     Now, 1 = 5 - 2(7-5) = 5 - 2 x 7 + 2 x 5 = 3 x 5 - 2 x 7.

     Therefore, A = 3 and B = 2.

     Example 2: Show that two integers C and D, can be found, such that,

13 C - 11 D = 1

 

      Solution.   13 = 11 x 1 + 2. 11 = 5 x 2 + 1. (We found the 1). Now because 11 and 13 are twin primes, like 5 and 7.

       Now, 1 = 11 - 5 x 2 = 11 - 5 (13 - 11) = 11 - 5 x 13 + 11 x 5 = 6 x 11 - 5 x 13. (66-65).

       Therefore, because (-5) x 13 - (-6) x 11 = 1, we have that C = -5 and D = -6.

Thanks,

Giovanni A. Orlando.

PS. Gauss was great ... Bach too ... but the Pasta is better Italian. And Galois was still greater. Spanish is the Second Language of the World ... all must learn ...

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Last Updated on Wednesday, 01 August 2012 18:45