On a Quick Expansion Method for Pythagorus Triples

by Frank C. Fung ( 1st published in December, 2021. )

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Section III - Fermat's { Sum of 2 Squares } Revisited

Summary for the Section :

• In this Section , we shall demonstrate that :
  • each-and-every { prime number of the format [ 4*K + 1 ] } is a Pythagorus Triple .

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Bringing in the { [ 4*K +1 ] prime number } :

• Let us bring in a prime number of the format [ 4*K + 1 ] , i.e. :
  • 

  The Fermat's Sum-Of-Two-Squares Theorem then states that :

  • all { [ 4*K + 1 ] prime numbers } can always be expressed as the sum of 2 squares ;

    i.e. :

    • 

      where [ alpha ] and [ beta ] are positive integers .

• Let us now take a quick look at the square of this { [ 4*K + 1 ] prime number } :
  • 

  Consequently , we can now write :

  • 

    yielding :

  • 

  This then tell us that :

  • all { [ 4*K + 1 ] prime numbers } can be expressed as a Pythagorus Triple .

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What's Next :

• With this in hand , we shall look at the usual procedure for expanding on 2 Pythagorus Triples , next .

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go to the next Section : Section IV - The Usual Procedure for Expanding Pythagorus Triples

go to the last Section : Section II - An Overview of the Paper

return to the Pythagorus Triples Homepage

Original dated 2021-12-27