On a Quick Expansion Method for Pythagorus Triples

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

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Section XXXII - Developing the One-On-One Expansion Formula ( II )

Summary for the Section :

• In this Section , we shall break-down [ P ] and [ Q ] into their component parts ,
  • ahead of the { conversion process } in the next Section and the { squaring process } in the Section thereafter .

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Concentrating on [ P ] :

• Let us bring in the value of [ P ] from the last Section - Section XXXI :
  • 
    

  For our analysis purposes here , let us bring in this set of 4 relations below :

  • FIRST :
    • We bring in this relation for the 1st and 2nd term on the first row :
      • 

      Thus , we have :

      • 

        yielding :

      • 

  • SECONDLY :
    • We have this relation for the 3rd and 4th term in the first row :
      • 

  • THIRDLY :
    • We have this relation for the 5th and 6th term in the first row :
      • 
        

      Consequently :

      • 
        

  • FOURTH and FINALLY :
    • We have this relation for the 1st and 2nd term in the second row :
      • 

• Let us bring in the original equation for [ P ] from above :
  • 
    

  And based on the 4 relations above , we can now re-express [ P ] in this format :

  • 
    
    
    
    

  On consolidation , we then have this expression for [ P ] :

  • 
    

    for our use in the next Section .

  And we also set up this expression for [ P ] :

  • 

    for our further evaluation at a later stage .

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Concentrating on [ Q ] :

• Let us bring in the value of [ Q ] from the last Section - Section XXXI :
  • 
    

  And on consolidating terms , we have :

  • 
    

• Let us now bring in this relation for the terms in the 2nd row :
  • 
    
    

    yielding :

  • 
    

• And on applying this to the expression for [ Q ] immediately above , we then have :
  • 
    

    And this then will be the expression for [ Q ] we shall be using in the next Section .

• We also set up at this point an alternate format for expressing [ Q ] as follows :
  • 

    for our further evaluation at a later stage .

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What's Next :

• And with this in-hand , we shall take a closer look at { [ P-Square ] + [ Q-Square ] } , next .

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go to the next Section : Section XXXIII - Developing the One-On-One Expansion Formula ( III )

go to the last Section : Section XXXI - Developing the One-On-One Expansion Formula ( I )

return to the Pythagorus Triples Homepage

Original dated 2021-12-27