On a Quick Expansion Method for Pythagorus Triples

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

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Section XXXVIII - Structure of the One-On-One Expansion Formula

Summary for the Section :

• In this Section , we shall revise the expressions for the { [ X ] / [ Y ] / [ Z ] Pythagorus Triple } into this format :
  • 

  • 

  • 

    for a better understanding of the structural relations among [ X ] / [ Y ] / [ Z ] .

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Structure of [ Z ] :

• Let us first bring back this expression for [ Z ] from before :
  • 

  We then set up [ GAMMA ] being :

  • 

  so that :

  • we can now re-write the above expression of [ Z ] as :
    • 

• Let us now take a closer look at [ GAMMA ] and :
  • we introduce [ Alpha ] and [ Beta ] for our use in this Section only ,

    i.e. :

    • 

      and

    • 

  We then have :

  • 

  And based on this relation , we can now re-write this expression for [ Z ] above :

  • 

    as :

  • 

• Let us now take a quick look at [ GAMMA-Square ] .
  • We then have :
    • 

      yielding :

    • 

    Consequently , we have :

    • 

      so that :

      • [ GAMMA ] is expressible as the { hypotunuse of a right-angle triangle } .

• We also take note here that :
  • 

    arising from :

    • 

    so that :

    • [ Lambda ] is also expressible as the { hypotunuse of a right-angle triangle } .

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Structure of [ X ] :

• Let us bring in again this expression of [ X ] from before :
  • 

  And on expanding terms , we have :

  • 
    
    

  Let us now split up the term in the middle-row into 2 terms , to yield :

  • 
    

  And on consolidating terms , we have :

  • 
    

• Let us recall that we have defined above the values of [ Alpha ] and [ Beta ] as :
  • 

    and

  • 

  We can therefore express [ X ] in this final format :

  • 

    for our comparison purposes below .

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Structure of [ Y ] :

• Let us bring in again this expression of [ Y ] from before :
  • 

  And on expanding terms , we have :

  • 

  Further expansion then yields us this expression below :

  • 
    

  Let us now split up the 1st row in the above expression into 2 rows :

  • 
    
    

• Let us now bring in this special relation for [ Lambda ] :
  • 

    arising from :

    • 

  Applying this to the previous expression for [ Y ] above then yields us this new expression :

  • 
    
    

  And , on consolidating terms and some further expansions , we have :

  • 
    

  Further re-arranging terms then yields us this expression below :

  • 
    

  And on re-grouping , we have :

  • 
    

• Let us recall that we have defined above the values of [ Alpha ] and [ Beta ] as :
  • 

    and

  • 

  We can therefore express [ Y ] in this final format :

  • 

    for our comparison purposes immediately below .

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Comparing [ X ] / [ Y ] / [ Z ] :

• Let us bring in [ X ] / [ Y ] / [ Z ] from above :
  • 

  • 

  • 

• Let us now set up the angle [ Angle PSI ] as :
  • 

  In order to set up { cosine of PSI } and { sine of PSI } , we then bring in :

  • 

    yielding :

  • 

    and finally :

  • 

  Thus , we have :

  • 

  • 

• Let us recall we have set up above :
  • 

  Consequently , we have :

  • 

  • 

  And based on this , we can now re-write the above expressions for [ X ] / [ Y ] / [ Z ] as :

  • 

  • 

  • 

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What's Next :

• Let us now go to the final Part of the Paper for some observations and concluding remarks .

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go to the next Section : Section XXXIX - Some Observations on Pythatgorus Triples

go to the last Section : Section XXXVII - Checking Out the One-On-One Expansion Formula

return to the Pythagorus Triples Homepage

Original dated 2021-12-27