On a Quick Expansion Method for Pythagorus Triples

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

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Section XXXIV - Developing the One-On-One Expansion Formula ( IV )

Summary for the Section :

• In this Section , we shall express the sum of { P-Square } and { Q-Square } in terms of :
  • [ Lambda ] , [ Omega ] and the { sine and cosine of [ THETA ] } .

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Calculating { P-Square } :

• Let us bring back this expression for [ P ] from the last Section - Section XXXIII :
  • 
    

  Consequently , we have :

  • 
    
    
    
    
    
    

  And on consolidating terms , we have :

  • 
    
    
    
    
    
    

  yielding this final relation for { P-Square } for our use below :

  • 
    
    
    
    

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Calculating { Q-Square } :

• Let us bring back this expression for [ Q ] from the last Section - Section XXXIII :
  • 
    

  Consequently , we have :

  • 
    
    
    
    

  And on consolidating terms , we have :

  • 
    
    
    
    

  yielding this final relation for { Q-Square } for our use below :

  • 
    
    
    
    

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Sum of { P-Square } and { Q-Square } :

• We then bring back from above :
  • this expression for { P-Square } :
    • 
      
      
      
      

    and also this expression for { Q-Square } :

    • 
      
      
      
      

  Consequently , we have :

  • 
    
    
    
    
    

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What's Next :

• And with this in-hand , we shall next attempt to establish that :
  • the above expression for { [ P-Square ] + [ Q-Square ] } is always a perfect square .

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go to the next Section : Section XXXV - Developing the One-On-One Expansion Formula ( V )

go to the last Section : Section XXXIII - Developing the One-On-One Expansion Formula ( VI )

return to the Pythagorus Triples Homepage

Original dated 2021-12-27