On a Quick Expansion Method for Pythagorus Triples

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

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Section XXXVII - Checking Out the One-On-One Expansion Formula

Summary for the Section :

• In this Section , we shall do a numerical example :
  • to demonstrate that the One-On-One Expansion Formula actually works .

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Bringing in the new { Triangle A-B-C } for checking purposes :

• Let us first bring in the new { Triangle A-B-C } for our checking purposes :

  

  • arising from the { 36-15-39 Pythagorus-Triple Triangle } and the { 20-15-25 Pythagorus-Triple Triangle } .

• We then set up the values for [ W ] / [ D ] / [ H ] :
  • 

  • 

  • 

  Consequently :

  • 

  • 

  • 

  • 

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Checking on [ X ] :

• Let us bring in this expression for [ X ] from the last Section - Section XXXVI :
  • 

  We then set up :

  • 

    yielding :

  • 

  and

  • 

  • 

• We then have :
  • 

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Checking on [ Y ] :

• Let us bring in this expression for [ Y ] from the last Section - Section XXXVI :
  • 

  We then set up :

  • 

    yielding :

  • 

  and

  • 

• Consequently :
  • 

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Checking on [ Z ] :

• We then bring in this expression for [ Z ] from the last Section - Section XXXVI :
  • 

  We then set up :

  • 

    yielding :

  • 

  and

  • 

• And the final valeu for [ Z ] is then :
  • 

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Checking [ X ] / [ Y ] / [ Z ] :

• We then bring in [ Case 3 ] from Section XI in Part IV for a quick comparison :

  	Value of Number	Square of Number	Factors of Number
  X	7,545,027,329	56,927,437,395,356,874,241	17 x 17 x 7 x 1409 x 2647
  Y	13,044,327,120	170,154,470,013,567,494,400	17 x 17 x 16 x 9 x 5 x 11 x 41 x 139
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	15,069,237,121	227,081,907,408,924,368,641	17 x 17 x 17 x 353 x 8689

  The One-On-One Expansion formula does seem to be working fine .

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What's Next :

• Let us take quick look at some structural aspects associated with the One-On-One Expansion Formula , next .

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go to the next Section : Section XXXVIII - Structure of the One-On-One Expansion Formula

go to the last Section : Section XXXVI - Confirming the On-On-One Expansion Formula

return to the Pythagorus Triples Homepage

Original dated 2021-12-27