On a Quick Expansion Method for Pythagorus Triples

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

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Section XXIV - First Try at the Doubling-Up Process

Summary for the Section :

• In this Section , we shall do the calculations for our 1st try at the Doubling-Up Process ;
  • using the { 3-4-5 Pythagorus-Triple Triangle } as the base Triangle .

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Bring-in the base { Triangle A-B-C } for the Doubling-Up Process :

• Let us bring-in again the base { Triangle A-B-C } for the Doubling-Up Process from the last Section :
  • as per the diagram on-the-right below .

  

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Establishing the numeric values for [ K2 ] / [ K1 ] / [ K0 ] :

• We then start-off with :
  • 
  • 
  • 

  Consequently , we have :

  • First :
    • 

  • Secondly :
    • 

    • 

    • 

      yielding :

    • 

  • Third-and-Finally :
    • 

• Let us recall that :
  • In order to solve for the 2 mirror-imaged Equilateral Triangles associated with the given { Triangle A-B-C } ,
    • we shall assume that the complex values [ A ] / [ B ] / [ C ] are the solution values
      • for a { 3rd-Order Polynomial Equation } of the format :
        • 

      so that :

      • these 2 equations below shall be considered equivalent throughout this Section here .
        • 

        • 

  As such :

  • upon expansion of the 2nd equation above , we have :
    • 

    • 

    • 

• Let us recall that we have established above :
  • 

  • 

  • 

  Therefore :

  • 

  • 

  • 

  Consequently we can now re-write the original { 3rd-Order Polynomial Equation } ,

  • i.e. this equation :
    • 

    in this revised format :

    • 

  And based on this , we can now proceed to calculate the numeric value for the { Value-In-Question }, [ V-sub-IQ ] ;

  • using the procedure we have outlined in Part III of this paper .

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Calculating [ V-sub-IQ ] :

• Let us now do the calculations :
  • For [ K1 ] :
    • We start-off with :
      • 

      Consequently :

      • 

        yielding :

      • 

  • For [ KO ] :
    • We start-off with :
      • 

      Consequently :

      • 

      We then have :

      • 

• Let us recall this definition of [ V-sub-IQ ] from Part III :
  • 

  We then bring back from above :

  • 

  • 

  Consequently , we have :

  • 

  And we can now confirm that :

  • the [ real-part ] and the [ imaginary-part ] of { V-sub-IQ } do form a Pythagorus Triple ,
    • as per this table below .

  	Value of Number	Square of Number	Factors of Number
  X	4,123	16,999,129	7 x 19 x 31
  Y	23,364	545,876,496	4 x 9 x 11 x 59
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	23,725	562,875,625	5 x 5 x 13 x 73

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What's Next :

• This then prompted us try this Doubling-Up Process for { [ 4K + 1 ] prime numbers } up to [ 233 ] , next .

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go to the next Section : Section XXV - Results for the Doubling-Up Process thru to the prime number [ 233 ]

go to the last Section : Section XXIII - Setting up the new { Triangle A-B-C } for the Doubling-Up Process

return to the Pythagorus Triples Homepage

Original dated 2021-12-27