On a Quick Expansion Method for Pythagorus Triples

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

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Section XI - Results for the [ 3-4-5 / 5-12-13 ] Pythagorus-Triple Pair

Summary for the Section :

• In this Section , we shall present the results for the [ 3-4-5 / 5-12-13 ] pair ,
  • arising from the { Quick Expansion Procedure } as discussed in Part III of this paper .

Case 1 :

• For Case 1 , the values for { A, B, C } are :
  • A = +14 + 0i
  • B = -9 + 12i
  • C = -5 - 12i

  And the resultant Pythagorus Triple { X, Y, Z } is then :

  	Value of Number	Square of Number	Factors of Number
  X	44,258,732	1,958,835,358,247,824	4 x 7 x 59 x 73 x 367
  Y	23,900,976	571,256,653,752,576	4 x 4 x 9 x 11 x 79 x 191
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	50,300,020	2,530,092, 012,000,400	4 x 5 x 37 x 101 x 673

Case 2 :

• For Case 2, the values for { A, B, C } are :
  • A = +63 + 0i
  • B = -15 + 20i
  • C = -48 - 20i

  And the resultant Pythagorus Triple { X, Y, Z } is then :

  	Value of Number	Square of Number	Factors of Number
  X	2,534,573,251	6,424,061,564,684,709,001	7 x 23 x 199 x 239 x 331
  Y	55,391,214,780	3,068,186,674,804,090,448,400	4 x 9 x 5 x 11 x 31 x 53
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	55,449,172,549	3,074,610,736,368,775,157,401	1621 x 2689 x 12721

Case 3 :

• For Case 3 , the values for { A, B, C } are :
  • A = +56 + 0i
  • B = -20 + 15i
  • C = -36 - 15i

  And the resultant Pythagorus Triple { X, Y, Z } is then :

  	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

Case 4 :

• For Case 4 , the values for { A, B, C } are :
  • A = +21 + 0i
  • B = -16 + 12i
  • C = -5 - 12i

  And the resultant Pythagorus Triple { X, Y, Z } is then :

  	Value of Number	Square of Number	Factors of Number
  X	98,619,227	9,725,751,934,077,529	17 x 17 x 41 x 41 x 7 x 29
  Y	192,380,364	37,010,204,452,772,496	17 x 17 x 41 x 41 x 4 x 9 x 11
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	216,185,005	46,735,956,386,850,025	17 x 17 x 41 x 41 x 5 x 89

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go to the next Section : Section XII - Results for the [ 3-4-5 / 8-15-17 ] Pythagorus-Triple Pair

go to the last Section : Section X - Outline and Purpose for Part IV

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Original dated 2021-12-27