On a Quick Expansion Method for Pythagorus Triples

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

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Section XIII - Results for the [ 3-4-5 / 20-21-29 ] Pythagorus-Triple Pair

Summary for the Section :

• In this Section , we shall present the results for the [ 3-4-5 / 20-21-29 ] pair ,
  • arising from the { Quick Expansion Procedure } as discussed in Part III of this paper .

Case 9 :

• For Case 9 , the values for { A, B, C } are :
  • A = +36 + 0i
  • B = -15 + 20i
  • C = -21 - 20i

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

  	Value of Number	Square of Number	Factors of Number
  X	4,175,201,359	17,432,306,388,195,446,881	7 x 7 x 1091 x 78101
  Y	1,620,950,760	2,627,481,366,344,577,600	8 x 9 x 5 x 11 x 13 x 23 x 37 x 37
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	4,478,815,441	20,059,787,754,540,024,481	41 x 89 x 409 x 3001

Case 10 :

• For Case 10 , the values for { A, B, C } are :
  • A = +143 + 0i
  • B = -63 + 84i
  • C = -80 - 84i

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

  	Value of Number	Square of Number	Factors of Number
  X	19,382,542,151,363	375,682,940,249,363,432,402,757,769	97 x 51647 x 3868957
  Y	5,064,428,485,116	25,648,435,880,854,342,633,533,456	4 x 9 x 7 x 11 x 13 x 17 x 29 x 89 x 3203
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	20,033,256,752,965	401,331,376,130,217,775,036,291,225	5 x 41 x 277 x 12373 x 28513

Case 11 :

• For Case 11 , the values for { A, B, C } are :
  • A = +48 + 0i
  • B = -28 + 21i
  • C = -20 - 21i

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

  	Value of Number	Square of Number	Factors of Number
  X	13,042,262,393	170,100,608,327,862,086,449	199 x 719 x 91153
  Y	6,081,651,576	36,986,485,891,863,283,776	8 x 9 x 7 x 11 x 13 x 13 x 6491
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	14,390,520,985	207,087,094,219,725,370,225	5 x 457 x 1013 x 6217

Case 12 :

• For Case 12 , the values for { A, B, C } are :
  • A = +143 + 0i
  • B = -80 + 60i
  • C = -63 - 60i

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

  	Value of Number	Square of Number	Factors of Number
  X	8,550,396,985,571	73,109,288,610,861,643,582,196,041	17 x 17 x 7 x 23 x 227 x 643 x 1259
  Y	2,858,538,171,060	8,171,240,475,407,049,821,523,600	17 x 17 x 4 x 9 x 5 x 11 x 13 x 149 x 2579
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	9,015,571,478,629	81,280,529,086,268,693,403,719,641	17 x 17 x 37 x 397 x 677 x 3137

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go to the next Section : Section XIV - Results for the [ 3-4-5 / 12-35-37 ] Pythagorus-Triple Pair

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Original dated 2021-12-27