On a Quick Expansion Method for Pythagorus Triples

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

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Section XVI - Results for the [ 5-12-13 / 20-21-29 ] Pythagorus-Triple Pair

Summary for the Section :

• In this Section , we shall present the results for the [ 5-12-13 / 20-21-29 ] pair ,
  • arising from the { Quick Expansion Procedure } as discussed in Part III of this paper .

Case 21 :

• For Case 21 , the values for { A, B, C } are :
  • A = +88 + 0i
  • B = -25 + 60i
  • C = -63 - 60i

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

  	Value of Number	Square of Number	Factors of Number
  X	1,151,255,347,991	1,325,388,876,277,878,507,736,081	17 x 17 x 7 x 14503 x 39239
  Y	1,266,804,669,240	1,604,794,070,008,265,802,177,600	17 x 17 x 8 x 9 x 5 x 11 x 17 x 19 x 23 x 149
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	1,711,777,715,209	2,930,182,946,286,144,309,913,681	17 x 17 x 233 x 1553 x 16369

Case 22 :

• For Case 22, the values for { A, B, C } are :
  • A = +115 + 0i
  • B = -35 + 84i
  • C = -80 - 84i

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

  	Value of Number	Square of Number	Factors of Number
  X	7,502,298,635,691	56,284,484,819,091,039,939,047,481	729 x 53 x 1279 x 151817
  Y	6,726,236,641,740	45,242,259,360,685,793,110,227,600	729 x 4 x 3 x 5 x 7 x 11 x 23 x 31 x 2801
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	10,076,048,043,741	101,526,744,179,776,833,049,275,081	729 x 821 x 3361 x 5009

Case 23 :

• For Case 23 , the values for the A, B and C are :
  • A = +352 + 0i
  • B = -252 + 105i
  • C = -100 - 105i

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

  	Value of Number	Square of Number	Factors of Number
  X	277,426,022,868,161	76,965,198,164,445,390,124,787,521,921	71 x 730603 x 5348197
  Y	1,331,092,758,691,080	1,771,807,932,239,829,730,874,871,566,400	8 x 9 x 5 x 7 x 11 x 19 x 23 x 359 x 306083
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	1,359,695,969,841,889	1,848,773,130,404,275,120,999,659,088,321	53 x 317 x 215329 x 375841

Case 24 :

• For Case 24, the values for the A, B and C are :
  • A = +69 + 0i
  • B = -48 + 20i
  • C = -21 - 20i

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

  	Value of Number	Square of Number	Factors of Number
  X	19,575,610,075	383,204,509,808,441,505,625	25 x 17 x 17 x 7 x 89 x 4349
  Y	66,923,990,100	4,478,820,450,904,898,010,000	25 x 17 x 17 x 4 x 81 x 11 x 23 x 113
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	69,728,222,125	4,862,024,960,713,339,515,625	25 x 17 x 17 x 5 x 73 x 137 x 193

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go to the next Section : Section XVII - Results for the [ 5-12-13 / 12-35-37 ] Pythagorus-Triple Pair

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Original dated 2021-12-27