On a Quick Expansion Method for Pythagorus Triples

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

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Section XV - Results for the [ 5-12-13 / 8-15-17 ] Pythagorus-Triple Pair

Summary for the Section :

• In this Section , we shall present the results for the [ 5-12-13 / 8-15-17 ] pair ,
  • arising from the { Quick Expansion Procedure } as discussed in Part III of this paper .

Case 17 :

• For Case 17 , the values for { A, B, C } are :
  • A = +44 + 0i
  • B = -9 + 12i
  • C = -35 - 12i

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

  	Value of Number	Square of Number	Factors of Number
  X	3,330,335,276	11,091,133,050,569,996,176	4 x 19 x 293 x 347 x 431
  Y	44,672,609,520	1,995,642,041,326,394,630,400	4 x 4 x 9 x 5 x 7 x 11 x 11 x 17 x 31 x 139
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	44,796,575,476	2,006,733,174,376,964,626,576	4 x 661 x 3529 x 4801

Case 18 :

• For Case 18, the values for { A, B, C } are :
  • A = +171 + 0i
  • B = -96 + 40i
  • C = -75 - 40i

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

  	Value of Number	Square of Number	Factors of Number
  X	6,647,901,367,879	44,194,592,597,047,479,292,958,641	1942399 x 3422521
  Y	3,970,368,060,120	15,763,822,532,821,051,934,414,400	8 x 9 x 5 x 7 x 11 x 17 x 89 x 137 x 691
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	7,743,281,935,321	59,958,415,129,868,531,227,373,041	53 x 293 x 6841 x 72889

Case 19 :

• For Case 19 , the values for the A, B and C are :
  • A = +44 + 0i
  • B = -36 + 15i
  • C = -8 - 15i

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

  	Value of Number	Square of Number	Factors of Number
  X	994,525,075	989,080,124,803,755,625	25 x 17 x 19 x 79 x 1559
  Y	8,108,585,100	65,749,152,323,942,010,000	25 x 4 x 9 x 7 x 11 x 11 x 11 x 967
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	8,169,347,125	66,738,232,448,745,765,625	25 x 5 x 29 x 53 x 101 x 421

Case 20 :

• For Case 20, the values for the A, B and C are :
  • A = +57 + 0i
  • B = -25 + 60i
  • C = -32 - 60i

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

  	Value of Number	Square of Number	Factors of Number
  X	421,626,472,451	177,768,882,271,473,861,947,401	17 x 34651 x 715753
  Y	82,651,213,260	6,831,223,053,349,999,827,600	4 x 9 x 5 x 7 x 11 x 659 x 9049
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	429,651,143,749	184,600,105,324,823,861,775,001	29 x 41 x 89 x 281 x 14449

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

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

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Original dated 2021-12-27