On a Quick Expansion Method for Pythagorus Triples

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

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Section XVIII - Results for the [ 8-15-17 / 20-21-29 ] Pythagorus-Triple Pair

Summary for the Section :

• In this Section , we shall present the results for the [ 8-15-17 / 20-21-29 ] pair ,
  • arising from the { Quick Expansion Procedure } as discussed in Part III of this paper .

Case 29 :

• For Case 29 , the values for { A, B, C } are :
  • A = +117 + 0i
  • B = -75 + 40i
  • C = -42 - 40i

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

  	Value of Number	Square of Number	Factors of Number
  X	1,256,241,555,856	1,578,142,846,659,503,567,892,736	16 x 7 x 19 x 67 x 83 x 83 x 1279
  Y	1,472,832,092,160	2,169,234,371,696,402,733,465,600	16 x 128 x 9 x 5 x 11 x 13 x 23 x 43 x 113
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	1,935,814,355,344	3,747,377,218,355,906,301,358,336	16 x 601 x 7489 x 26881

Case 30 :

• For Case 30, the values for { A, B, C } are :
  • A = +475 + 0i
  • B = -315 + 168i
  • C = -160 - 168i

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

  	Value of Number	Square of Number	Factors of Number
  X	5,504,337,858,133,799	30,297,735,256,484,977,966,142,986,172,401	463 x 286049 x 41560777
  Y	7,906,618,694,094,120	62,514,619,173,798,607,538,967,418,574,400	8 x 9 x 5 x 7 x 11 x 13 x 29 x 31 x 919 x 26557
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	9,633,916,878,937,849	92,812,354,430,283,585,505,110,404,746,801	17 x 53 x 181 x 269 x 509 x 431449

Case 31 :

• For Case 31 , the values for the A, B and C are :
  • A = +156 + 0i
  • B = -56 + 105i
  • C = -100 - 105i

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

  	Value of Number	Square of Number	Factors of Number
  X	41,931,200,927,069	1,758,225,611,186,232,165,056,930,761	19 x 53 x 67 x 1361 x 456641
  Y	25,959,526,653,060	673,897,024,050,932,525,607,363,600	4 x 9 x 5 x 7 x 11 x 13 x 41 x 523 x 6719
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	49,316,555,386,981	2,432,122,635,237,164,690,664,294,361	17 x 97 x 577 x 677 x 76561

Case 32 :

• For Case 32, the values for the A, B and C are :
  • A = +95 + 0i
  • B = -32 + 60i
  • C = -63 - 60i

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

  	Value of Number	Square of Number	Factors of Number
  X	1,706,354,109,971	2,911,644,348,614,923,561,620,841	7 x 103 x 17 x 823 x 1109 x 2593
  Y	1,331,097,175,980	1,771,819,691,901,931,088,960,400	4 x 9 x 5 x 11 x 13 x 23 x 29 x 31 x 41 x 61
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	2,164,131,243,829	4,683,464,040,516,854,650,581,241	37 x 109 x 181 x 193 x 15361

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go to the next Section : Section XIX - Results for the [ 8-15-17 / 12-35-37 ] Pythagorus-Triple Pair

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