On a Quick Expansion Method for Pythagorus Triples

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

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

Summary for the Section :

• In this Section , we shall present the results for the [ 3-4-5 / 8-15-17 ] pair ,
  • arising from the { Quick Expansion Procedure } as discussed in Part III of this paper .

Case 5 :

• For Case 5 , the values for { A, B, C } are :
  • A = +21 + 0i
  • B = -6 + 8i
  • C = -15 - 8i

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

  	Value of Number	Square of Number	Factors of Number
  X	35,988,652	1,295,183,072,777,104	4 x 7 x 23 x 29 x 41 x 47
  Y	83,430,864	6,960,709,067,786,496	4 x 4 x 27 x 11 x 97 x 181
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	90,861,940	8,255,892,140,563,600	4 x 5 x 13 x 17 x 61 x 337

Case 6 :

• For Case 6 , the values for { A, B, C } are :
  • A = +77 + 0i
  • B = -45 + 60i
  • C = -32 - 60i

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

  	Value of Number	Square of Number	Factors of Number
  X	993,162,466,331	986,371,684,528,674,708,601,561	7 x 23 x 67 x 79
  Y	316,033,943,940	99,877,453,722,271,062,600	4 x 9 x 5 x 11 x 13 x 1019 x 12049
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	1,042,232,765,869	1,086,249,138,250,945,771,325,161	17 x 113 x 137 x 857 x 4621

Case 7 :

• For Case 7 , the values for { A, B, C } are :
  • A = +28 + 0i
  • B = -20 + 15i
  • C = -8 - 15i

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

  	Value of Number	Square of Number	Factors of Number
  X	588,207,501	345,988,064,232,665,001	729 x 7 x 73 x 1579
  Y	813,607,740	661,957,554,587,907,600	729 x 4 x 3 x 5 x 11 x 19 x 89
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	1,003,964,949	1,007,945,618,820,572,601	729 x 13 x 13 x 29 x 281

Case 8:

• For Case 8, the values for { A, B, C }are :
  • A = +61 + 0i
  • B = -16 + 24i
  • C = -45 - 24i

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

  	Value of Number	Square of Number	Factors of Number
  X	19,283,150,375	371,839,888,384,862,640,625	25 x 5 x 11243 x 13721
  Y	57,253,012,200	3,214,974,294,353,348,840,000	25 x 8 x 9 x 23 x 29 x 43 x 1109
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	60,413,138,425	3,649,747,294,358,211,480,625	25 x 17 x 37 x 1301 x 2953

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

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

return to Pythagorus Triples Homepage

Original dated 2021-12-27