On a Quick Expansion Method for Pythagorus Triples

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

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

Summary for the Section :

• In this Section , we shall present the results for the [ 3-4-5 / 12-35-37 ] pair ,
  • arising from the { Quick Expansion Procedure } as discussed in Part III of this paper .

Case 13 :

• For Case 13 , 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	1,309,699,153	1,715,311,871,368,917,409	7 x 107 x 1009 x 1733
  Y	5,754,424,104	33,113,396,768,696,202,816	8 x 9 x 11 x 11 x 11 x 13 x 31 x 149
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	5,901,585,265	34,828,708,640,065,120,225	5 x 313 x 1277 x 2953

Case 14 :

• For Case 14 , the values for { A, B, C } are :
  • A = +153 + 0i
  • B = -105 + 140i
  • C = -48 - 140i

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

  	Value of Number	Square of Number	Factors of Number
  X	85,637,509,371,091	7,333,783,011,283,698,842,346,530,281	409 x 18539 x 20641
  Y	61,666,530,835,140	3,802,761,025,241,272,425,858,819,600	4 x 9 x 5 x 7 x 11 x 19 x 12697 x 18443
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	105,529,825,341,109	11,136,544,036,524,971,268,205,349,881	13 x 29 x 1657 x 2069 x 81649

Case 15 :

• For Case 15 , the values for { A, B, C } are :
  • A = +176 + 0i
  • B = -140 + 105i
  • C = -36 - 105i

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

  	Value of Number	Square of Number	Factors of Number
  X	31,048,291,624,769	963,996,412,816,700,829,894,303,361	71 x 73 x 359 x 467 x 35731
  Y	79,342,151,648,760	6,295,177,028,254,829,186,409,537,600	8 x 9 x 5 x 7 x 11 x 11 x 13 x 827 x 24203
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	85,200,783,101,281	7,259,173,441,071,530,016,303,840,961	97 x 577 x 13729 x 110881

Case 16 :

• For Case 16 , the values for { A, B, C } are :
  • A = +153 + 0i
  • B = -48 + 36i
  • C = -105 - 36i

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

  	Value of Number	Square of Number	Factors of Number
  X	917,093,508,723	841,060,503,741,863,277,090,729	729 x 7 x 1409 x 127,549
  Y	5,894,500,316,364	34,745,133,979,615,296,086,180,496	729 x 4 x 9 x 11 x 19 x 19 x 163 x 347
  xxxxx	xxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx	xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  Z	5,965,416,538,965	35,586,194,483,357,159,363,271,225	729 x 5 x 13 x 29 x 41 x 113 x 937

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

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

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Original dated 2021-12-27