Pythagorean triples – Equal products of a leg and hypotenuse

 
 
The following integers

a_1 \; = \; p^2 \; - \; q^2
b_1 \; = \; 2 \, p \, q
c_1 \; = \; p^2 \; + \; q^2

a_2 \; = \; r^2 \; - \; s^2
b_2 \; = \; 2 \, r \, s
c_2 \; = \; r^2 \; + \; s^2

define two right triangles   (a_1, \; b_1, \; c_1),    (a_2, \; b_2, \; c_2).
 

Equal products of a leg and hypotenuse :

(2 \, p \, q) \,(p^2 + q^2) \; = \; (2 \, r \, s) \,(r^2 + s^2)

 
Here are some solutions:
 

(3504, 21172, 21460),    (7104, 7847, 10585)

3504^2 + 21172^2 = 21460^2 ….. 7104^2 + 7847^2 = 10585^2

3504 \times 21460 \; = \; 7104 \times 10585 \; = \; 75195840
 

(278588, 14784, 278980),   (36977, 59136, 69745)

278588^2 \; + \; 14784^2 \; = \; 278980^2 ….. 36977^2 \; + \; 59136^2 \; = \; 69745^2

14784 \times 278980 \; = \; 59136 \times 69745 \; = \; 4124440320
 

(5658912, 128466, 5660370),    (198512, 841266, 864370)

5658912^2 + 128466^2 = 5660370^2 ….. 198512^2 + 841266^2 = 864370^2

128466 \times 5660370 \; = \; 841266 \times 864370 \; = \; 727165092420
 

(298734352, 1659264, 298738960),    (3530017, 22124544, 22404385)

298734352^2 + 1659264^2 = 298738960^2 ….. 3530017^2 + 22124544^2 = 22404385^2

1659264 \times 298738960 \; = \; 22124544 \times 22404385 \; = \; 495686801725440
 

(6561804400, 12150750, 6561815650),    (119498176, 270018750, 295279426)

6561804400^2 + 12150750^2 = 6561815650^2 ….. 119498176^2 + 270018750^2 = 295279426^2

12150750  \times 6561815650 \; = \; 270018750  \times 295279426 \; = \; 79730981509237500

 
(82533234132, 62053776, 82533257460),  
(1400702617, 2057576256, 2489093785)

82533234132^2 \; + \; 62053776^2 \; = \; 82533257460^2
1400702617^2 \; + \; 2057576256^2 \; = \; 2489093785^2

62053776 \times  82533257460 \; = \; 2057576256 \times 2489093785 \; = \; 5121500270973168960

 

(705047366632, 246863274, 705047409850)
(10203455368, 11384759274, 15288009850)

705047366632^2 \; + \; 246863274^2 \; = \; 705047409850
10203455368^2 \; + \; 11384759274^2 \; = \; 15288009850

246863274 \times 705047409850 \; = \; 11384759274 \times 15288009850
= \; 174050311920790848900

 

(4532232204352, 817499136, 4532232278080)
(54860959873, 49941774336, 74188312705)

4532232204352^2 \; + \; 817499136^2 \; = \; 4532232278080^2
54860959873^2 \; + \; 49941774336^2 \; = \; 74188312705^2

817499136 \times 4532232278080 \; = \; 49941774336 \times 74188312705
= \; 3705095971481711738880

 

(23431108184712, 2352516534, 23431108302810)
(237358537192, 183666363894, 300120656410)

23431108184712^2 \; + \; 2352516534^2 \; = \; 23431108302810^2
237358537192^2 \; + \; 183666363894^2 \; = \; 300120656410^2

2352516534 \times 23431108302810 \; = \; 183666363894 \times 300120656410
= \; 55122069692305203660540

 

(28075678656192, 8566932018744, 29353637870280)
(394642839154432, 637209819576, 394643353589320)

28075678656192^2 \; + \; 8566932018744^2 \; = \; 29353637870280^2
394642839154432^2 \; + \; 637209819576^2 \; = \; 394643353589320^2

8566932018744 \times 29353637870280 \; = \; 637209819576 \times 394643353589320
= \; 251470620137518169200528320

 

(101969805870100, 6058806000, 101969806050100)
(870597970201, 588060600000, 1050597970201)

101969805870100^2 \; + \; 6058806000^2 \; = \; 101969806050100^2
870597970201^2 \; + \; 588060600000^2 \; = \; 1050597970201^2

6058806000 \times 101969806050100 \; = \; 588060600000 \times 1050597970201
= \; 617815272715182180600000

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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