Pythagorean triples – Equal products of two legs 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)

 

Can you find a pair of Pythagorean triples with equal products of two legs and hypotenuse:

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

 
Is it possible to find a pair of primitive Pythagorean triples?
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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