Make {(x^2 + 3xy + y^2), (x^2 + 4xz + z^2), (y^2 + 5yz + z^2)} squares

 
 
 
Find positive integers   x, \; y, \; z   such that

x^2 \; + \; 3 \, x \, y \; + \; y^2
x^2 \; + \; 4 \, x \, z \; + \; z^2
y^2 \; + \; 5 \, y \, z \; + \; z^2

 
Here’s one solution:

 
(x, \; y, \; z) \; = \; (32, \; 21, \; 40)

32^2 \; + \; 3 \,(32) \,(21) \; + \; 21^2 \; = \; 59^2
32^2 \; + \; 4 \,(32) \,(40) \; + \; 40^2 \; = \; 88^2
21^2 \; + \; 5 \,(21) \,(40) \; + \; 40^2 \; = \; 79^2

 
 
Find other solutions.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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