Make {2x^2 + 2y^2 – z^2, 2x^2 – y^2 + 2z^2, -x^2 + 2*y^2 + 2z^2} squares

 
 

we can make the three expressions squares

(1)    2 \, x^2 \; + \; 2 \, y^2 \; - \; z^2
(2)    2 \, x^2 \; - \; y^2 \; + \; 2 \, z^2
(3)    -x^2 \; + \; 2 \, y^2 \; + \; 2 \, z^2

with a simple parameterization:

x \; = \; m \; - \; n
y \; = \; m \; + \; 2 \, n
z \; = \; 2 \, m \; + \; n

(1)
2 \, x^2 \; + 2 \, y^2 \; - \; z^2
= \; 2 \,(m - n)^2 \; + \; 2 \,(m + 2 \, n)^2 \; - \; (2 \, m + n)^2 \; = \; 9 \, n^2

(2)
2 \, x^2 \; - \; y^2 \; + \; 2 \, z^2
= \; 2 \,(m - n)^2 - (m + 2 \, n)^2 + 2 \,(2 \, m + n)^2 \; = \; 9 \, m^2

(3)
-x^2 \; + \; 2 \, y^2 \; + \; 2 \, z^2
= \; -(m - n)^2 + 2 \,(m + 2 \, n)^2 \; + \; 2 \,(2 \, m + n)^2 \; = \; 9 \, (m + n)^2

Interestingly,

(2) + (1)
(2 \, x^2 \; - \; y^2 \; + \; 2 \, z^2) \; + \; (2 \, x^2 \; + \; 2 \, y^2 \; - \; z^2) \; = \; 9 \, (m^2 + n^2)

(3)
-x^2 \; + \; 2 \, y^2 \; + \; 2 \, z^2 \; = \; 9 \, (m + n)^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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