Make {(x^2 – (xy)/2 + y^2), (x^2 – (xz)/2 + z^2), (y^2 + (yz)/2 + z^2)} squares

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

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

are square numbers.

where   (x, \; y, \; z) \; = \; 1

 
for example,
 

(x, \; y, \; z) \; = \; (96, \; 153, \; 112)

96^2 \; - \; (96) \,(153)/2 \; + \; 153^2 \; = \; 159^2
96^2 \; - \; (96) \,(112)/2 \; + \; 112^2 \; = \; 128^2
153^2 \; + \; (153) \,(112)/2 \; + \; 112^2 \; = \; 211^2

(x, \; y, \; z) \; = \; (9728, \; 1113, \; 15504)

9728^2 \; - \; (9728) \,(1113)/2 \; + \; 1113^2 \; = \; 9511^2
9728^2 \; - \; (9728) \,(15504)/2 \; + \; 15504^2 \; = \; 16112^2
1113^2 \; + \; (1113) \,(15504)/2 \; + \; 15504^2 \; = \; 15819^2

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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