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

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

x^2 \; + \; x \, y \; + \; y^2
x^2 \; + \; x \, z \; + \; z^2
y^2 \; - \; y \, z \; + \; z^2

are square numbers

 
for example,
 

(x, \; y, \; z) \; = \; (5, \; 3, \; 3)

5^2 \; + \; (5)(3) \; + \; 3^2 \; = \; 7^2
5^2 \; + \; (5)(3) \; + \; 3^2 \; = \; 7^2
3^2 \; - \; (3)(3) \; + \; 3^2 \; = \; 3^2

(x, y, \; z) \; = \; (64, \; 221, \; 56)

64^2 \; + \; (64)(221) \; + \; 221^2 \; = \; 259^2
64^2 \; + \; (64)(56) \; + \; 56^2 \; = \; 104^2
221^2 \; - \; (221)(56) \; + \; 56^2 \; = \; 199^2

(x, \; y, \; z) \; = \; (2496, \; 549, \; 1144)

2496^2 \; + \; (2496)(549) \; + \; 549^2 \; = \; 2811^2
2496^2 \; + \; (2496)(1144) \; + \; 1144^2 \; = \; 3224^2
549^2 \; - \; (549)(1144) \; + \; 1144^2 \; = \; 991^2

 
Find more solutions
 

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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