## 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,

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

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

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

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

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

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

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