## Make {x^2 + xy + y^2, x^2 + xz + z^2, y^2 + yz + z^2} squares … Part 1

Can you find 3 distinct rational numbers    $x, \; y, \; z$    to make the three expressions squares

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

pipo found:

$(x, \; y, \; z) \; = \; (195, \; 264, \; 325)$
$195^2 \; + \; (195\times 264) \; + \; 264^2 \; = \; 399^2$
$195^2 \; + \; (195\times 325) \; + \; 325^2 \; = \; 455^2$
$264^2 \; + \; (264\times 325) \; + \; 325^2 \; = \; 511^2$

$(x, \; y, \; z) = (264, \; 325, \; 440)$
$264^2 \; + \; (264\times 325) \; + \; 325^2 \; = \; 511^2$
$264^2 \; + \; (264\times 440) \; + \; 440^2 \; = \; 616^2$
$325^2 \; + \; (325\times 440) \; + \; 440^2 \; = \; 665^2$

math grad - Interest: Number theory
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### 2 Responses to Make {x^2 + xy + y^2, x^2 + xz + z^2, y^2 + yz + z^2} squares … Part 1

1. pipo says:

Found five integer solutions for x<y<z<100:
Format (x,y,z,square1,square2,square3)
195 264 325 159201 261121 207025
264 325 440 261121 442225 379456
390 528 650 636804 1044484 828100 (this is 2 x first solution)
528 650 880 1044484 1768900 1517824 (this is 2 x second solution)
585 792 975 1432809 2350089 1863225 (this is 3 x first solution)
pipo

• benvitalis says: