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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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

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