Make {x^2 + y^2 ± xy, x^2 + z^2 ± xy, y^2 + z^2 ± xy} squares

 
 

                                                                        (1)                   

 

x^2 \; + \; y^2 \; - \; x \, y \; = \; A^2
x^2 \; + \; z^2 \; - \; x \, y \; = \; B^2
y^2 \; + \; z^2 \; - \; x \, y \; = \; C^2

 
Here are some solutions:
 

\{ \, x, \; y, \; z, \; A, \; B, \; C \, \}

\{ \, 22, \; 70, \; 41, \; 62, \; 25, \; 71 \, \}
\{ \, 468, \; 660, \; 316, \; 588, \; 100, \; 476 \, \}
\{ \, 2444, \; 2924, \; 1121, \; 2716, \; 289, \; 1631 \, \}
\{ \, 8140, \; 9100, \; 2876, \; 8660, \; 676, \; 4124 \, \}
\{ \, 21186, \; 22866, \; 6121, \; 22074, \; 1369, \; 8711 \, \}
\{ \, 47012, \; 49700, \; 11516, \; 48412, \; 2500, \; 16316 \, \}
\{ \, 93208, \; 97240, \; 19841, \; 95288, \; 4225, \; 28031 \, \}
\{ \, 169884, \; 175644, \; 31996, \; 172836, \; 6724, \; 45116 \, \}
\{ \, 290030, \; 297950, \; 49001, \; 294070, \; 10201, \; 68999 \, \}

 
Find other solutions where   x < 290030

 

                                                                        (2)                   

 
Find positive integers   (x, y, z)   such that

x^2 \; + \; y^2 \; + \; x \, y \; = \; A^2
x^2 \; + \; z^2 \; + \; x \, y \; = \; B^2
y^2 \; + \; z^2 \; + \; x \, y \; = \; C^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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