Make {(x^2 – xy + y^2), (x^2 – xz + z^2), (y^2 – yz + z^2)} squares … Part 4

 
 

Find distinct integers   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

 
 
 

(1)

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

Pipo found:

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

\{195, \; 459, \; 819, \; 399, \; 711, \; 741 \}
\{231, \; 351, \; 495, \; 309, \; 441, \; 429 \}
\{253, \; 493, \; 805, \; 427, \; 703, \; 713 \}
\{264, \; 459, \; 704, \; 399, \; 619, \; 616 \}
\{273, \; 385, \; 585, \; 343, \; 515, \; 507 \}
\{273, \; 425, \; 480, \; 373, \; 455, \; 417 \}
\{308, \; 468, \; 660, \; 412, \; 588, \; 572 \}
\{357, \; 637, \; 792, \; 553, \; 727, \; 687 \}
\{384, \; 560, \; 650, \; 496, \; 610, \; 566 \}
\{385, \; 585, \; 825, \; 515, \; 735, \; 715 \}
\{462, \; 702, \; 990, \; 618, \; 882, \; 858 \}
\{520, \; 667, \; 832, \; 607, \; 763, \; 728 \}
\{546, \; 850, \; 960, \; 746, \; 910, \; 834 \}
\{576, \; 840, \; 975, \; 744, \; 915, \; 849 \}

 
 

(2)   Here’s another set of solutions
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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 4

  1. pipo says:

    Found here a little bit more solutions for x<y<z<1000:
    Format {x, y, z, square1, square2, square3}
    195 459 819 159201 505521 549081
    231 351 495 95481 194481 184041
    253 493 805 182329 494209 508369
    264 459 704 159201 383161 379456
    273 385 585 117649 265225 257049
    273 425 480 139129 207025 173889
    308 468 660 169744 345744 327184
    357 637 792 305809 528529 471969
    384 560 650 246016 372100 320356
    385 585 825 265225 540225 511225
    462 702 990 381924 777924 736164
    520 667 832 368449 582169 529984
    546 850 960 556516 828100 695556
    576 840 975 553536 837225 720801
    pipo

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