## 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

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 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