When x ± y, x ± z, y ± z are squares

Find positive integers $x, \; y, \; z$ so that the expressions

$x \; \pm \; y$
$x \; \pm \; z$
$y \; \pm \; z$

are to made squares.

Here are some solutions:

$x = 40606322$ , $y = 29316722$ , $z = 26633678$

$40606322 \; + \; 29316722 \; = \; 69923044 \; = \; 8362^2$
$40606322 \; - \; 29316722 \; = \; 11289600 \; = \; 3360^2$

$40606322 \; + \; 26633678 \; = \; 67240000 \; = \; 8200^2$
$40606322 \; - \; 26633678 \; = \; 13972644 \; = \; 3738^2$

$29316722 \; + \; 26633678 \; = \; 55950400 \; = \; 7480^2$
$29316722 \; - \; 26633678 \; = \; 2683044 \; = \; 1638^2$

$x = 27122258$ , $y = 15832658$ , $z = 1010158$

$27122258 \; + \; 15832658 \; = \; 42954916 \; = \; 6554^2$
$27122258 \; - \; 15832658 \; = \; 11289600 \; = \; 3360^2$

$27122258 \; + \; 1010158 \; = \; 28132416 \; = \; 5304^2$
$27122258 \; - \; 1010158 \; = \; 26112100 \; = \; 5110^2$

$15832658 \; + \; 1010158 \; = \; 16842816 \; = \; 4104^2$
$15832658 \; - \; 1010158 \; = \; 14822500 \; = \; 3850^2$

$x = 59421728$ , $y = 14263328$ , $z = 1418272$

$59421728 \; + \; 14263328 \; = \; 73685056 \; = \; 8584^2$
$59421728 \; - \; 14263328 \; = \; 45158400 \; = \; 6720^2$

$59421728 \; + \; 1418272 \; = \; 60840000 \; = \; 7800^2$
$59421728 \; - \; 1418272 \; = \; 58003456 \; = \; 7616^2$

$14263328 \; + \; 1418272 \; = \; 15681600 \; = \; 3960^2$
$14263328 \; - \; 1418272 \; = \; 12845056 \; = \; 3584^2$

$x = 214390472$ , $y = 102876872$ , $z = 20066872$

$214390472 \; + \; 102876872 \; = \; 317267344 \; = \; 17812^2$
$214390472 \; - \; 102876872 \; = \; 111513600 \; = \; 10560^2$

$214390472 \; + \; 20066872 \; = \; 234457344 \; = \; 15312^2$
$214390472 \; - \; 20066872 \; = \; 194323600 \; = \; 13940^2$

$102876872 \; + \; 20066872 \; = \; 122943744 \; = \; 11088^2$
$102876872 \; - \; 20066872 \; = \; 82810000 \; = \; 9100^2$

$x = 18098627922$ , $y = 16553194578$ , $z = 16334627922$

$18098627922 \; + \; 16553194578 \; = \; 34651822500 \; = \; 186150^2$
$18098627922 \; - \; 16553194578 \; = \; 1545433344 \; = \; 39312^2$

$18098627922 \; + \; 16334627922 \; = \; 34433255844 \; = \; 185562^2$
$18098627922 \; - \; 16334627922 \; = \; 1764000000 \; = \; 42000^2$

$16553194578 \; + \; 16334627922 \; = \; 32887822500 \; = \; 181350^2$
$16553194578 \; - \; 16334627922 \; = \; 218566656 \; = \; 14784^2$

$x = 1160672258$ , $y = 683686658$ , $z = 348650242$

$1160672258 \; + \; 683686658 \; = \; 1844358916 \; = \; 42946^2$
$1160672258 \; - \; 683686658 \; = \; 476985600 \; = \; 21840^2$

$1160672258 \; + \; 348650242 \; = \; 1509322500 \; = \; 38850^2$
$1160672258 \; - \; 348650242 \; = \; 812022016 \; = \; 28496^2$

$683686658 \; + \; 348650242 \; = \; 1032336900 \; = \; 32130^2$
$683686658 \; - \; 348650242 \; = \; 335036416 \; = \; 18304^2$

$x = 2598804608$ , $y = 1876270208$ , $z = 1704555392$

$2598804608 \; + \; 1876270208 \; = \; 4475074816 \; = \; 66896^2$
$2598804608 \; - \; 1876270208 \; = \; 722534400 \; = \; 26880^2$

$2598804608 \; + \; 1704555392 \; = \; 4303360000 \; = \; 65600^2$
$2598804608 \; - \; 1704555392 \; = \; 894249216 \; = \; 29904^2$

$1876270208 \; + \; 1704555392 \; = \; 3580825600 \; = \; 59840^2$
$1876270208 \; - \; 1704555392 \; = \; 171714816 \; = \; 13104^2$

Find more solutions.

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