To make {(x±y), (y±z), (z±x)} all squares

Find three positive integers   $A, B, C$   such that

$A \; \pm \; B$
$A \; \pm \; C$
$B \; \pm \; C$

are all squares.

Here are some solutions:

(A, B, C) = (2399057, 2288168, 1873432)

$A + B = 2399057 + 2288168 = 2165^2$
$A - B = 2399057 - 2288168 = 333^2$
$A + C = 2399057 + 1873432 = 2067^2$
$A - C = 2399057 - 1873432 = 725^2$
$B + C = 2288168 + 1873432 = 2040^2$
$B - C = 2288168 - 1873432 = 644^2$

(A, B, C) = (4387539232, 3762939168, 2433899232)

$A + B = 4387539232 + 3762939168 = 90280^2$
$A - B = 4387539232 - 3762939168 = 24992^2$
$A + C = 4387539232 + 2433899232 = 82592^2$
$A - C = 4387539232 - 2433899232 = 44200^2$
$B + C = 3762939168 + 2433899232 = 78720^2$
$B - C = 3762939168 - 2433899232 = 36456^2$

(A, B, C) = (1189604889857, 680815132832, 418662940768)

$A + B = 1189604889857 + 680815132832 = 1367633^2$
$A - B = 1189604889857 - 680815132832 = 713295^2$
$A + C = 1189604889857 + 418662940768 = 1268175^2$
$A - C = 1189604889857 - 418662940768 = 878033^2$
$B + C = 680815132832 + 418662940768 = 1048560^2$
$B - C = 680815132832 - 418662940768 = 512008^2$

(A, B, C) = (98570310169376, 38001073844000, 23033769906400)

$A + B = 98570310169376 + 38001073844000 = 11686376^2$
$A - B = 98570310169376 - 38001073844000 = 7782624^2$
$A + C = 98570310169376 + 23033769906400 = 11027424^2$
$A - C = 98570310169376 - 23033769906400 = 8691176^2$
$B + C = 38001073844000 + 23033769906400 = 7812480^2$
$B - C = 38001073844000 - 23033769906400 = 3868760^2$

(A, B, C) = (3715388677375057, 1013722453597032, 611234918180568)

$A + B = 3715388677375057 + 1013722453597032 = 68768533^2$
$A - B = 3715388677375057 - 1013722453597032 = 51977555^2$
$A + C = 3715388677375057 + 611234918180568 = 65777075^2$
$A - C = 3715388677375057 - 611234918180568 = 55714933^2$
$B + C = 1013722453597032 + 611234918180568 = 40310760^2$
$B - C = 1013722453597032 - 611234918180568 = 20062092^2$

(A, B, C) = (80490074757312032, 16267875571047968, 9786732956312032)

$A + B = 80490074757312032 + 16267875571047968 = 311059400^2$
$A - B = 80490074757312032 - 16267875571047968 = 253420992^2$
$A + C = 80490074757312032 + 9786732956312032 = 300460992^2$
$A - C = 80490074757312032 - 9786732956312032 = 265901000^2$
$B + C = 16267875571047968 + 9786732956312032 = 161414400^2$
$B - C = 16267875571047968 - 9786732956312032 = 80505544^2$

(A, B, C) = (1158832066700333057, 180038496396771968, 108191845395202432)

$A + B = 1158832066700333057 + 180038496396771968 = 1157095745^2$
$A - B = 1158832066700333057 - 180038496396771968 = 989339967^2$
$A + C = 1158832066700333057 + 108191845395202432 = 1125621567^2$
$A - C = 1158832066700333057 - 108191845395202432 = 1025007425^2$
$B + C = 180038496396771968 + 108191845395202432 = 536870880^2$
$B - C = 180038496396771968 - 108191845395202432 = 268042256^2$

(A, B, C) = (12196577188187456032, 1500397519145382432, 901116534029503968)

$A + B = 12196577188187456032 + 1500397519145382432 = 3700942408^2$
$A - B = 12196577188187456032 - 1500397519145382432 = 3270501440^2$
$A + C = 12196577188187456032 + 901116534029503968 = 3619073600^2$
$A - C = 12196577188187456032 - 901116534029503968 = 3360872008^2$
$B + C = 1500397519145382432 + 901116534029503968 = 1549681920^2$
$B - C = 1500397519145382432 - 901116534029503968 = 774132408^2$

A = 100209993999400210001,
B = 9997600599976001000,
C = 6002399080024000600

$A + B = 100209993999400210001 + 9997600599976001000 = 10497980501^2$
$A - B = 100209993999400210001 - 9997600599976001000 = 9498020499^2$
$A + C = 100209993999400210001 + 6002399080024000600 = 10305939699^2$
$A - C = 100209993999400210001 - 6002399080024000600 = 9706059701^2$
$B + C = 9997600599976001000 + 6002399080024000600 = 3999999960^2$
$B - C = 9997600599976001000 - 6002399080024000600 = 1998800020^2$

A = 673714920427838918432
B = 55590060695119276832
C = 33368615490758841568

$A + B = 673714920427838918432 + 55590060695119276832 = 27005647208^2$
$A - B = 673714920427838918432 - 55590060695119276832 = 24862116960^2$
$A + C = 673714920427838918432 + 33368615490758841568 = 26591042400^2$
$A - C = 673714920427838918432 - 33368615490758841568 = 25305064808^2$
$B + C = 55590060695119276832 + 33368615490758841568 = 9431790720^2$
$B - C = 55590060695119276832 - 33368615490758841568 = 4713962792^2$

A = 3837642508385088742657,
B = 266202522480673457568,
C = 159770808032134792032

$A + B = 3837642508385088742657 + 266202522480673457568 = 64061259985^2$
$A - B = 3837642508385088742657 - 266202522480673457568 = 59761525967^2$
$A + C = 3837642508385088742657 + 159770808032134792032 = 63225100367^2$
$A - C = 3837642508385088742657 - 159770808032134792032 = 60645459025^2$
$B + C = 266202522480673457568 + 159770808032134792032 = 20639121360^2$
$B - C = 266202522480673457568 - 159770808032134792032 = 10316574744^2$

A = 19018937383880241696032,
B = 1124459580757710817568,
C = 674826926062135736032

$A + B = 19018937383880241696032 + 1124459580757710817568 = 141927435560^2$
$A - B = 19018937383880241696032 - 1124459580757710817568 = 133770242592^2$
$A + C = 19018937383880241696032 + 674826926062135736032 = 140334472992^2$
$A - C = 19018937383880241696032 - 674826926062135736032 = 135440431400^2$
$B + C = 1124459580757710817568 + 674826926062135736032 = 42417997440^2$
$B - C = 1124459580757710817568 - 674826926062135736032 = 21204543256^2$

A = 83713992105206470175057,
B = 4268521871297652625832,
C = 2561539786816386300568

$A + B = 83713992105206470175057 + 4268521871297652625832 = 296618465333^2$
$A - B = 83713992105206470175057 - 4268521871297652625832 = 281860728435^2$
$A + C = 83713992105206470175057 + 2561539786816386300568 = 293726968275^2$
$A - C = 83713992105206470175057 - 2561539786816386300568 = 284872694933^2$
$B + C = 4268521871297652625832 + 2561539786816386300568 = 82644187080^2$
$B - C = 4268521871297652625832 - 2561539786816386300568 = 41315639708^2$

A = 332663608804941654969376,
B = 14778218204728144596000,
C = 8868051857385449157600

$A + B = 332663608804941654969376 + 14778218204728144596000 = 589441962376^2$
$A - B = 332663608804941654969376 - 14778218204728144596000 = 563813258624^2$
$A + C = 332663608804941654969376 + 8868051857385449157600 = 584407101824^2$
$A - C = 332663608804941654969376 - 8868051857385449157600 = 569030365576^2$
$B + C = 14778218204728144596000 + 8868051857385449157600 = 153773437440^2$
$B - C = 14778218204728144596000 - 8868051857385449157600 = 76877606280^2$

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