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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About benvitalis

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

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