To make {(x^2 + 3xy + y^2), (x^2 + 5xz + z^2), (y^2 + 7yz + z^2)} squares

 
 
To find integers   x, \; y, \; z   such that

x^2 \; + \; 3 \, x \, y \; + \; y^2
x^2 \; + \; 5 \, x \, z \; + \; z^2
y^2 \; + \; 7 \, y \, z \; + \; z^2

are all squares.
 

Here are some solutions:

(x,   y,   z)   =   (72,   -27,   216)

72^2 \; + \; 3 \,(72) \,(-27) \; + \; (-27)^2 \; = \; 9^2
72^2 \; + \; 5 \,(72) \,(216) \; + \; 216^2 \; = \; 360^2
(-27)^2 \; + \; 7 \,(-27) \,(216) \; + \; 216^2 \; = \; 81^2

(x,   y,   z)   =   (-1240,   6765,   14136)

(-1240)^2 \; + \; 3 \,(-1240) \,(6765) \; + \; 6765^2 \; = \; 4705^2
(-1240)^2 \; + \; 5 \,(-1240) \,(14136) \; + \; 14136^2 \; = \; 10664^2
6765^2 \; + \; 7 \,(6765) \,(14136) \; + \; 14136^2 \; = \; 30249^2

(x,   y,   z)   =   (-5184,   22932,   41472)

(-5184)^2 \; + \; 3 \,(-5184) \,(22932) \; + \; 22932^2 \; = \; 14004^2
(-5184)^2 \; + \; 5 \,(-5184) \,(41472) \; + \; 41472^2 \; = \; 25920^2
22932^2 \; + \; 7 \,(22932) \,(41472) \; + \; 41472^2 \; = \; 94356^2

(x,   y,   z)   =   (-13944,   57125,   96280)

(-13944)^2 \; + \; 3 \,(-13944) \,(57125) \; + \; 57125^2 \; = \; 32681^2
(-13944)^2 \; + \; 5 \,(-13944) \,(96280) \; + \; 96280^2 \; = \; 52456^2
57125^2 \; + \; 7 \,(57125) \,(96280) \; + \; 96280^2 \; = \; 225905^2

(x,   y,   z)   =   (-30208,   119028,   192576)

(-30208)^2 \; + \; 3 \,(-30208) \,(119028) \; + \; 119028^2 \; = \; 65524^2
(-30208)^2 \; + \; 5 \,(-30208) \,(192576) \; + \; 192576^2 \; = \; 94400^2
119028^2 \; + \; 7 \,(119028) \,(192576) \; + \; 192576^2 \; = \; 460116^2

(x,   y,   z)   =   (-57240,   220365,   347256)

(-57240)^2 \; + \; 3 \,(-57240) \,(220365) \; + \; 220365^2 \; = \; 118305^2
(-57240)^2 \; + \; 5 \,(-57240) \,(347256) \; + \; 347256^2 \; = \; 156456^2
220365^2 \; + \; 7 \,(220365) \,(347256) \; + \; 347256^2 \; = \; 839529^2

(x,   y,   z)   =   (-98880,   374900,   580096)

(-98880)^2 \; + \; 3 \,(-98880) \,(374900) \; + \; 374900^2 \; = \; 197780^2
(-98880)^2 \; + \; 5 \,(-98880) \,(580096) \; + \; 580096^2 \; = \; 243904^2
374900^2 \; + \; 7 \,(374900) \,(580096) \; + \; 580096^2 \; = \; 1414004^2

 

Find other solutions.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

One Response to To make {(x^2 + 3xy + y^2), (x^2 + 5xz + z^2), (y^2 + 7yz + z^2)} squares

  1. paul says:

    Here are a few all positive solutions

    (x y z) = (9, 21, 40)
    9^2 + 3(9)(21) + 21^2 = 33^2
    9^2 + 5(9)(40) + 40^2 = 59^2
    21^2 + 7(21)(40) + 40^2 = 89^2

    (x y z) = (13, 24, 39)
    13^2 + 3(13)(24) + 24^2 = 41^2
    13^2 + 5(13)(39) + 39^2 = 65^2
    24^2 + 7(24)(39) + 39^2 = 93^2

    (x y z) = (17, 48, 176)
    17^2 + 3(17)(48) + 48^2 = 71^2
    17^2 + 5(17)(176) + 176^2 = 215^2
    48^2 + 7(48)(176) + 176^2 = 304^2

    (x y z) = (18, 42, 80)
    18^2 + 3(18)(42) + 42^2 = 66^2
    18^2 + 5(18)(80) + 80^2 = 118^2
    42^2 + 7(42)(80) + 80^2 = 178^2

    (x y z) = (26, 48, 78)
    26^2 + 3(26)(48) + 48^2 = 82^2
    26^2 + 5(26)(78) + 78^2 = 130^2
    48^2 + 7(48)(78) + 78^2 = 186^2

    (x y z) = (27, 63, 120)
    27^2 + 3(27)(63) + 63^2 = 99^2
    27^2 + 5(27)(120) + 120^2 = 177^2
    63^2 + 7(63)(120) + 120^2 = 267^2

    (x y z) = (36, 84, 160)
    36^2 + 3(36)(84) + 84^2 = 132^2
    36^2 + 5(36)(160) + 160^2 = 236^2
    84^2 + 7(84)(160) + 160^2 = 356^2

    (x y z) = (39, 72, 117)
    39^2 + 3(39)(72) + 72^2 = 123^2
    39^2 + 5(39)(117) + 117^2 = 195^2
    72^2 + 7(72)(117) + 117^2 = 279^2

    (x y z) = (45, 105, 200)
    45^2 + 3(45)(105) + 105^2 = 165^2
    45^2 + 5(45)(200) + 200^2 = 295^2
    105^2 + 7(105)(200) + 200^2 = 445^2

    (x y z) = (52, 96, 156)
    52^2 + 3(52)(96) + 96^2 = 164^2
    52^2 + 5(52)(156) + 156^2 = 260^2
    96^2 + 7(96)(156) + 156^2 = 372^2

    (x y z) = (54, 126, 240)
    54^2 + 3(54)(126) + 126^2 = 198^2
    54^2 + 5(54)(240) + 240^2 = 354^2
    126^2 + 7(126)(240) + 240^2 = 534^2

    (x y z) = (63, 147, 280)
    63^2 + 3(63)(147) + 147^2 = 231^2
    63^2 + 5(63)(280) + 280^2 = 413^2
    147^2 + 7(147)(280) + 280^2 = 623^2

    (x y z) = (64, 72, 117)
    64^2 + 3(64)(72) + 72^2 = 152^2
    64^2 + 5(64)(117) + 117^2 = 235^2
    72^2 + 7(72)(117) + 117^2 = 279^2

    (x y z) = (65, 120, 195)
    65^2 + 3(65)(120) + 120^2 = 205^2
    65^2 + 5(65)(195) + 195^2 = 325^2
    120^2 + 7(120)(195) + 195^2 = 465^2

    (x y z) = (78, 144, 234)
    78^2 + 3(78)(144) + 144^2 = 246^2
    78^2 + 5(78)(234) + 234^2 = 390^2
    144^2 + 7(144)(234) + 234^2 = 558^2

    (x y z) = (91, 168, 273)
    91^2 + 3(91)(168) + 168^2 = 287^2
    91^2 + 5(91)(273) + 273^2 = 455^2
    168^2 + 7(168)(273) + 273^2 = 651^2

    (x y z) = (128, 144, 234)
    128^2 + 3(128)(144) + 144^2 = 304^2
    128^2 + 5(128)(234) + 234^2 = 470^2
    144^2 + 7(144)(234) + 234^2 = 558^2

    Paul.

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