To make {(x^2 + 11xy + y^2), (x^2 + 13xz + z^2), (y^2 + 17yz + z^2)} squares

 
 
Find integers   x, \; y, \; z   such that each of

x^2 \; + \; 11 \, x \, y \; + \; y^2
x^2 \; + \; 13 \, x \, z \; + \; z^2
y^2 \; + \; 17 \, y \, z \; + \; z^2

is a square number
 

For example,

 
(x,   y,   z)   =   (704,   1615,   1672)

704^2 \; + \; 11 \,(704) \,(1615) \; + \; 1615^2 \; = \; 3951^2
704^2 \; + \; 13 \,(704) \,(1672) \; + \; 1672^2 \; = \; 4312^2
1615^2 \; + \; 17 \,(1615) \,(1672) \; + \; 1672^2 \; = \; 7163^2

 
 
Find other solutions.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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One Response to To make {(x^2 + 11xy + y^2), (x^2 + 13xz + z^2), (y^2 + 17yz + z^2)} squares

  1. paul says:

    Some more

    (x y z) = (1, 5, 35)
    1^2 + 11(1)(5) + 5^2 = 9^2
    1^2 + 13(1)(35) + 35^2 = 41^2
    5^2 + 17(5)(35) + 35^2 = 65^2

    (x y z) = (2, 10, 70)
    2^2 + 11(2)(10) + 10^2 = 18^2
    2^2 + 13(2)(70) + 70^2 = 82^2
    10^2 + 17(10)(70) + 70^2 = 130^2

    (x y z) = (3, 7, 24)
    3^2 + 11(3)(7) + 7^2 = 17^2
    3^2 + 13(3)(24) + 24^2 = 39^2
    7^2 + 17(7)(24) + 24^2 = 59^2

    (x y z) = (3, 15, 105)
    3^2 + 11(3)(15) + 15^2 = 27^2
    3^2 + 13(3)(105) + 105^2 = 123^2
    15^2 + 17(15)(105) + 105^2 = 195^2

    (x y z) = (4, 20, 140)
    4^2 + 11(4)(20) + 20^2 = 36^2
    4^2 + 13(4)(140) + 140^2 = 164^2
    20^2 + 17(20)(140) + 140^2 = 260^2

    (x y z) = (5, 25, 175)
    5^2 + 11(5)(25) + 25^2 = 45^2
    5^2 + 13(5)(175) + 175^2 = 205^2
    25^2 + 17(25)(175) + 175^2 = 325^2

    (x y z) = (6, 14, 48)
    6^2 + 11(6)(14) + 14^2 = 34^2
    6^2 + 13(6)(48) + 48^2 = 78^2
    14^2 + 17(14)(48) + 48^2 = 118^2

    (x y z) = (6, 30, 210)
    6^2 + 11(6)(30) + 30^2 = 54^2
    6^2 + 13(6)(210) + 210^2 = 246^2
    30^2 + 17(30)(210) + 210^2 = 390^2

    (x y z) = (7, 8, 56)
    7^2 + 11(7)(8) + 8^2 = 27^2
    7^2 + 13(7)(56) + 56^2 = 91^2
    8^2 + 17(8)(56) + 56^2 = 104^2

    (x y z) = (7, 35, 245)
    7^2 + 11(7)(35) + 35^2 = 63^2
    7^2 + 13(7)(245) + 245^2 = 287^2
    35^2 + 17(35)(245) + 245^2 = 455^2

    (x y z) = (8, 40, 280)
    8^2 + 11(8)(40) + 40^2 = 72^2
    8^2 + 13(8)(280) + 280^2 = 328^2
    40^2 + 17(40)(280) + 280^2 = 520^2

    (x y z) = (9, 21, 72)
    9^2 + 11(9)(21) + 21^2 = 51^2
    9^2 + 13(9)(72) + 72^2 = 117^2
    21^2 + 17(21)(72) + 72^2 = 177^2

    (x y z) = (12, 28, 96)
    12^2 + 11(12)(28) + 28^2 = 68^2
    12^2 + 13(12)(96) + 96^2 = 156^2
    28^2 + 17(28)(96) + 96^2 = 236^2

    (x y z) = (14, 16, 112)
    14^2 + 11(14)(16) + 16^2 = 54^2
    14^2 + 13(14)(112) + 112^2 = 182^2
    16^2 + 17(16)(112) + 112^2 = 208^2

    (x y z) = (15, 35, 120)
    15^2 + 11(15)(35) + 35^2 = 85^2
    15^2 + 13(15)(120) + 120^2 = 195^2
    35^2 + 17(35)(120) + 120^2 = 295^2

    (x y z) = (15, 168, 280)
    15^2 + 11(15)(168) + 168^2 = 237^2
    15^2 + 13(15)(280) + 280^2 = 365^2
    168^2 + 17(168)(280) + 280^2 = 952^2

    (x y z) = (18, 42, 144)
    18^2 + 11(18)(42) + 42^2 = 102^2
    18^2 + 13(18)(144) + 144^2 = 234^2
    42^2 + 17(42)(144) + 144^2 = 354^2

    (x y z) = (21, 24, 168)
    21^2 + 11(21)(24) + 24^2 = 81^2
    21^2 + 13(21)(168) + 168^2 = 273^2
    24^2 + 17(24)(168) + 168^2 = 312^2

    (x y z) = (21, 49, 168)
    21^2 + 11(21)(49) + 49^2 = 119^2
    21^2 + 13(21)(168) + 168^2 = 273^2
    49^2 + 17(49)(168) + 168^2 = 413^2

    (x y z) = (21, 64, 72)
    21^2 + 11(21)(64) + 64^2 = 139^2
    21^2 + 13(21)(72) + 72^2 = 159^2
    64^2 + 17(64)(72) + 72^2 = 296^2

    (x y z) = (24, 56, 161)
    24^2 + 11(24)(56) + 56^2 = 136^2
    24^2 + 13(24)(161) + 161^2 = 277^2
    56^2 + 17(56)(161) + 161^2 = 427^2

    (x y z) = (24, 56, 192)
    24^2 + 11(24)(56) + 56^2 = 136^2
    24^2 + 13(24)(192) + 192^2 = 312^2
    56^2 + 17(56)(192) + 192^2 = 472^2

    (x y z) = (27, 63, 216)
    27^2 + 11(27)(63) + 63^2 = 153^2
    27^2 + 13(27)(216) + 216^2 = 351^2
    63^2 + 17(63)(216) + 216^2 = 531^2

    (x y z) = (28, 32, 224)
    28^2 + 11(28)(32) + 32^2 = 108^2
    28^2 + 13(28)(224) + 224^2 = 364^2
    32^2 + 17(32)(224) + 224^2 = 416^2

    (x y z) = (30, 70, 240)
    30^2 + 11(30)(70) + 70^2 = 170^2
    30^2 + 13(30)(240) + 240^2 = 390^2
    70^2 + 17(70)(240) + 240^2 = 590^2

    (x y z) = (33, 77, 264)
    33^2 + 11(33)(77) + 77^2 = 187^2
    33^2 + 13(33)(264) + 264^2 = 429^2
    77^2 + 17(77)(264) + 264^2 = 649^2

    (x y z) = (35, 40, 280)
    35^2 + 11(35)(40) + 40^2 = 135^2
    35^2 + 13(35)(280) + 280^2 = 455^2
    40^2 + 17(40)(280) + 280^2 = 520^2

    (x y z) = (36, 84, 288)
    36^2 + 11(36)(84) + 84^2 = 204^2
    36^2 + 13(36)(288) + 288^2 = 468^2
    84^2 + 17(84)(288) + 288^2 = 708^2

    (x y z) = (40, 57, 95)
    40^2 + 11(40)(57) + 57^2 = 173^2
    40^2 + 13(40)(95) + 95^2 = 245^2
    57^2 + 17(57)(95) + 95^2 = 323^2

    (x y z) = (42, 128, 144)
    42^2 + 11(42)(128) + 128^2 = 278^2
    42^2 + 13(42)(144) + 144^2 = 318^2
    128^2 + 17(128)(144) + 144^2 = 592^2

    (x y z) = (63, 192, 216)
    63^2 + 11(63)(192) + 192^2 = 417^2
    63^2 + 13(63)(216) + 216^2 = 477^2
    192^2 + 17(192)(216) + 216^2 = 888^2

    (x y z) = (80, 114, 190)
    80^2 + 11(80)(114) + 114^2 = 346^2
    80^2 + 13(80)(190) + 190^2 = 490^2
    114^2 + 17(114)(190) + 190^2 = 646^2

    (x y z) = (84, 256, 288)
    84^2 + 11(84)(256) + 256^2 = 556^2
    84^2 + 13(84)(288) + 288^2 = 636^2
    256^2 + 17(256)(288) + 288^2 = 1184^2

    (x y z) = (112, 128, 221)
    112^2 + 11(112)(128) + 128^2 = 432^2
    112^2 + 13(112)(221) + 221^2 = 619^2
    128^2 + 17(128)(221) + 221^2 = 739^2

    (x y z) = (120, 171, 285)
    120^2 + 11(120)(171) + 171^2 = 519^2
    120^2 + 13(120)(285) + 285^2 = 735^2
    171^2 + 17(171)(285) + 285^2 = 969^2

    Paul.

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