Make {(x^2 + 13xy + y^2), (x^2 + 17xz + z^2), (y^2 + 19yz + z^2)} squares

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

x^2 \; + \; 13 \, x \, y \; + \; y^2
x^2 \; + \; 17 \, x \, z \; + \; z^2
y^2 \; + \; 19 \, y \, z \; + \; z^2

are square numbers.
 

for example,

 
(x,   y,   z)   =   (15,   5,   25)

15^2 \; + \; 13 \,(15) \,(5) \; + \; 5^2 \; = \; 35^2
15^2 \; + \; 17 \,(15) \,(25) \; + \; 25^2 \; = \; 85^2
5^2 \; + \; 19 \,(5) \,(25) \; + \; 25^2 \; = \; 55^2

(x,   y,   z)   =   (144,   5040,   1008)

144^2 \; + \; 13 \,(144) \,(5040) \; + \; 5040^2 \; = \; 5904^2
144^2 \; + \; 17 \,(144) \,(1008) \; + \; 1008^2 \; = \; 1872^2
5040^2 \; + \; 19 \,(5040) \,(1008) \; + \; 1008^2 \; = \; 11088^2

(x,   y,   z)   =   (320,   2301,   920)

320^2 \; + \; 13 \,(320) \,(2301) \; + \; 2301^2 \; = \; 3869^2
320^2 \; + \; 17 \,(320) \,(920) \; + \; 920^2 \; = \; 2440^2
2301^2 \; + \; 19 \,(2301) \,(920) \; + \; 920^2 \; = \; 6809^2

(x,   y,   z)   =   (595,   95513,   15045)

595^2 \; + \; 13 \,(595) \,(95513) \; + \; 95513^2 \; = \; 99307^2
595^2 \; + \; 17 \,(595) \,(15045) \; + \; 15045^2 \; = \; 19465^2
95513^2 \; + \; 19 \,(95513) \,(15045) \; + \; 15045^2 \; = \; 191447^2

(x,   y,   z)   =   (1000,   96725,   16000)

1000^2 \; + \; 13 \,(1000) \,(96725) \; + \; 96725^2 \; = \; 103025^2
1000^2 \; + \; 17 \,(1000) \,(16000) \; + \; 16000^2 \; = \; 23000^2
96725^2 \; + \; 19 \,(96725) \,(16000) \; + \; 16000^2 \; = \; 197525^2

 
 
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 Make {(x^2 + 13xy + y^2), (x^2 + 17xz + z^2), (y^2 + 19yz + z^2)} squares

  1. paul says:

    and some more

    (x y z) = (1, 3, 63)
    1^2 + 13(1)(3) + 3^2 = 7^2
    1^2 + 17(1)(63) + 63^2 = 71^2
    3^2 + 19(3)(63) + 63^2 = 87^2

    (x y z) = (1, 35, 63)
    1^2 + 13(1)(35) + 35^2 = 41^2
    1^2 + 17(1)(63) + 63^2 = 71^2
    35^2 + 19(35)(63) + 63^2 = 217^2

    (x y z) = (2, 6, 126)
    2^2 + 13(2)(6) + 6^2 = 14^2
    2^2 + 17(2)(126) + 126^2 = 142^2
    6^2 + 19(6)(126) + 126^2 = 174^2

    (x y z) = (2, 70, 126)
    2^2 + 13(2)(70) + 70^2 = 82^2
    2^2 + 17(2)(126) + 126^2 = 142^2
    70^2 + 19(70)(126) + 126^2 = 434^2

    (x y z) = (3, 9, 189)
    3^2 + 13(3)(9) + 9^2 = 21^2
    3^2 + 17(3)(189) + 189^2 = 213^2
    9^2 + 19(9)(189) + 189^2 = 261^2

    (x y z) = (3, 105, 189)
    3^2 + 13(3)(105) + 105^2 = 123^2
    3^2 + 17(3)(189) + 189^2 = 213^2
    105^2 + 19(105)(189) + 189^2 = 651^2

    (x y z) = (4, 12, 252)
    4^2 + 13(4)(12) + 12^2 = 28^2
    4^2 + 17(4)(252) + 252^2 = 284^2
    12^2 + 19(12)(252) + 252^2 = 348^2

    (x y z) = (4, 140, 252)
    4^2 + 13(4)(140) + 140^2 = 164^2
    4^2 + 17(4)(252) + 252^2 = 284^2
    140^2 + 19(140)(252) + 252^2 = 868^2

    (x y z) = (5, 7, 35)
    5^2 + 13(5)(7) + 7^2 = 23^2
    5^2 + 17(5)(35) + 35^2 = 65^2
    7^2 + 19(7)(35) + 35^2 = 77^2

    (x y z) = (5, 15, 56)
    5^2 + 13(5)(15) + 15^2 = 35^2
    5^2 + 17(5)(56) + 56^2 = 89^2
    15^2 + 19(15)(56) + 56^2 = 139^2

    (x y z) = (8, 15, 56)
    8^2 + 13(8)(15) + 15^2 = 43^2
    8^2 + 17(8)(56) + 56^2 = 104^2
    15^2 + 19(15)(56) + 56^2 = 139^2

    (x y z) = (8, 24, 39)
    8^2 + 13(8)(24) + 24^2 = 56^2
    8^2 + 17(8)(39) + 39^2 = 83^2
    24^2 + 19(24)(39) + 39^2 = 141^2

    (x y z) = (8, 24, 165)
    8^2 + 13(8)(24) + 24^2 = 56^2
    8^2 + 17(8)(165) + 165^2 = 223^2
    24^2 + 19(24)(165) + 165^2 = 321^2

    (x y z) = (10, 14, 70)
    10^2 + 13(10)(14) + 14^2 = 46^2
    10^2 + 17(10)(70) + 70^2 = 130^2
    14^2 + 19(14)(70) + 70^2 = 154^2

    (x y z) = (10, 30, 112)
    10^2 + 13(10)(30) + 30^2 = 70^2
    10^2 + 17(10)(112) + 112^2 = 178^2
    30^2 + 19(30)(112) + 112^2 = 278^2

    (x y z) = (13, 39, 40)
    13^2 + 13(13)(39) + 39^2 = 91^2
    13^2 + 17(13)(40) + 40^2 = 103^2
    39^2 + 19(39)(40) + 40^2 = 181^2

    (x y z) = (13, 56, 91)
    13^2 + 13(13)(56) + 56^2 = 113^2
    13^2 + 17(13)(91) + 91^2 = 169^2
    56^2 + 19(56)(91) + 91^2 = 329^2

    (x y z) = (15, 21, 105)
    15^2 + 13(15)(21) + 21^2 = 69^2
    15^2 + 17(15)(105) + 105^2 = 195^2
    21^2 + 19(21)(105) + 105^2 = 231^2

    (x y z) = (15, 45, 168)
    15^2 + 13(15)(45) + 45^2 = 105^2
    15^2 + 17(15)(168) + 168^2 = 267^2
    45^2 + 19(45)(168) + 168^2 = 417^2

    (x y z) = (16, 30, 112)
    16^2 + 13(16)(30) + 30^2 = 86^2
    16^2 + 17(16)(112) + 112^2 = 208^2
    30^2 + 19(30)(112) + 112^2 = 278^2

    (x y z) = (16, 48, 78)
    16^2 + 13(16)(48) + 48^2 = 112^2
    16^2 + 17(16)(78) + 78^2 = 166^2
    48^2 + 19(48)(78) + 78^2 = 282^2

    (x y z) = (16, 77, 112)
    16^2 + 13(16)(77) + 77^2 = 149^2
    16^2 + 17(16)(112) + 112^2 = 208^2
    77^2 + 19(77)(112) + 112^2 = 427^2

    (x y z) = (16, 128, 165)
    16^2 + 13(16)(128) + 128^2 = 208^2
    16^2 + 17(16)(165) + 165^2 = 269^2
    128^2 + 19(128)(165) + 165^2 = 667^2

    (x y z) = (17, 120, 231)
    17^2 + 13(17)(120) + 120^2 = 203^2
    17^2 + 17(17)(231) + 231^2 = 347^2
    120^2 + 19(120)(231) + 231^2 = 771^2

    (x y z) = (20, 28, 140)
    20^2 + 13(20)(28) + 28^2 = 92^2
    20^2 + 17(20)(140) + 140^2 = 260^2
    28^2 + 19(28)(140) + 140^2 = 308^2

    (x y z) = (20, 60, 224)
    20^2 + 13(20)(60) + 60^2 = 140^2
    20^2 + 17(20)(224) + 224^2 = 356^2
    60^2 + 19(60)(224) + 224^2 = 556^2

    (x y z) = (24, 45, 104)
    24^2 + 13(24)(45) + 45^2 = 129^2
    24^2 + 17(24)(104) + 104^2 = 232^2
    45^2 + 19(45)(104) + 104^2 = 319^2

    (x y z) = (24, 45, 168)
    24^2 + 13(24)(45) + 45^2 = 129^2
    24^2 + 17(24)(168) + 168^2 = 312^2
    45^2 + 19(45)(168) + 168^2 = 417^2

    (x y z) = (24, 72, 117)
    24^2 + 13(24)(72) + 72^2 = 168^2
    24^2 + 17(24)(117) + 117^2 = 249^2
    72^2 + 19(72)(117) + 117^2 = 423^2

    (x y z) = (25, 35, 175)
    25^2 + 13(25)(35) + 35^2 = 115^2
    25^2 + 17(25)(175) + 175^2 = 325^2
    35^2 + 19(35)(175) + 175^2 = 385^2

    (x y z) = (25, 75, 280)
    25^2 + 13(25)(75) + 75^2 = 175^2
    25^2 + 17(25)(280) + 280^2 = 445^2
    75^2 + 19(75)(280) + 280^2 = 695^2

    (x y z) = (26, 78, 80)
    26^2 + 13(26)(78) + 78^2 = 182^2
    26^2 + 17(26)(80) + 80^2 = 206^2
    78^2 + 19(78)(80) + 80^2 = 362^2

    (x y z) = (26, 112, 182)
    26^2 + 13(26)(112) + 112^2 = 226^2
    26^2 + 17(26)(182) + 182^2 = 338^2
    112^2 + 19(112)(182) + 182^2 = 658^2

    (x y z) = (30, 42, 210)
    30^2 + 13(30)(42) + 42^2 = 138^2
    30^2 + 17(30)(210) + 210^2 = 390^2
    42^2 + 19(42)(210) + 210^2 = 462^2

    (x y z) = (32, 60, 224)
    32^2 + 13(32)(60) + 60^2 = 172^2
    32^2 + 17(32)(224) + 224^2 = 416^2
    60^2 + 19(60)(224) + 224^2 = 556^2

    (x y z) = (32, 96, 156)
    32^2 + 13(32)(96) + 96^2 = 224^2
    32^2 + 17(32)(156) + 156^2 = 332^2
    96^2 + 19(96)(156) + 156^2 = 564^2

    (x y z) = (32, 154, 224)
    32^2 + 13(32)(154) + 154^2 = 298^2
    32^2 + 17(32)(224) + 224^2 = 416^2
    154^2 + 19(154)(224) + 224^2 = 854^2

    (x y z) = (35, 49, 245)
    35^2 + 13(35)(49) + 49^2 = 161^2
    35^2 + 17(35)(245) + 245^2 = 455^2
    49^2 + 19(49)(245) + 245^2 = 539^2

    (x y z) = (39, 117, 120)
    39^2 + 13(39)(117) + 117^2 = 273^2
    39^2 + 17(39)(120) + 120^2 = 309^2
    117^2 + 19(117)(120) + 120^2 = 543^2

    (x y z) = (39, 168, 273)
    39^2 + 13(39)(168) + 168^2 = 339^2
    39^2 + 17(39)(273) + 273^2 = 507^2
    168^2 + 19(168)(273) + 273^2 = 987^2

    (x y z) = (40, 56, 280)
    40^2 + 13(40)(56) + 56^2 = 184^2
    40^2 + 17(40)(280) + 280^2 = 520^2
    56^2 + 19(56)(280) + 280^2 = 616^2

    (x y z) = (40, 75, 280)
    40^2 + 13(40)(75) + 75^2 = 215^2
    40^2 + 17(40)(280) + 280^2 = 520^2
    75^2 + 19(75)(280) + 280^2 = 695^2

    (x y z) = (40, 120, 195)
    40^2 + 13(40)(120) + 120^2 = 280^2
    40^2 + 17(40)(195) + 195^2 = 415^2
    120^2 + 19(120)(195) + 195^2 = 705^2

    (x y z) = (40, 273, 280)
    40^2 + 13(40)(273) + 273^2 = 467^2
    40^2 + 17(40)(280) + 280^2 = 520^2
    273^2 + 19(273)(280) + 280^2 = 1267^2

    (x y z) = (45, 255, 272)
    45^2 + 13(45)(255) + 255^2 = 465^2
    45^2 + 17(45)(272) + 272^2 = 533^2
    255^2 + 19(255)(272) + 272^2 = 1207^2

    (x y z) = (48, 90, 208)
    48^2 + 13(48)(90) + 90^2 = 258^2
    48^2 + 17(48)(208) + 208^2 = 464^2
    90^2 + 19(90)(208) + 208^2 = 638^2

    (x y z) = (48, 144, 234)
    48^2 + 13(48)(144) + 144^2 = 336^2
    48^2 + 17(48)(234) + 234^2 = 498^2
    144^2 + 19(144)(234) + 234^2 = 846^2

    (x y z) = (52, 156, 160)
    52^2 + 13(52)(156) + 156^2 = 364^2
    52^2 + 17(52)(160) + 160^2 = 412^2
    156^2 + 19(156)(160) + 160^2 = 724^2

    (x y z) = (55, 77, 105)
    55^2 + 13(55)(77) + 77^2 = 253^2
    55^2 + 17(55)(105) + 105^2 = 335^2
    77^2 + 19(77)(105) + 105^2 = 413^2

    (x y z) = (56, 168, 273)
    56^2 + 13(56)(168) + 168^2 = 392^2
    56^2 + 17(56)(273) + 273^2 = 581^2
    168^2 + 19(168)(273) + 273^2 = 987^2

    (x y z) = (65, 195, 200)
    65^2 + 13(65)(195) + 195^2 = 455^2
    65^2 + 17(65)(200) + 200^2 = 515^2
    195^2 + 19(195)(200) + 200^2 = 905^2

    (x y z) = (78, 234, 240)
    78^2 + 13(78)(234) + 234^2 = 546^2
    78^2 + 17(78)(240) + 240^2 = 618^2
    234^2 + 19(234)(240) + 240^2 = 1086^2

    (x y z) = (91, 273, 280)
    91^2 + 13(91)(273) + 273^2 = 637^2
    91^2 + 17(91)(280) + 280^2 = 721^2
    273^2 + 19(273)(280) + 280^2 = 1267^2

    (x y z) = (110, 154, 210)
    110^2 + 13(110)(154) + 154^2 = 506^2
    110^2 + 17(110)(210) + 210^2 = 670^2
    154^2 + 19(154)(210) + 210^2 = 826^2

    Paul.

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