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

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

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

 
For example,
 

(x,   y,   z)   =   (8,   40,   9)

8^2 \; + \; 2 \,(8) \,(40) \; + \; 40^2 \; = \; 48^2
8^2 \; + \; 3 \,(8) \,(9) \; + \; 9^2 \; = \; 19^2
40^2 \; + \; 5 \,(40) \,(9) \; + \; 9^2 \; = \; 59^2

(x,   y,   z)   =   (48,   -21,   112)

48^2 \; + \; 2 \,(48) \,(-21) \; + \; (-21)^2 \; = \; 27^2
48^2 \; + \; 3 \,(48) \,(112) \; + \; 112^2 \; = \; 176^2
(-21)^2 \; + \; 5 \,(-21) \,(112) \; + \; 112^2 \; = \; 35^2

(x,   y,   z)   =   (656,   -464,   2337)

656^2 \; + \; 2 \,(656) \,(-464) \; + \; (-464)^2 \; = \; 192^2
656^2 \; + \; 3 \,(656) \,(2337) \; + \; 2337^2 \; = \; 3239^2
(-464)^2 \; + \; 5 \,(-464) \,(2337) \; + \; 2337^2 \; = \; 505^2

(x,   y,   z)   =   (924,   399,   1408)

924^2 \; + \; 2 \,(924) \,(399) \; + \; 399^2 \; = \; 1323^2
924^2 \; + \; 3 \,(924) \,(1408) \; + \; 1408^2 \; = \; 2596^2
399^2 \; + \; 5 \,(399) \,(1408) \; + \; 1408^2 \; = \; 2225^2

(x,   y,   z)   =   (1232,   -1085,   8976)

1232^2 \; + \; 2 \,(1232) \,(-1085) \; + \; (-1085)^2 \; = \; 147^2
1232^2 \; + \; 3 \,(1232) \,(8976) \; + \; 8976^2 \; = \; 10736^2
(-1085)^2 \; + \; 5 \,(-1085) \,(8976) \; + \; 8976^2 \; = \; 5749^2

(x,   y,   z)   =   (2844,   -2601,   27808)

2844^2 \; + \; 2 \,(2844) \,(-2601) \; + \; (-2601)^2 \; = \; 243^2
2844^2 \; + \; 3 \,(2844) \,(27808) \; + \; 27808^2 \; = \; 31916^2
(-2601)^2 \; + \; 5 \,(-2601) \,(27808) \; + \; 27808^2 \; = \; 20455^2

 
 
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 + 2xyy + y^2), (x^2 + 3xz + z^2), (y^2 + 5yz + z^2)} squares

  1. paul says:

    Here’s some more

    (x y z) = (5, 8, 24)
    5^2 + 2(5)(8) + 8^2 = 13^2
    5^2 + 3(5)(24) + 24^2 = 31^2
    8^2 + 5(8)(24) + 24^2 = 40^2

    (x y z) = (5, 15, 24)
    5^2 + 2(5)(15) + 15^2 = 20^2
    5^2 + 3(5)(24) + 24^2 = 31^2
    15^2 + 5(15)(24) + 24^2 = 51^2

    (x y z) = (7, 17, 51)
    7^2 + 2(7)(17) + 17^2 = 24^2
    7^2 + 3(7)(51) + 51^2 = 61^2
    17^2 + 5(17)(51) + 51^2 = 85^2

    (x y z) = (9, 55, 88)
    9^2 + 2(9)(55) + 55^2 = 64^2
    9^2 + 3(9)(88) + 88^2 = 101^2
    55^2 + 5(55)(88) + 88^2 = 187^2

    (x y z) = (9, 64, 88)
    9^2 + 2(9)(64) + 64^2 = 73^2
    9^2 + 3(9)(88) + 88^2 = 101^2
    64^2 + 5(64)(88) + 88^2 = 200^2

    (x y z) = (10, 16, 48)
    10^2 + 2(10)(16) + 16^2 = 26^2
    10^2 + 3(10)(48) + 48^2 = 62^2
    16^2 + 5(16)(48) + 48^2 = 80^2

    (x y z) = (10, 19, 48)
    10^2 + 2(10)(19) + 19^2 = 29^2
    10^2 + 3(10)(48) + 48^2 = 62^2
    19^2 + 5(19)(48) + 48^2 = 85^2

    (x y z) = (10, 30, 48)
    10^2 + 2(10)(30) + 30^2 = 40^2
    10^2 + 3(10)(48) + 48^2 = 62^2
    30^2 + 5(30)(48) + 48^2 = 102^2

    (x y z) = (10, 45, 48)
    10^2 + 2(10)(45) + 45^2 = 55^2
    10^2 + 3(10)(48) + 48^2 = 62^2
    45^2 + 5(45)(48) + 48^2 = 123^2

    (x y z) = (11, 13, 15)
    11^2 + 2(11)(13) + 13^2 = 24^2
    11^2 + 3(11)(15) + 15^2 = 29^2
    13^2 + 5(13)(15) + 15^2 = 37^2

    (x y z) = (11, 45, 135)
    11^2 + 2(11)(45) + 45^2 = 56^2
    11^2 + 3(11)(135) + 135^2 = 151^2
    45^2 + 5(45)(135) + 135^2 = 225^2

    (x y z) = (11, 117, 135)
    11^2 + 2(11)(117) + 117^2 = 128^2
    11^2 + 3(11)(135) + 135^2 = 151^2
    117^2 + 5(117)(135) + 135^2 = 333^2

    (x y z) = (13, 15, 24)
    13^2 + 2(13)(15) + 15^2 = 28^2
    13^2 + 3(13)(24) + 24^2 = 41^2
    15^2 + 5(15)(24) + 24^2 = 51^2

    (x y z) = (13, 26, 192)
    13^2 + 2(13)(26) + 26^2 = 39^2
    13^2 + 3(13)(192) + 192^2 = 211^2
    26^2 + 5(26)(192) + 192^2 = 250^2

    (x y z) = (13, 30, 192)
    13^2 + 2(13)(30) + 30^2 = 43^2
    13^2 + 3(13)(192) + 192^2 = 211^2
    30^2 + 5(30)(192) + 192^2 = 258^2

    (x y z) = (13, 64, 192)
    13^2 + 2(13)(64) + 64^2 = 77^2
    13^2 + 3(13)(192) + 192^2 = 211^2
    64^2 + 5(64)(192) + 192^2 = 320^2

    (x y z) = (13, 76, 192)
    13^2 + 2(13)(76) + 76^2 = 89^2
    13^2 + 3(13)(192) + 192^2 = 211^2
    76^2 + 5(76)(192) + 192^2 = 340^2

    (x y z) = (13, 120, 192)
    13^2 + 2(13)(120) + 120^2 = 133^2
    13^2 + 3(13)(192) + 192^2 = 211^2
    120^2 + 5(120)(192) + 192^2 = 408^2

    (x y z) = (13, 180, 192)
    13^2 + 2(13)(180) + 180^2 = 193^2
    13^2 + 3(13)(192) + 192^2 = 211^2
    180^2 + 5(180)(192) + 192^2 = 492^2

    (x y z) = (14, 34, 102)
    14^2 + 2(14)(34) + 34^2 = 48^2
    14^2 + 3(14)(102) + 102^2 = 122^2
    34^2 + 5(34)(102) + 102^2 = 170^2

    (x y z) = (15, 17, 35)
    15^2 + 2(15)(17) + 17^2 = 32^2
    15^2 + 3(15)(35) + 35^2 = 55^2
    17^2 + 5(17)(35) + 35^2 = 67^2

    (x y z) = (15, 24, 72)
    15^2 + 2(15)(24) + 24^2 = 39^2
    15^2 + 3(15)(72) + 72^2 = 93^2
    24^2 + 5(24)(72) + 72^2 = 120^2

    (x y z) = (15, 45, 72)
    15^2 + 2(15)(45) + 45^2 = 60^2
    15^2 + 3(15)(72) + 72^2 = 93^2
    45^2 + 5(45)(72) + 72^2 = 153^2

    (x y z) = (16, 19, 57)
    16^2 + 2(16)(19) + 19^2 = 35^2
    16^2 + 3(16)(57) + 57^2 = 79^2
    19^2 + 5(19)(57) + 57^2 = 95^2

    (x y z) = (17, 19, 48)
    17^2 + 2(17)(19) + 19^2 = 36^2
    17^2 + 3(17)(48) + 48^2 = 71^2
    19^2 + 5(19)(48) + 48^2 = 85^2

    (x y z) = (17, 30, 48)
    17^2 + 2(17)(30) + 30^2 = 47^2
    17^2 + 3(17)(48) + 48^2 = 71^2
    30^2 + 5(30)(48) + 48^2 = 102^2

    (x y z) = (17, 45, 48)
    17^2 + 2(17)(45) + 45^2 = 62^2
    17^2 + 3(17)(48) + 48^2 = 71^2
    45^2 + 5(45)(48) + 48^2 = 123^2

    (x y z) = (18, 35, 176)
    18^2 + 2(18)(35) + 35^2 = 53^2
    18^2 + 3(18)(176) + 176^2 = 202^2
    35^2 + 5(35)(176) + 176^2 = 251^2

    (x y z) = (18, 110, 176)
    18^2 + 2(18)(110) + 110^2 = 128^2
    18^2 + 3(18)(176) + 176^2 = 202^2
    110^2 + 5(110)(176) + 176^2 = 374^2

    (x y z) = (18, 128, 176)
    18^2 + 2(18)(128) + 128^2 = 146^2
    18^2 + 3(18)(176) + 176^2 = 202^2
    128^2 + 5(128)(176) + 176^2 = 400^2

    (x y z) = (18, 165, 176)
    18^2 + 2(18)(165) + 165^2 = 183^2
    18^2 + 3(18)(176) + 176^2 = 202^2
    165^2 + 5(165)(176) + 176^2 = 451^2

    (x y z) = (19, 21, 63)
    19^2 + 2(19)(21) + 21^2 = 40^2
    19^2 + 3(19)(63) + 63^2 = 89^2
    21^2 + 5(21)(63) + 63^2 = 105^2

    (x y z) = (20, 32, 96)
    20^2 + 2(20)(32) + 32^2 = 52^2
    20^2 + 3(20)(96) + 96^2 = 124^2
    32^2 + 5(32)(96) + 96^2 = 160^2

    (x y z) = (20, 38, 96)
    20^2 + 2(20)(38) + 38^2 = 58^2
    20^2 + 3(20)(96) + 96^2 = 124^2
    38^2 + 5(38)(96) + 96^2 = 170^2

    (x y z) = (20, 60, 96)
    20^2 + 2(20)(60) + 60^2 = 80^2
    20^2 + 3(20)(96) + 96^2 = 124^2
    60^2 + 5(60)(96) + 96^2 = 204^2

    (x y z) = (20, 90, 96)
    20^2 + 2(20)(90) + 90^2 = 110^2
    20^2 + 3(20)(96) + 96^2 = 124^2
    90^2 + 5(90)(96) + 96^2 = 246^2

    (x y z) = (21, 23, 80)
    21^2 + 2(21)(23) + 23^2 = 44^2
    21^2 + 3(21)(80) + 80^2 = 109^2
    23^2 + 5(23)(80) + 80^2 = 127^2

    (x y z) = (21, 30, 32)
    21^2 + 2(21)(30) + 30^2 = 51^2
    21^2 + 3(21)(32) + 32^2 = 59^2
    30^2 + 5(30)(32) + 32^2 = 82^2

    (x y z) = (21, 42, 80)
    21^2 + 2(21)(42) + 42^2 = 63^2
    21^2 + 3(21)(80) + 80^2 = 109^2
    42^2 + 5(42)(80) + 80^2 = 158^2

    (x y z) = (21, 50, 80)
    21^2 + 2(21)(50) + 50^2 = 71^2
    21^2 + 3(21)(80) + 80^2 = 109^2
    50^2 + 5(50)(80) + 80^2 = 170^2

    (x y z) = (21, 51, 153)
    21^2 + 2(21)(51) + 51^2 = 72^2
    21^2 + 3(21)(153) + 153^2 = 183^2
    51^2 + 5(51)(153) + 153^2 = 255^2

    (x y z) = (21, 75, 80)
    21^2 + 2(21)(75) + 75^2 = 96^2
    21^2 + 3(21)(80) + 80^2 = 109^2
    75^2 + 5(75)(80) + 80^2 = 205^2

    (x y z) = (21, 115, 153)
    21^2 + 2(21)(115) + 115^2 = 136^2
    21^2 + 3(21)(153) + 153^2 = 183^2
    115^2 + 5(115)(153) + 153^2 = 353^2

    (x y z) = (22, 26, 30)
    22^2 + 2(22)(26) + 26^2 = 48^2
    22^2 + 3(22)(30) + 30^2 = 58^2
    26^2 + 5(26)(30) + 30^2 = 74^2

    (x y z) = (22, 90, 270)
    22^2 + 2(22)(90) + 90^2 = 112^2
    22^2 + 3(22)(270) + 270^2 = 302^2
    90^2 + 5(90)(270) + 270^2 = 450^2

    (x y z) = (22, 234, 270)
    22^2 + 2(22)(234) + 234^2 = 256^2
    22^2 + 3(22)(270) + 270^2 = 302^2
    234^2 + 5(234)(270) + 270^2 = 666^2

    (x y z) = (23, 25, 99)
    23^2 + 2(23)(25) + 25^2 = 48^2
    23^2 + 3(23)(99) + 99^2 = 131^2
    25^2 + 5(25)(99) + 99^2 = 151^2

    (x y z) = (23, 33, 99)
    23^2 + 2(23)(33) + 33^2 = 56^2
    23^2 + 3(23)(99) + 99^2 = 131^2
    33^2 + 5(33)(99) + 99^2 = 165^2

    (x y z) = (23, 72, 99)
    23^2 + 2(23)(72) + 72^2 = 95^2
    23^2 + 3(23)(99) + 99^2 = 131^2
    72^2 + 5(72)(99) + 99^2 = 225^2

    (x y z) = (24, 35, 56)
    24^2 + 2(24)(35) + 35^2 = 59^2
    24^2 + 3(24)(56) + 56^2 = 88^2
    35^2 + 5(35)(56) + 56^2 = 119^2

    (x y z) = (25, 27, 120)
    25^2 + 2(25)(27) + 27^2 = 52^2
    25^2 + 3(25)(120) + 120^2 = 155^2
    27^2 + 5(27)(120) + 120^2 = 177^2

    (x y z) = (25, 40, 120)
    25^2 + 2(25)(40) + 40^2 = 65^2
    25^2 + 3(25)(120) + 120^2 = 155^2
    40^2 + 5(40)(120) + 120^2 = 200^2

    (x y z) = (25, 63, 120)
    25^2 + 2(25)(63) + 63^2 = 88^2
    25^2 + 3(25)(120) + 120^2 = 155^2
    63^2 + 5(63)(120) + 120^2 = 237^2

    (x y z) = (25, 75, 120)
    25^2 + 2(25)(75) + 75^2 = 100^2
    25^2 + 3(25)(120) + 120^2 = 155^2
    75^2 + 5(75)(120) + 120^2 = 255^2

    (x y z) = (25, 104, 120)
    25^2 + 2(25)(104) + 104^2 = 129^2
    25^2 + 3(25)(120) + 120^2 = 155^2
    104^2 + 5(104)(120) + 120^2 = 296^2

    (x y z) = (26, 30, 48)
    26^2 + 2(26)(30) + 30^2 = 56^2
    26^2 + 3(26)(48) + 48^2 = 82^2
    30^2 + 5(30)(48) + 48^2 = 102^2

    (x y z) = (26, 45, 48)
    26^2 + 2(26)(45) + 45^2 = 71^2
    26^2 + 3(26)(48) + 48^2 = 82^2
    45^2 + 5(45)(48) + 48^2 = 123^2

    (x y z) = (27, 29, 143)
    27^2 + 2(27)(29) + 29^2 = 56^2
    27^2 + 3(27)(143) + 143^2 = 181^2
    29^2 + 5(29)(143) + 143^2 = 205^2

    (x y z) = (27, 88, 264)
    27^2 + 2(27)(88) + 88^2 = 115^2
    27^2 + 3(27)(264) + 264^2 = 303^2
    88^2 + 5(88)(264) + 264^2 = 440^2

    (x y z) = (27, 104, 143)
    27^2 + 2(27)(104) + 104^2 = 131^2
    27^2 + 3(27)(143) + 143^2 = 181^2
    104^2 + 5(104)(143) + 143^2 = 325^2

    (x y z) = (27, 165, 264)
    27^2 + 2(27)(165) + 165^2 = 192^2
    27^2 + 3(27)(264) + 264^2 = 303^2
    165^2 + 5(165)(264) + 264^2 = 561^2

    (x y z) = (27, 192, 264)
    27^2 + 2(27)(192) + 192^2 = 219^2
    27^2 + 3(27)(264) + 264^2 = 303^2
    192^2 + 5(192)(264) + 264^2 = 600^2

    (x y z) = (27, 217, 264)
    27^2 + 2(27)(217) + 217^2 = 244^2
    27^2 + 3(27)(264) + 264^2 = 303^2
    217^2 + 5(217)(264) + 264^2 = 635^2

    (x y z) = (28, 68, 204)
    28^2 + 2(28)(68) + 68^2 = 96^2
    28^2 + 3(28)(204) + 204^2 = 244^2
    68^2 + 5(68)(204) + 204^2 = 340^2

    (x y z) = (29, 31, 168)
    29^2 + 2(29)(31) + 31^2 = 60^2
    29^2 + 3(29)(168) + 168^2 = 209^2
    31^2 + 5(31)(168) + 168^2 = 235^2

    (x y z) = (29, 56, 168)
    29^2 + 2(29)(56) + 56^2 = 85^2
    29^2 + 3(29)(168) + 168^2 = 209^2
    56^2 + 5(56)(168) + 168^2 = 280^2

    (x y z) = (29, 105, 168)
    29^2 + 2(29)(105) + 105^2 = 134^2
    29^2 + 3(29)(168) + 168^2 = 209^2
    105^2 + 5(105)(168) + 168^2 = 357^2

    (x y z) = (30, 34, 70)
    30^2 + 2(30)(34) + 34^2 = 64^2
    30^2 + 3(30)(70) + 70^2 = 110^2
    34^2 + 5(34)(70) + 70^2 = 134^2

    (x y z) = (30, 48, 144)
    30^2 + 2(30)(48) + 48^2 = 78^2
    30^2 + 3(30)(144) + 144^2 = 186^2
    48^2 + 5(48)(144) + 144^2 = 240^2

    (x y z) = (30, 57, 144)
    30^2 + 2(30)(57) + 57^2 = 87^2
    30^2 + 3(30)(144) + 144^2 = 186^2
    57^2 + 5(57)(144) + 144^2 = 255^2

    (x y z) = (30, 90, 144)
    30^2 + 2(30)(90) + 90^2 = 120^2
    30^2 + 3(30)(144) + 144^2 = 186^2
    90^2 + 5(90)(144) + 144^2 = 306^2

    (x y z) = (30, 135, 144)
    30^2 + 2(30)(135) + 135^2 = 165^2
    30^2 + 3(30)(144) + 144^2 = 186^2
    135^2 + 5(135)(144) + 144^2 = 369^2

    (x y z) = (31, 33, 195)
    31^2 + 2(31)(33) + 33^2 = 64^2
    31^2 + 3(31)(195) + 195^2 = 239^2
    33^2 + 5(33)(195) + 195^2 = 267^2

    (x y z) = (31, 65, 195)
    31^2 + 2(31)(65) + 65^2 = 96^2
    31^2 + 3(31)(195) + 195^2 = 239^2
    65^2 + 5(65)(195) + 195^2 = 325^2

    (x y z) = (31, 169, 195)
    31^2 + 2(31)(169) + 169^2 = 200^2
    31^2 + 3(31)(195) + 195^2 = 239^2
    169^2 + 5(169)(195) + 195^2 = 481^2

    (x y z) = (32, 38, 114)
    32^2 + 2(32)(38) + 38^2 = 70^2
    32^2 + 3(32)(114) + 114^2 = 158^2
    38^2 + 5(38)(114) + 114^2 = 190^2

    (x y z) = (32, 91, 273)
    32^2 + 2(32)(91) + 91^2 = 123^2
    32^2 + 3(32)(273) + 273^2 = 319^2
    91^2 + 5(91)(273) + 273^2 = 455^2

    (x y z) = (32, 216, 273)
    32^2 + 2(32)(216) + 216^2 = 248^2
    32^2 + 3(32)(273) + 273^2 = 319^2
    216^2 + 5(216)(273) + 273^2 = 645^2

    (x y z) = (33, 35, 224)
    33^2 + 2(33)(35) + 35^2 = 68^2
    33^2 + 3(33)(224) + 224^2 = 271^2
    35^2 + 5(35)(224) + 224^2 = 301^2

    (x y z) = (33, 39, 45)
    33^2 + 2(33)(39) + 39^2 = 72^2
    33^2 + 3(33)(45) + 45^2 = 87^2
    39^2 + 5(39)(45) + 45^2 = 111^2

    (x y z) = (33, 56, 77)
    33^2 + 2(33)(56) + 56^2 = 89^2
    33^2 + 3(33)(77) + 77^2 = 121^2
    56^2 + 5(56)(77) + 77^2 = 175^2

    (x y z) = (33, 65, 104)
    33^2 + 2(33)(65) + 65^2 = 98^2
    33^2 + 3(33)(104) + 104^2 = 149^2
    65^2 + 5(65)(104) + 104^2 = 221^2

    (x y z) = (33, 140, 224)
    33^2 + 2(33)(140) + 140^2 = 173^2
    33^2 + 3(33)(224) + 224^2 = 271^2
    140^2 + 5(140)(224) + 224^2 = 476^2

    (x y z) = (33, 210, 224)
    33^2 + 2(33)(210) + 210^2 = 243^2
    33^2 + 3(33)(224) + 224^2 = 271^2
    210^2 + 5(210)(224) + 224^2 = 574^2

    (x y z) = (33, 222, 224)
    33^2 + 2(33)(222) + 222^2 = 255^2
    33^2 + 3(33)(224) + 224^2 = 271^2
    222^2 + 5(222)(224) + 224^2 = 590^2

    (x y z) = (34, 38, 96)
    34^2 + 2(34)(38) + 38^2 = 72^2
    34^2 + 3(34)(96) + 96^2 = 142^2
    38^2 + 5(38)(96) + 96^2 = 170^2

    (x y z) = (34, 60, 96)
    34^2 + 2(34)(60) + 60^2 = 94^2
    34^2 + 3(34)(96) + 96^2 = 142^2
    60^2 + 5(60)(96) + 96^2 = 204^2

    (x y z) = (34, 90, 96)
    34^2 + 2(34)(90) + 90^2 = 124^2
    34^2 + 3(34)(96) + 96^2 = 142^2
    90^2 + 5(90)(96) + 96^2 = 246^2

    (x y z) = (35, 37, 255)
    35^2 + 2(35)(37) + 37^2 = 72^2
    35^2 + 3(35)(255) + 255^2 = 305^2
    37^2 + 5(37)(255) + 255^2 = 337^2

    (x y z) = (35, 56, 168)
    35^2 + 2(35)(56) + 56^2 = 91^2
    35^2 + 3(35)(168) + 168^2 = 217^2
    56^2 + 5(56)(168) + 168^2 = 280^2

    (x y z) = (35, 85, 255)
    35^2 + 2(35)(85) + 85^2 = 120^2
    35^2 + 3(35)(255) + 255^2 = 305^2
    85^2 + 5(85)(255) + 255^2 = 425^2

    (x y z) = (35, 105, 168)
    35^2 + 2(35)(105) + 105^2 = 140^2
    35^2 + 3(35)(168) + 168^2 = 217^2
    105^2 + 5(105)(168) + 168^2 = 357^2

    (x y z) = (35, 144, 255)
    35^2 + 2(35)(144) + 144^2 = 179^2
    35^2 + 3(35)(255) + 255^2 = 305^2
    144^2 + 5(144)(255) + 255^2 = 519^2

    (x y z) = (35, 221, 255)
    35^2 + 2(35)(221) + 221^2 = 256^2
    35^2 + 3(35)(255) + 255^2 = 305^2
    221^2 + 5(221)(255) + 255^2 = 629^2

    (x y z) = (37, 39, 288)
    37^2 + 2(37)(39) + 39^2 = 76^2
    37^2 + 3(37)(288) + 288^2 = 341^2
    39^2 + 5(39)(288) + 288^2 = 375^2

    (x y z) = (37, 45, 288)
    37^2 + 2(37)(45) + 45^2 = 82^2
    37^2 + 3(37)(288) + 288^2 = 341^2
    45^2 + 5(45)(288) + 288^2 = 387^2

    (x y z) = (37, 96, 288)
    37^2 + 2(37)(96) + 96^2 = 133^2
    37^2 + 3(37)(288) + 288^2 = 341^2
    96^2 + 5(96)(288) + 288^2 = 480^2

    (x y z) = (37, 114, 288)
    37^2 + 2(37)(114) + 114^2 = 151^2
    37^2 + 3(37)(288) + 288^2 = 341^2
    114^2 + 5(114)(288) + 288^2 = 510^2

    (x y z) = (37, 180, 288)
    37^2 + 2(37)(180) + 180^2 = 217^2
    37^2 + 3(37)(288) + 288^2 = 341^2
    180^2 + 5(180)(288) + 288^2 = 612^2

    (x y z) = (37, 270, 288)
    37^2 + 2(37)(270) + 270^2 = 307^2
    37^2 + 3(37)(288) + 288^2 = 341^2
    270^2 + 5(270)(288) + 288^2 = 738^2

    (x y z) = (38, 42, 126)
    38^2 + 2(38)(42) + 42^2 = 80^2
    38^2 + 3(38)(126) + 126^2 = 178^2
    42^2 + 5(42)(126) + 126^2 = 210^2

    (x y z) = (39, 45, 72)
    39^2 + 2(39)(45) + 45^2 = 84^2
    39^2 + 3(39)(72) + 72^2 = 123^2
    45^2 + 5(45)(72) + 72^2 = 153^2

    (x y z) = (39, 72, 91)
    39^2 + 2(39)(72) + 72^2 = 111^2
    39^2 + 3(39)(91) + 91^2 = 143^2
    72^2 + 5(72)(91) + 91^2 = 215^2

    (x y z) = (40, 64, 192)
    40^2 + 2(40)(64) + 64^2 = 104^2
    40^2 + 3(40)(192) + 192^2 = 248^2
    64^2 + 5(64)(192) + 192^2 = 320^2

    (x y z) = (40, 76, 192)
    40^2 + 2(40)(76) + 76^2 = 116^2
    40^2 + 3(40)(192) + 192^2 = 248^2
    76^2 + 5(76)(192) + 192^2 = 340^2

    (x y z) = (40, 120, 192)
    40^2 + 2(40)(120) + 120^2 = 160^2
    40^2 + 3(40)(192) + 192^2 = 248^2
    120^2 + 5(120)(192) + 192^2 = 408^2

    (x y z) = (40, 180, 192)
    40^2 + 2(40)(180) + 180^2 = 220^2
    40^2 + 3(40)(192) + 192^2 = 248^2
    180^2 + 5(180)(192) + 192^2 = 492^2

    (x y z) = (42, 46, 160)
    42^2 + 2(42)(46) + 46^2 = 88^2
    42^2 + 3(42)(160) + 160^2 = 218^2
    46^2 + 5(46)(160) + 160^2 = 254^2

    (x y z) = (42, 60, 64)
    42^2 + 2(42)(60) + 60^2 = 102^2
    42^2 + 3(42)(64) + 64^2 = 118^2
    60^2 + 5(60)(64) + 64^2 = 164^2

    (x y z) = (42, 84, 160)
    42^2 + 2(42)(84) + 84^2 = 126^2
    42^2 + 3(42)(160) + 160^2 = 218^2
    84^2 + 5(84)(160) + 160^2 = 316^2

    (x y z) = (42, 100, 160)
    42^2 + 2(42)(100) + 100^2 = 142^2
    42^2 + 3(42)(160) + 160^2 = 218^2
    100^2 + 5(100)(160) + 160^2 = 340^2

    (x y z) = (42, 123, 160)
    42^2 + 2(42)(123) + 123^2 = 165^2
    42^2 + 3(42)(160) + 160^2 = 218^2
    123^2 + 5(123)(160) + 160^2 = 373^2

    (x y z) = (42, 150, 160)
    42^2 + 2(42)(150) + 150^2 = 192^2
    42^2 + 3(42)(160) + 160^2 = 218^2
    150^2 + 5(150)(160) + 160^2 = 410^2

    (x y z) = (44, 52, 60)
    44^2 + 2(44)(52) + 52^2 = 96^2
    44^2 + 3(44)(60) + 60^2 = 116^2
    52^2 + 5(52)(60) + 60^2 = 148^2

    (x y z) = (45, 51, 105)
    45^2 + 2(45)(51) + 51^2 = 96^2
    45^2 + 3(45)(105) + 105^2 = 165^2
    51^2 + 5(51)(105) + 105^2 = 201^2

    (x y z) = (45, 72, 216)
    45^2 + 2(45)(72) + 72^2 = 117^2
    45^2 + 3(45)(216) + 216^2 = 279^2
    72^2 + 5(72)(216) + 216^2 = 360^2

    (x y z) = (45, 91, 105)
    45^2 + 2(45)(91) + 91^2 = 136^2
    45^2 + 3(45)(105) + 105^2 = 165^2
    91^2 + 5(91)(105) + 105^2 = 259^2

    (x y z) = (45, 135, 216)
    45^2 + 2(45)(135) + 135^2 = 180^2
    45^2 + 3(45)(216) + 216^2 = 279^2
    135^2 + 5(135)(216) + 216^2 = 459^2

    (x y z) = (46, 50, 198)
    46^2 + 2(46)(50) + 50^2 = 96^2
    46^2 + 3(46)(198) + 198^2 = 262^2
    50^2 + 5(50)(198) + 198^2 = 302^2

    (x y z) = (46, 66, 198)
    46^2 + 2(46)(66) + 66^2 = 112^2
    46^2 + 3(46)(198) + 198^2 = 262^2
    66^2 + 5(66)(198) + 198^2 = 330^2

    (x y z) = (46, 144, 198)
    46^2 + 2(46)(144) + 144^2 = 190^2
    46^2 + 3(46)(198) + 198^2 = 262^2
    144^2 + 5(144)(198) + 198^2 = 450^2

    (x y z) = (48, 56, 77)
    48^2 + 2(48)(56) + 56^2 = 104^2
    48^2 + 3(48)(77) + 77^2 = 139^2
    56^2 + 5(56)(77) + 77^2 = 175^2

    (x y z) = (48, 57, 171)
    48^2 + 2(48)(57) + 57^2 = 105^2
    48^2 + 3(48)(171) + 171^2 = 237^2
    57^2 + 5(57)(171) + 171^2 = 285^2

    (x y z) = (48, 70, 112)
    48^2 + 2(48)(70) + 70^2 = 118^2
    48^2 + 3(48)(112) + 112^2 = 176^2
    70^2 + 5(70)(112) + 112^2 = 238^2

    (x y z) = (48, 88, 171)
    48^2 + 2(48)(88) + 88^2 = 136^2
    48^2 + 3(48)(171) + 171^2 = 237^2
    88^2 + 5(88)(171) + 171^2 = 335^2

    (x y z) = (48, 105, 112)
    48^2 + 2(48)(105) + 105^2 = 153^2
    48^2 + 3(48)(112) + 112^2 = 176^2
    105^2 + 5(105)(112) + 112^2 = 287^2

    (x y z) = (48, 111, 112)
    48^2 + 2(48)(111) + 111^2 = 159^2
    48^2 + 3(48)(112) + 112^2 = 176^2
    111^2 + 5(111)(112) + 112^2 = 295^2

    (x y z) = (50, 54, 240)
    50^2 + 2(50)(54) + 54^2 = 104^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    54^2 + 5(54)(240) + 240^2 = 354^2

    (x y z) = (50, 69, 240)
    50^2 + 2(50)(69) + 69^2 = 119^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    69^2 + 5(69)(240) + 240^2 = 381^2

    (x y z) = (50, 80, 240)
    50^2 + 2(50)(80) + 80^2 = 130^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    80^2 + 5(80)(240) + 240^2 = 400^2

    (x y z) = (50, 95, 240)
    50^2 + 2(50)(95) + 95^2 = 145^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    95^2 + 5(95)(240) + 240^2 = 425^2

    (x y z) = (50, 126, 240)
    50^2 + 2(50)(126) + 126^2 = 176^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    126^2 + 5(126)(240) + 240^2 = 474^2

    (x y z) = (50, 150, 240)
    50^2 + 2(50)(150) + 150^2 = 200^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    150^2 + 5(150)(240) + 240^2 = 510^2

    (x y z) = (50, 187, 240)
    50^2 + 2(50)(187) + 187^2 = 237^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    187^2 + 5(187)(240) + 240^2 = 563^2

    (x y z) = (50, 208, 240)
    50^2 + 2(50)(208) + 208^2 = 258^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    208^2 + 5(208)(240) + 240^2 = 592^2

    (x y z) = (50, 225, 240)
    50^2 + 2(50)(225) + 225^2 = 275^2
    50^2 + 3(50)(240) + 240^2 = 310^2
    225^2 + 5(225)(240) + 240^2 = 615^2

    (x y z) = (51, 57, 144)
    51^2 + 2(51)(57) + 57^2 = 108^2
    51^2 + 3(51)(144) + 144^2 = 213^2
    57^2 + 5(57)(144) + 144^2 = 255^2

    (x y z) = (51, 90, 144)
    51^2 + 2(51)(90) + 90^2 = 141^2
    51^2 + 3(51)(144) + 144^2 = 213^2
    90^2 + 5(90)(144) + 144^2 = 306^2

    (x y z) = (51, 135, 144)
    51^2 + 2(51)(135) + 135^2 = 186^2
    51^2 + 3(51)(144) + 144^2 = 213^2
    135^2 + 5(135)(144) + 144^2 = 369^2

    (x y z) = (52, 60, 96)
    52^2 + 2(52)(60) + 60^2 = 112^2
    52^2 + 3(52)(96) + 96^2 = 164^2
    60^2 + 5(60)(96) + 96^2 = 204^2

    (x y z) = (52, 90, 96)
    52^2 + 2(52)(90) + 90^2 = 142^2
    52^2 + 3(52)(96) + 96^2 = 164^2
    90^2 + 5(90)(96) + 96^2 = 246^2

    (x y z) = (54, 58, 286)
    54^2 + 2(54)(58) + 58^2 = 112^2
    54^2 + 3(54)(286) + 286^2 = 362^2
    58^2 + 5(58)(286) + 286^2 = 410^2

    (x y z) = (54, 208, 286)
    54^2 + 2(54)(208) + 208^2 = 262^2
    54^2 + 3(54)(286) + 286^2 = 362^2
    208^2 + 5(208)(286) + 286^2 = 650^2

    (x y z) = (55, 65, 75)
    55^2 + 2(55)(65) + 65^2 = 120^2
    55^2 + 3(55)(75) + 75^2 = 145^2
    65^2 + 5(65)(75) + 75^2 = 185^2

    (x y z) = (55, 88, 264)
    55^2 + 2(55)(88) + 88^2 = 143^2
    55^2 + 3(55)(264) + 264^2 = 341^2
    88^2 + 5(88)(264) + 264^2 = 440^2

    (x y z) = (55, 165, 264)
    55^2 + 2(55)(165) + 165^2 = 220^2
    55^2 + 3(55)(264) + 264^2 = 341^2
    165^2 + 5(165)(264) + 264^2 = 561^2

    (x y z) = (55, 192, 264)
    55^2 + 2(55)(192) + 192^2 = 247^2
    55^2 + 3(55)(264) + 264^2 = 341^2
    192^2 + 5(192)(264) + 264^2 = 600^2

    (x y z) = (55, 217, 264)
    55^2 + 2(55)(217) + 217^2 = 272^2
    55^2 + 3(55)(264) + 264^2 = 341^2
    217^2 + 5(217)(264) + 264^2 = 635^2

    (x y z) = (56, 64, 117)
    56^2 + 2(56)(64) + 64^2 = 120^2
    56^2 + 3(56)(117) + 117^2 = 191^2
    64^2 + 5(64)(117) + 117^2 = 235^2

    (x y z) = (56, 72, 165)
    56^2 + 2(56)(72) + 72^2 = 128^2
    56^2 + 3(56)(165) + 165^2 = 241^2
    72^2 + 5(72)(165) + 165^2 = 303^2

    (x y z) = (56, 120, 165)
    56^2 + 2(56)(120) + 120^2 = 176^2
    56^2 + 3(56)(165) + 165^2 = 241^2
    120^2 + 5(120)(165) + 165^2 = 375^2

    (x y z) = (56, 143, 165)
    56^2 + 2(56)(143) + 143^2 = 199^2
    56^2 + 3(56)(165) + 165^2 = 241^2
    143^2 + 5(143)(165) + 165^2 = 407^2

    (x y z) = (57, 63, 189)
    57^2 + 2(57)(63) + 63^2 = 120^2
    57^2 + 3(57)(189) + 189^2 = 267^2
    63^2 + 5(63)(189) + 189^2 = 315^2

    (x y z) = (57, 160, 189)
    57^2 + 2(57)(160) + 160^2 = 217^2
    57^2 + 3(57)(189) + 189^2 = 267^2
    160^2 + 5(160)(189) + 189^2 = 461^2

    (x y z) = (60, 68, 140)
    60^2 + 2(60)(68) + 68^2 = 128^2
    60^2 + 3(60)(140) + 140^2 = 220^2
    68^2 + 5(68)(140) + 140^2 = 268^2

    (x y z) = (60, 96, 288)
    60^2 + 2(60)(96) + 96^2 = 156^2
    60^2 + 3(60)(288) + 288^2 = 372^2
    96^2 + 5(96)(288) + 288^2 = 480^2

    (x y z) = (60, 114, 288)
    60^2 + 2(60)(114) + 114^2 = 174^2
    60^2 + 3(60)(288) + 288^2 = 372^2
    114^2 + 5(114)(288) + 288^2 = 510^2

    (x y z) = (60, 180, 288)
    60^2 + 2(60)(180) + 180^2 = 240^2
    60^2 + 3(60)(288) + 288^2 = 372^2
    180^2 + 5(180)(288) + 288^2 = 612^2

    (x y z) = (60, 270, 288)
    60^2 + 2(60)(270) + 270^2 = 330^2
    60^2 + 3(60)(288) + 288^2 = 372^2
    270^2 + 5(270)(288) + 288^2 = 738^2

    (x y z) = (63, 69, 240)
    63^2 + 2(63)(69) + 69^2 = 132^2
    63^2 + 3(63)(240) + 240^2 = 327^2
    69^2 + 5(69)(240) + 240^2 = 381^2

    (x y z) = (63, 80, 240)
    63^2 + 2(63)(80) + 80^2 = 143^2
    63^2 + 3(63)(240) + 240^2 = 327^2
    80^2 + 5(80)(240) + 240^2 = 400^2

    (x y z) = (63, 90, 96)
    63^2 + 2(63)(90) + 90^2 = 153^2
    63^2 + 3(63)(96) + 96^2 = 177^2
    90^2 + 5(90)(96) + 96^2 = 246^2

    (x y z) = (63, 95, 240)
    63^2 + 2(63)(95) + 95^2 = 158^2
    63^2 + 3(63)(240) + 240^2 = 327^2
    95^2 + 5(95)(240) + 240^2 = 425^2

    (x y z) = (63, 126, 240)
    63^2 + 2(63)(126) + 126^2 = 189^2
    63^2 + 3(63)(240) + 240^2 = 327^2
    126^2 + 5(126)(240) + 240^2 = 474^2

    (x y z) = (63, 150, 240)
    63^2 + 2(63)(150) + 150^2 = 213^2
    63^2 + 3(63)(240) + 240^2 = 327^2
    150^2 + 5(150)(240) + 240^2 = 510^2

    (x y z) = (63, 187, 240)
    63^2 + 2(63)(187) + 187^2 = 250^2
    63^2 + 3(63)(240) + 240^2 = 327^2
    187^2 + 5(187)(240) + 240^2 = 563^2

    (x y z) = (63, 208, 240)
    63^2 + 2(63)(208) + 208^2 = 271^2
    63^2 + 3(63)(240) + 240^2 = 327^2
    208^2 + 5(208)(240) + 240^2 = 592^2

    (x y z) = (63, 225, 240)
    63^2 + 2(63)(225) + 225^2 = 288^2
    63^2 + 3(63)(240) + 240^2 = 327^2
    225^2 + 5(225)(240) + 240^2 = 615^2

    (x y z) = (64, 72, 165)
    64^2 + 2(64)(72) + 72^2 = 136^2
    64^2 + 3(64)(165) + 165^2 = 251^2
    72^2 + 5(72)(165) + 165^2 = 303^2

    (x y z) = (64, 76, 228)
    64^2 + 2(64)(76) + 76^2 = 140^2
    64^2 + 3(64)(228) + 228^2 = 316^2
    76^2 + 5(76)(228) + 228^2 = 380^2

    (x y z) = (64, 120, 165)
    64^2 + 2(64)(120) + 120^2 = 184^2
    64^2 + 3(64)(165) + 165^2 = 251^2
    120^2 + 5(120)(165) + 165^2 = 375^2

    (x y z) = (64, 143, 165)
    64^2 + 2(64)(143) + 143^2 = 207^2
    64^2 + 3(64)(165) + 165^2 = 251^2
    143^2 + 5(143)(165) + 165^2 = 407^2

    (x y z) = (65, 75, 120)
    65^2 + 2(65)(75) + 75^2 = 140^2
    65^2 + 3(65)(120) + 120^2 = 205^2
    75^2 + 5(75)(120) + 120^2 = 255^2

    (x y z) = (65, 104, 120)
    65^2 + 2(65)(104) + 104^2 = 169^2
    65^2 + 3(65)(120) + 120^2 = 205^2
    104^2 + 5(104)(120) + 120^2 = 296^2

    (x y z) = (66, 78, 90)
    66^2 + 2(66)(78) + 78^2 = 144^2
    66^2 + 3(66)(90) + 90^2 = 174^2
    78^2 + 5(78)(90) + 90^2 = 222^2

    (x y z) = (66, 99, 208)
    66^2 + 2(66)(99) + 99^2 = 165^2
    66^2 + 3(66)(208) + 208^2 = 298^2
    99^2 + 5(99)(208) + 208^2 = 395^2

    (x y z) = (66, 112, 154)
    66^2 + 2(66)(112) + 112^2 = 178^2
    66^2 + 3(66)(154) + 154^2 = 242^2
    112^2 + 5(112)(154) + 154^2 = 350^2

    (x y z) = (66, 130, 208)
    66^2 + 2(66)(130) + 130^2 = 196^2
    66^2 + 3(66)(208) + 208^2 = 298^2
    130^2 + 5(130)(208) + 208^2 = 442^2

    (x y z) = (66, 195, 208)
    66^2 + 2(66)(195) + 195^2 = 261^2
    66^2 + 3(66)(208) + 208^2 = 298^2
    195^2 + 5(195)(208) + 208^2 = 533^2

    (x y z) = (68, 76, 192)
    68^2 + 2(68)(76) + 76^2 = 144^2
    68^2 + 3(68)(192) + 192^2 = 284^2
    76^2 + 5(76)(192) + 192^2 = 340^2

    (x y z) = (68, 120, 192)
    68^2 + 2(68)(120) + 120^2 = 188^2
    68^2 + 3(68)(192) + 192^2 = 284^2
    120^2 + 5(120)(192) + 192^2 = 408^2

    (x y z) = (68, 180, 192)
    68^2 + 2(68)(180) + 180^2 = 248^2
    68^2 + 3(68)(192) + 192^2 = 284^2
    180^2 + 5(180)(192) + 192^2 = 492^2

    (x y z) = (69, 75, 297)
    69^2 + 2(69)(75) + 75^2 = 144^2
    69^2 + 3(69)(297) + 297^2 = 393^2
    75^2 + 5(75)(297) + 297^2 = 453^2

    (x y z) = (69, 80, 161)
    69^2 + 2(69)(80) + 80^2 = 149^2
    69^2 + 3(69)(161) + 161^2 = 253^2
    80^2 + 5(80)(161) + 161^2 = 311^2

    (x y z) = (69, 99, 297)
    69^2 + 2(69)(99) + 99^2 = 168^2
    69^2 + 3(69)(297) + 297^2 = 393^2
    99^2 + 5(99)(297) + 297^2 = 495^2

    (x y z) = (69, 216, 297)
    69^2 + 2(69)(216) + 216^2 = 285^2
    69^2 + 3(69)(297) + 297^2 = 393^2
    216^2 + 5(216)(297) + 297^2 = 675^2

    (x y z) = (72, 80, 221)
    72^2 + 2(72)(80) + 80^2 = 152^2
    72^2 + 3(72)(221) + 221^2 = 319^2
    80^2 + 5(80)(221) + 221^2 = 379^2

    (x y z) = (72, 105, 168)
    72^2 + 2(72)(105) + 105^2 = 177^2
    72^2 + 3(72)(168) + 168^2 = 264^2
    105^2 + 5(105)(168) + 168^2 = 357^2

    (x y z) = (75, 85, 175)
    75^2 + 2(75)(85) + 85^2 = 160^2
    75^2 + 3(75)(175) + 175^2 = 275^2
    85^2 + 5(85)(175) + 175^2 = 335^2

    (x y z) = (75, 117, 175)
    75^2 + 2(75)(117) + 117^2 = 192^2
    75^2 + 3(75)(175) + 175^2 = 275^2
    117^2 + 5(117)(175) + 175^2 = 383^2

    (x y z) = (76, 84, 252)
    76^2 + 2(76)(84) + 84^2 = 160^2
    76^2 + 3(76)(252) + 252^2 = 356^2
    84^2 + 5(84)(252) + 252^2 = 420^2

    (x y z) = (77, 91, 105)
    77^2 + 2(77)(91) + 91^2 = 168^2
    77^2 + 3(77)(105) + 105^2 = 203^2
    91^2 + 5(91)(105) + 105^2 = 259^2

    (x y z) = (78, 90, 144)
    78^2 + 2(78)(90) + 90^2 = 168^2
    78^2 + 3(78)(144) + 144^2 = 246^2
    90^2 + 5(90)(144) + 144^2 = 306^2

    (x y z) = (78, 135, 144)
    78^2 + 2(78)(135) + 135^2 = 213^2
    78^2 + 3(78)(144) + 144^2 = 246^2
    135^2 + 5(135)(144) + 144^2 = 369^2

    (x y z) = (78, 144, 182)
    78^2 + 2(78)(144) + 144^2 = 222^2
    78^2 + 3(78)(182) + 182^2 = 286^2
    144^2 + 5(144)(182) + 182^2 = 430^2

    (x y z) = (80, 88, 285)
    80^2 + 2(80)(88) + 88^2 = 168^2
    80^2 + 3(80)(285) + 285^2 = 395^2
    88^2 + 5(88)(285) + 285^2 = 463^2

    (x y z) = (80, 95, 285)
    80^2 + 2(80)(95) + 95^2 = 175^2
    80^2 + 3(80)(285) + 285^2 = 395^2
    95^2 + 5(95)(285) + 285^2 = 475^2

    (x y z) = (80, 231, 285)
    80^2 + 2(80)(231) + 231^2 = 311^2
    80^2 + 3(80)(285) + 285^2 = 395^2
    231^2 + 5(231)(285) + 285^2 = 681^2

    (x y z) = (80, 247, 285)
    80^2 + 2(80)(247) + 247^2 = 327^2
    80^2 + 3(80)(285) + 285^2 = 395^2
    247^2 + 5(247)(285) + 285^2 = 703^2

    (x y z) = (81, 160, 189)
    81^2 + 2(81)(160) + 160^2 = 241^2
    81^2 + 3(81)(189) + 189^2 = 297^2
    160^2 + 5(160)(189) + 189^2 = 461^2

    (x y z) = (84, 120, 128)
    84^2 + 2(84)(120) + 120^2 = 204^2
    84^2 + 3(84)(128) + 128^2 = 236^2
    120^2 + 5(120)(128) + 128^2 = 328^2

    (x y z) = (85, 95, 240)
    85^2 + 2(85)(95) + 95^2 = 180^2
    85^2 + 3(85)(240) + 240^2 = 355^2
    95^2 + 5(95)(240) + 240^2 = 425^2

    (x y z) = (85, 126, 240)
    85^2 + 2(85)(126) + 126^2 = 211^2
    85^2 + 3(85)(240) + 240^2 = 355^2
    126^2 + 5(126)(240) + 240^2 = 474^2

    (x y z) = (85, 150, 240)
    85^2 + 2(85)(150) + 150^2 = 235^2
    85^2 + 3(85)(240) + 240^2 = 355^2
    150^2 + 5(150)(240) + 240^2 = 510^2

    (x y z) = (85, 187, 240)
    85^2 + 2(85)(187) + 187^2 = 272^2
    85^2 + 3(85)(240) + 240^2 = 355^2
    187^2 + 5(187)(240) + 240^2 = 563^2

    (x y z) = (85, 208, 240)
    85^2 + 2(85)(208) + 208^2 = 293^2
    85^2 + 3(85)(240) + 240^2 = 355^2
    208^2 + 5(208)(240) + 240^2 = 592^2

    (x y z) = (85, 225, 240)
    85^2 + 2(85)(225) + 225^2 = 310^2
    85^2 + 3(85)(240) + 240^2 = 355^2
    225^2 + 5(225)(240) + 240^2 = 615^2

    (x y z) = (88, 104, 120)
    88^2 + 2(88)(104) + 104^2 = 192^2
    88^2 + 3(88)(120) + 120^2 = 232^2
    104^2 + 5(104)(120) + 120^2 = 296^2

    (x y z) = (90, 102, 210)
    90^2 + 2(90)(102) + 102^2 = 192^2
    90^2 + 3(90)(210) + 210^2 = 330^2
    102^2 + 5(102)(210) + 210^2 = 402^2

    (x y z) = (90, 182, 210)
    90^2 + 2(90)(182) + 182^2 = 272^2
    90^2 + 3(90)(210) + 210^2 = 330^2
    182^2 + 5(182)(210) + 210^2 = 518^2

    (x y z) = (91, 105, 168)
    91^2 + 2(91)(105) + 105^2 = 196^2
    91^2 + 3(91)(168) + 168^2 = 287^2
    105^2 + 5(105)(168) + 168^2 = 357^2

    (x y z) = (93, 105, 112)
    93^2 + 2(93)(105) + 105^2 = 198^2
    93^2 + 3(93)(112) + 112^2 = 229^2
    105^2 + 5(105)(112) + 112^2 = 287^2

    (x y z) = (93, 111, 112)
    93^2 + 2(93)(111) + 111^2 = 204^2
    93^2 + 3(93)(112) + 112^2 = 229^2
    111^2 + 5(111)(112) + 112^2 = 295^2

    (x y z) = (96, 112, 154)
    96^2 + 2(96)(112) + 112^2 = 208^2
    96^2 + 3(96)(154) + 154^2 = 278^2
    112^2 + 5(112)(154) + 154^2 = 350^2

    (x y z) = (96, 140, 224)
    96^2 + 2(96)(140) + 140^2 = 236^2
    96^2 + 3(96)(224) + 224^2 = 352^2
    140^2 + 5(140)(224) + 224^2 = 476^2

    (x y z) = (96, 210, 224)
    96^2 + 2(96)(210) + 210^2 = 306^2
    96^2 + 3(96)(224) + 224^2 = 352^2
    210^2 + 5(210)(224) + 224^2 = 574^2

    (x y z) = (96, 222, 224)
    96^2 + 2(96)(222) + 222^2 = 318^2
    96^2 + 3(96)(224) + 224^2 = 352^2
    222^2 + 5(222)(224) + 224^2 = 590^2

    (x y z) = (99, 117, 135)
    99^2 + 2(99)(117) + 117^2 = 216^2
    99^2 + 3(99)(135) + 135^2 = 261^2
    117^2 + 5(117)(135) + 135^2 = 333^2

    (x y z) = (99, 168, 231)
    99^2 + 2(99)(168) + 168^2 = 267^2
    99^2 + 3(99)(231) + 231^2 = 363^2
    168^2 + 5(168)(231) + 231^2 = 525^2

    (x y z) = (102, 114, 288)
    102^2 + 2(102)(114) + 114^2 = 216^2
    102^2 + 3(102)(288) + 288^2 = 426^2
    114^2 + 5(114)(288) + 288^2 = 510^2

    (x y z) = (102, 180, 288)
    102^2 + 2(102)(180) + 180^2 = 282^2
    102^2 + 3(102)(288) + 288^2 = 426^2
    180^2 + 5(180)(288) + 288^2 = 612^2

    (x y z) = (102, 270, 288)
    102^2 + 2(102)(270) + 270^2 = 372^2
    102^2 + 3(102)(288) + 288^2 = 426^2
    270^2 + 5(270)(288) + 288^2 = 738^2

    (x y z) = (104, 120, 192)
    104^2 + 2(104)(120) + 120^2 = 224^2
    104^2 + 3(104)(192) + 192^2 = 328^2
    120^2 + 5(120)(192) + 192^2 = 408^2

    (x y z) = (104, 180, 192)
    104^2 + 2(104)(180) + 180^2 = 284^2
    104^2 + 3(104)(192) + 192^2 = 328^2
    180^2 + 5(180)(192) + 192^2 = 492^2

    (x y z) = (105, 119, 245)
    105^2 + 2(105)(119) + 119^2 = 224^2
    105^2 + 3(105)(245) + 245^2 = 385^2
    119^2 + 5(119)(245) + 245^2 = 469^2

    (x y z) = (105, 123, 160)
    105^2 + 2(105)(123) + 123^2 = 228^2
    105^2 + 3(105)(160) + 160^2 = 295^2
    123^2 + 5(123)(160) + 160^2 = 373^2

    (x y z) = (105, 150, 160)
    105^2 + 2(105)(150) + 150^2 = 255^2
    105^2 + 3(105)(160) + 160^2 = 295^2
    150^2 + 5(150)(160) + 160^2 = 410^2

    (x y z) = (110, 130, 150)
    110^2 + 2(110)(130) + 130^2 = 240^2
    110^2 + 3(110)(150) + 150^2 = 290^2
    130^2 + 5(130)(150) + 150^2 = 370^2

    (x y z) = (111, 129, 187)
    111^2 + 2(111)(129) + 129^2 = 240^2
    111^2 + 3(111)(187) + 187^2 = 331^2
    129^2 + 5(129)(187) + 187^2 = 415^2

    (x y z) = (111, 136, 187)
    111^2 + 2(111)(136) + 136^2 = 247^2
    111^2 + 3(111)(187) + 187^2 = 331^2
    136^2 + 5(136)(187) + 187^2 = 425^2

    (x y z) = (112, 128, 234)
    112^2 + 2(112)(128) + 128^2 = 240^2
    112^2 + 3(112)(234) + 234^2 = 382^2
    128^2 + 5(128)(234) + 234^2 = 470^2

    (x y z) = (117, 135, 216)
    117^2 + 2(117)(135) + 135^2 = 252^2
    117^2 + 3(117)(216) + 216^2 = 369^2
    135^2 + 5(135)(216) + 216^2 = 459^2

    (x y z) = (117, 216, 273)
    117^2 + 2(117)(216) + 216^2 = 333^2
    117^2 + 3(117)(273) + 273^2 = 429^2
    216^2 + 5(216)(273) + 273^2 = 645^2

    (x y z) = (119, 169, 195)
    119^2 + 2(119)(169) + 169^2 = 288^2
    119^2 + 3(119)(195) + 195^2 = 349^2
    169^2 + 5(169)(195) + 195^2 = 481^2

    (x y z) = (120, 136, 280)
    120^2 + 2(120)(136) + 136^2 = 256^2
    120^2 + 3(120)(280) + 280^2 = 440^2
    136^2 + 5(136)(280) + 280^2 = 536^2

    (x y z) = (120, 147, 280)
    120^2 + 2(120)(147) + 147^2 = 267^2
    120^2 + 3(120)(280) + 280^2 = 440^2
    147^2 + 5(147)(280) + 280^2 = 553^2

    (x y z) = (120, 175, 280)
    120^2 + 2(120)(175) + 175^2 = 295^2
    120^2 + 3(120)(280) + 280^2 = 440^2
    175^2 + 5(175)(280) + 280^2 = 595^2

    (x y z) = (121, 143, 165)
    121^2 + 2(121)(143) + 143^2 = 264^2
    121^2 + 3(121)(165) + 165^2 = 319^2
    143^2 + 5(143)(165) + 165^2 = 407^2

    (x y z) = (123, 141, 247)
    123^2 + 2(123)(141) + 141^2 = 264^2
    123^2 + 3(123)(247) + 247^2 = 409^2
    141^2 + 5(141)(247) + 247^2 = 505^2

    (x y z) = (126, 180, 192)
    126^2 + 2(126)(180) + 180^2 = 306^2
    126^2 + 3(126)(192) + 192^2 = 354^2
    180^2 + 5(180)(192) + 192^2 = 492^2

    (x y z) = (128, 135, 144)
    128^2 + 2(128)(135) + 135^2 = 263^2
    128^2 + 3(128)(144) + 144^2 = 304^2
    135^2 + 5(135)(144) + 144^2 = 369^2

    (x y z) = (129, 136, 280)
    129^2 + 2(129)(136) + 136^2 = 265^2
    129^2 + 3(129)(280) + 280^2 = 451^2
    136^2 + 5(136)(280) + 280^2 = 536^2

    (x y z) = (129, 147, 280)
    129^2 + 2(129)(147) + 147^2 = 276^2
    129^2 + 3(129)(280) + 280^2 = 451^2
    147^2 + 5(147)(280) + 280^2 = 553^2

    (x y z) = (129, 175, 280)
    129^2 + 2(129)(175) + 175^2 = 304^2
    129^2 + 3(129)(280) + 280^2 = 451^2
    175^2 + 5(175)(280) + 280^2 = 595^2

    (x y z) = (130, 150, 240)
    130^2 + 2(130)(150) + 150^2 = 280^2
    130^2 + 3(130)(240) + 240^2 = 410^2
    150^2 + 5(150)(240) + 240^2 = 510^2

    (x y z) = (130, 187, 240)
    130^2 + 2(130)(187) + 187^2 = 317^2
    130^2 + 3(130)(240) + 240^2 = 410^2
    187^2 + 5(187)(240) + 240^2 = 563^2

    (x y z) = (130, 208, 240)
    130^2 + 2(130)(208) + 208^2 = 338^2
    130^2 + 3(130)(240) + 240^2 = 410^2
    208^2 + 5(208)(240) + 240^2 = 592^2

    (x y z) = (130, 225, 240)
    130^2 + 2(130)(225) + 225^2 = 355^2
    130^2 + 3(130)(240) + 240^2 = 410^2
    225^2 + 5(225)(240) + 240^2 = 615^2

    (x y z) = (132, 156, 180)
    132^2 + 2(132)(156) + 156^2 = 288^2
    132^2 + 3(132)(180) + 180^2 = 348^2
    156^2 + 5(156)(180) + 180^2 = 444^2

    (x y z) = (133, 165, 264)
    133^2 + 2(133)(165) + 165^2 = 298^2
    133^2 + 3(133)(264) + 264^2 = 439^2
    165^2 + 5(165)(264) + 264^2 = 561^2

    (x y z) = (133, 192, 264)
    133^2 + 2(133)(192) + 192^2 = 325^2
    133^2 + 3(133)(264) + 264^2 = 439^2
    192^2 + 5(192)(264) + 264^2 = 600^2

    (x y z) = (133, 217, 264)
    133^2 + 2(133)(217) + 217^2 = 350^2
    133^2 + 3(133)(264) + 264^2 = 439^2
    217^2 + 5(217)(264) + 264^2 = 635^2

    (x y z) = (143, 165, 264)
    143^2 + 2(143)(165) + 165^2 = 308^2
    143^2 + 3(143)(264) + 264^2 = 451^2
    165^2 + 5(165)(264) + 264^2 = 561^2

    (x y z) = (143, 169, 195)
    143^2 + 2(143)(169) + 169^2 = 312^2
    143^2 + 3(143)(195) + 195^2 = 377^2
    169^2 + 5(169)(195) + 195^2 = 481^2

    (x y z) = (143, 192, 264)
    143^2 + 2(143)(192) + 192^2 = 335^2
    143^2 + 3(143)(264) + 264^2 = 451^2
    192^2 + 5(192)(264) + 264^2 = 600^2

    (x y z) = (143, 217, 264)
    143^2 + 2(143)(217) + 217^2 = 360^2
    143^2 + 3(143)(264) + 264^2 = 451^2
    217^2 + 5(217)(264) + 264^2 = 635^2

    (x y z) = (144, 168, 231)
    144^2 + 2(144)(168) + 168^2 = 312^2
    144^2 + 3(144)(231) + 231^2 = 417^2
    168^2 + 5(168)(231) + 231^2 = 525^2

    (x y z) = (147, 210, 224)
    147^2 + 2(147)(210) + 210^2 = 357^2
    147^2 + 3(147)(224) + 224^2 = 413^2
    210^2 + 5(210)(224) + 224^2 = 574^2

    (x y z) = (147, 222, 224)
    147^2 + 2(147)(222) + 222^2 = 369^2
    147^2 + 3(147)(224) + 224^2 = 413^2
    222^2 + 5(222)(224) + 224^2 = 590^2

    (x y z) = (154, 182, 210)
    154^2 + 2(154)(182) + 182^2 = 336^2
    154^2 + 3(154)(210) + 210^2 = 406^2
    182^2 + 5(182)(210) + 210^2 = 518^2

    (x y z) = (156, 180, 288)
    156^2 + 2(156)(180) + 180^2 = 336^2
    156^2 + 3(156)(288) + 288^2 = 492^2
    180^2 + 5(180)(288) + 288^2 = 612^2

    (x y z) = (156, 270, 288)
    156^2 + 2(156)(270) + 270^2 = 426^2
    156^2 + 3(156)(288) + 288^2 = 492^2
    270^2 + 5(270)(288) + 288^2 = 738^2

    (x y z) = (165, 195, 225)
    165^2 + 2(165)(195) + 195^2 = 360^2
    165^2 + 3(165)(225) + 225^2 = 435^2
    195^2 + 5(195)(225) + 225^2 = 555^2

    (x y z) = (168, 200, 209)
    168^2 + 2(168)(200) + 200^2 = 368^2
    168^2 + 3(168)(209) + 209^2 = 421^2
    200^2 + 5(200)(209) + 209^2 = 541^2

    (x y z) = (168, 203, 209)
    168^2 + 2(168)(203) + 203^2 = 371^2
    168^2 + 3(168)(209) + 209^2 = 421^2
    203^2 + 5(203)(209) + 209^2 = 545^2

    (x y z) = (168, 240, 256)
    168^2 + 2(168)(240) + 240^2 = 408^2
    168^2 + 3(168)(256) + 256^2 = 472^2
    240^2 + 5(240)(256) + 256^2 = 656^2

    (x y z) = (176, 187, 240)
    176^2 + 2(176)(187) + 187^2 = 363^2
    176^2 + 3(176)(240) + 240^2 = 464^2
    187^2 + 5(187)(240) + 240^2 = 563^2

    (x y z) = (176, 208, 240)
    176^2 + 2(176)(208) + 208^2 = 384^2
    176^2 + 3(176)(240) + 240^2 = 464^2
    208^2 + 5(208)(240) + 240^2 = 592^2

    (x y z) = (176, 225, 240)
    176^2 + 2(176)(225) + 225^2 = 401^2
    176^2 + 3(176)(240) + 240^2 = 464^2
    225^2 + 5(225)(240) + 240^2 = 615^2

    (x y z) = (184, 216, 273)
    184^2 + 2(184)(216) + 216^2 = 400^2
    184^2 + 3(184)(273) + 273^2 = 509^2
    216^2 + 5(216)(273) + 273^2 = 645^2

    (x y z) = (186, 210, 224)
    186^2 + 2(186)(210) + 210^2 = 396^2
    186^2 + 3(186)(224) + 224^2 = 458^2
    210^2 + 5(210)(224) + 224^2 = 574^2

    (x y z) = (186, 222, 224)
    186^2 + 2(186)(222) + 222^2 = 408^2
    186^2 + 3(186)(224) + 224^2 = 458^2
    222^2 + 5(222)(224) + 224^2 = 590^2

    (x y z) = (187, 221, 255)
    187^2 + 2(187)(221) + 221^2 = 408^2
    187^2 + 3(187)(255) + 255^2 = 493^2
    221^2 + 5(221)(255) + 255^2 = 629^2

    (x y z) = (189, 270, 288)
    189^2 + 2(189)(270) + 270^2 = 459^2
    189^2 + 3(189)(288) + 288^2 = 531^2
    270^2 + 5(270)(288) + 288^2 = 738^2

    (x y z) = (198, 234, 270)
    198^2 + 2(198)(234) + 234^2 = 432^2
    198^2 + 3(198)(270) + 270^2 = 522^2
    234^2 + 5(234)(270) + 270^2 = 666^2

    (x y z) = (209, 231, 285)
    209^2 + 2(209)(231) + 231^2 = 440^2
    209^2 + 3(209)(285) + 285^2 = 551^2
    231^2 + 5(231)(285) + 285^2 = 681^2

    (x y z) = (209, 247, 285)
    209^2 + 2(209)(247) + 247^2 = 456^2
    209^2 + 3(209)(285) + 285^2 = 551^2
    247^2 + 5(247)(285) + 285^2 = 703^2

    (x y z) = (210, 255, 272)
    210^2 + 2(210)(255) + 255^2 = 465^2
    210^2 + 3(210)(272) + 272^2 = 538^2
    255^2 + 5(255)(272) + 272^2 = 697^2

    (x y z) = (220, 260, 300)
    220^2 + 2(220)(260) + 260^2 = 480^2
    220^2 + 3(220)(300) + 300^2 = 580^2
    260^2 + 5(260)(300) + 300^2 = 740^2

    (x y z) = (255, 272, 299)
    255^2 + 2(255)(272) + 272^2 = 527^2
    255^2 + 3(255)(299) + 299^2 = 619^2
    272^2 + 5(272)(299) + 299^2 = 755^2

    (x y z) = (256, 270, 288)
    256^2 + 2(256)(270) + 270^2 = 526^2
    256^2 + 3(256)(288) + 288^2 = 608^2
    270^2 + 5(270)(288) + 288^2 = 738^2

    Paul.

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