## 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.

math grad - Interest: Number theory
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### 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.