## Make {x^2 + y^2 + z^2, x^2 y^2 + x^2 z^2 + y^2 z^2} squares — Part 1

Find three positive integers   $(x, y, z)$   to can make the two expressions squares

$x^2 \; + \; y^2 \; + \; z^2$
$x^2 \, y^2 \; + \; x^2 \, z^2 \; + \; y^2 \, z^2$

Here are some solutions:

math grad - Interest: Number theory
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### 2 Responses to Make {x^2 + y^2 + z^2, x^2 y^2 + x^2 z^2 + y^2 z^2} squares — Part 1

1. Paul says:

Here is a list with x, y, z <=2000. Format is {x, y, z, A, B}

{35,72,96} , 125 , 8088
{70,144,192} , 250 , 32352
{105,216,288} , 375 , 72792
{140,288,384} , 500 , 129408
{175,360,480} , 625 , 202200
{210,432,576} , 750 , 291168
{245,504,672} , 875 , 396312
{280,576,768} , 1000 , 517632
{315,648,864} , 1125 , 655128
{350,720,960} , 1250 , 808800
{385,792,1056} , 1375 , 978648
{420,864,1152} , 1500 , 1164672
{455,936,1248} , 1625 , 1366872
{490,1008,1344} , 1750 , 1585248
{525,1080,1440} , 1875 , 1819800
{560,1152,1536} , 2000 , 2070528
{595,1224,1632} , 2125 , 2337432
{630,1296,1728} , 2250 , 2620512
{665,1368,1824} , 2375 , 2919768
{700,1440,1920} , 2500 , 3235200
{735,1512,2016} , 2625 , 3566808
{770,1584,2112} , 2750 , 3914592
{805,1656,2208} , 2875 , 4278552
{840,1728,2304} , 3000 , 4658688
{875,1800,2400} , 3125 , 5055000
{910,1872,2496} , 3250 , 5467488
{945,1944,2592} , 3375 , 5896152
{980,2016,2688} , 3500 , 6340992
{1015,2088,2784} , 3625 , 6802008
{1050,2160,2880} , 3750 , 7279200
{1085,2232,2976} , 3875 , 7772568
{1120,2304,3072} , 4000 , 8282112
{1155,2376,3168} , 4125 , 8807832
{1190,2448,3264} , 4250 , 9349728
{1225,2520,3360} , 4375 , 9907800
{1260,2592,3456} , 4500 , 10482048
{1295,2664,3552} , 4625 , 11072472
{1330,2736,3648} , 4750 , 11679072
{1365,2808,3744} , 4875 , 12301848
{1400,2880,3840} , 5000 , 12940800
{1435,2952,3936} , 5125 , 13595928
{1470,3024,4032} , 5250 , 14267232
{1505,3096,4128} , 5375 , 14954712
{1540,3168,4224} , 5500 , 15658368
{1575,3240,4320} , 5625 , 16378200
{1610,3312,4416} , 5750 , 17114208
{1645,3384,4512} , 5875 , 17866392
{1680,3456,4608} , 6000 , 18634752
{1715,3528,4704} , 6125 , 19419288
{1750,3600,4800} , 6250 , 20220000
{1785,3672,4896} , 6375 , 21036888
{1820,3744,4992} , 6500 , 21869952
{1855,3816,5088} , 6625 , 22719192
{1890,3888,5184} , 6750 , 23584608
{1925,3960,5280} , 6875 , 24466200
{1960,4032,5376} , 7000 , 25363968
{1995,4104,5472} , 7125 , 26277912
{2030,4176,5568} , 7250 , 27208032
{2065,4248,5664} , 7375 , 28154328
{2100,4320,5760} , 7500 , 29116800
{2135,4392,5856} , 7625 , 30095448
{2170,4464,5952} , 7750 , 31090272
{2205,4536,6048} , 7875 , 32101272
{2240,4608,6144} , 8000 , 33128448
{2275,4680,6240} , 8125 , 34171800
{2310,4752,6336} , 8250 , 35231328
{2345,4824,6432} , 8375 , 36307032
{2380,4896,6528} , 8500 , 37398912
{2415,4968,6624} , 8625 , 38506968
{2450,5040,6720} , 8750 , 39631200
{2485,5112,6816} , 8875 , 40771608
{2520,5184,6912} , 9000 , 41928192
{2555,5256,7008} , 9125 , 43100952
{2590,5328,7104} , 9250 , 44289888
{2625,5400,7200} , 9375 , 45495000
{2660,5472,7296} , 9500 , 46716288
{2695,5544,7392} , 9625 , 47953752
{2730,5616,7488} , 9750 , 49207392
{2765,5688,7584} , 9875 , 50477208
{2800,5760,7680} , 10000 , 51763200
{2835,5832,7776} , 10125 , 53065368
{2870,5904,7872} , 10250 , 54383712
{2905,5976,7968} , 10375 , 55718232
{2940,6048,8064} , 10500 , 57068928
{2975,6120,8160} , 10625 , 58435800
{3010,6192,8256} , 10750 , 59818848
{3045,6264,8352} , 10875 , 61218072
{3080,6336,8448} , 11000 , 62633472
{3115,6408,8544} , 11125 , 64065048
{3150,6480,8640} , 11250 , 65512800
{3185,6552,8736} , 11375 , 66976728
{3220,6624,8832} , 11500 , 68456832
{3255,6696,8928} , 11625 , 69953112
{3290,6768,9024} , 11750 , 71465568
{3325,6840,9120} , 11875 , 72994200
{3360,6912,9216} , 12000 , 74539008
{3395,6984,9312} , 12125 , 76099992
{3430,7056,9408} , 12250 , 77677152
{3465,7128,9504} , 12375 , 79270488
{3500,7200,9600} , 12500 , 80880000
{3535,7272,9696} , 12625 , 82505688
{3570,7344,9792} , 12750 , 84147552
{3605,7416,9888} , 12875 , 85805592
{3640,7488,9984} , 13000 , 87479808
{3675,7560,10080} , 13125 , 89170200
{3710,7632,10176} , 13250 , 90876768
{3745,7704,10272} , 13375 , 92599512
{3780,7776,10368} , 13500 , 94338432
{3815,7848,10464} , 13625 , 96093528
{3850,7920,10560} , 13750 , 97864800
{3885,7992,10656} , 13875 , 99652248
{3920,8064,10752} , 14000 , 101455872
{3955,8136,10848} , 14125 , 103275672
{3990,8208,10944} , 14250 , 105111648
{4025,8280,11040} , 14375 , 106963800
{4060,8352,11136} , 14500 , 108832128
{4095,8424,11232} , 14625 , 110716632
{4130,8496,11328} , 14750 , 112617312
{4165,8568,11424} , 14875 , 114534168
{4200,8640,11520} , 15000 , 116467200
{4235,8712,11616} , 15125 , 118416408
{4270,8784,11712} , 15250 , 120381792
{4305,8856,11808} , 15375 , 122363352
{4340,8928,11904} , 15500 , 124361088
{4375,9000,12000} , 15625 , 126375000
{4410,9072,12096} , 15750 , 128405088
{4445,9144,12192} , 15875 , 130451352
{4480,9216,12288} , 16000 , 132513792
{4515,9288,12384} , 16125 , 134592408
{4550,9360,12480} , 16250 , 136687200
{4585,9432,12576} , 16375 , 138798168
{4620,9504,12672} , 16500 , 140925312
{4655,9576,12768} , 16625 , 143068632
{4690,9648,12864} , 16750 , 145228128
{4725,9720,12960} , 16875 , 147403800
{4760,9792,13056} , 17000 , 149595648
{4795,9864,13152} , 17125 , 151803672
{4830,9936,13248} , 17250 , 154027872
{4865,10008,13344} , 17375 , 156268248
{4900,10080,13440} , 17500 , 158524800
{4935,10152,13536} , 17625 , 160797528
{4970,10224,13632} , 17750 , 163086432
{5005,10296,13728} , 17875 , 165391512
{5040,10368,13824} , 18000 , 167712768
{5075,10440,13920} , 18125 , 170050200
{5110,10512,14016} , 18250 , 172403808
{195,12740,14112} , 19013 , 179825100
{5145,10584,14112} , 18375 , 174773592
{5180,10656,14208} , 18500 , 177159552
{5215,10728,14304} , 18625 , 179561688
{5250,10800,14400} , 18750 , 181980000
{5285,10872,14496} , 18875 , 184414488
{5320,10944,14592} , 19000 , 186865152
{5355,11016,14688} , 19125 , 189331992
{5390,11088,14784} , 19250 , 191815008
{5425,11160,14880} , 19375 , 194314200
{5460,11232,14976} , 19500 , 196829568
{5495,11304,15072} , 19625 , 199361112
{5530,11376,15168} , 19750 , 201908832
{5565,11448,15264} , 19875 , 204472728
{5600,11520,15360} , 20000 , 207052800
{5635,11592,15456} , 20125 , 209649048
{5670,11664,15552} , 20250 , 212261472
{5705,11736,15648} , 20375 , 214890072
{5740,11808,15744} , 20500 , 217534848
{5775,11880,15840} , 20625 , 220195800
{5810,11952,15936} , 20750 , 222872928
{5845,12024,16032} , 20875 , 225566232
{5880,12096,16128} , 21000 , 228275712
{5915,12168,16224} , 21125 , 231001368
{5950,12240,16320} , 21250 , 233743200
{5985,12312,16416} , 21375 , 236501208
{6020,12384,16512} , 21500 , 239275392
{6055,12456,16608} , 21625 , 242065752
{6090,12528,16704} , 21750 , 244872288
{6125,12600,16800} , 21875 , 247695000
{6160,12672,16896} , 22000 , 250533888
{2397,7920,16940} , 18853 , 141454500
{6195,12744,16992} , 22125 , 253388952
{6230,12816,17088} , 22250 , 256260192
{6265,12888,17184} , 22375 , 259147608
{6300,12960,17280} , 22500 , 262051200
{6335,13032,17376} , 22625 , 264970968
{6370,13104,17472} , 22750 , 267906912
{6405,13176,17568} , 22875 , 270859032
{1189,16800,17640} , 24389 , 297764040
{6440,13248,17664} , 23000 , 273827328
{6475,13320,17760} , 23125 , 276811800
{6510,13392,17856} , 23250 , 279812448
{6545,13464,17952} , 23375 , 282829272
{6580,13536,18048} , 23500 , 285862272
{6615,13608,18144} , 23625 , 288911448
{6650,13680,18240} , 23750 , 291976800
{6685,13752,18336} , 23875 , 295058328
{6720,13824,18432} , 24000 , 298156032
{6755,13896,18528} , 24125 , 301269912
{7200,7735,18564} , 21361 , 203925540
{6790,13968,18624} , 24250 , 304399968
{6825,14040,18720} , 24375 , 307546200
{6860,14112,18816} , 24500 , 310708608
{6895,14184,18912} , 24625 , 313887192
{6930,14256,19008} , 24750 , 317081952
{6965,14328,19104} , 24875 , 320292888
{7000,14400,19200} , 25000 , 323520000
{7035,14472,19296} , 25125 , 326763288
{7070,14544,19392} , 25250 , 330022752
{7105,14616,19488} , 25375 , 333298392
{7140,14688,19584} , 25500 , 336590208
{7175,14760,19680} , 25625 , 339898200
{7210,14832,19776} , 25750 , 343222368
{7245,14904,19872} , 25875 , 346562712
{7280,14976,19968} , 26000 , 349919232

Paul

• benvitalis says:

Nice. In my next blog, I’ll post another family of solutions, solutions that include some of your results.