a^2+b^2+c^2+d^2 = e^2+f^2+g^2+h^2 and a*b*c*d = e*f*g*h

 
 

Find solutions in positive integers

 

a^2 \; + \; b^2 \; + \; c^2 \; + \; d^2 \; = \; e^2 \; + \; f^2 \; + \; g^2 \; + \; h^2

                     a \cdot b \cdot c \cdot d \; = \; e \cdot f \cdot g \cdot h

 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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2 Responses to a^2+b^2+c^2+d^2 = e^2+f^2+g^2+h^2 and a*b*c*d = e*f*g*h

  1. paul says:

    Here are those with a b c d <=10000

    1^2 + 6^2 + 7^2 + 20^2 = 2^2 + 4^2 + 5^2 + 21^2 = 486
    1 x 6 x 7 x 20 = 2 x 4 x 5 x 21 = 840

    1^2 + 5^2 + 12^2 + 18^2 = 2^2 + 3^2 + 9^2 + 20^2 = 494
    1 x 5 x 12 x 18 = 2 x 3 x 9 x 20 = 1080

    1^2 + 5^2 + 10^2 + 24^2 = 2^2 + 3^2 + 8^2 + 25^2 = 702
    1 x 5 x 10 x 24 = 2 x 3 x 8 x 25 = 1200

    1^2 + 10^2 + 12^2 + 15^2 = 3^2 + 5^2 + 6^2 + 20^2 = 470
    1 x 10 x 12 x 15 = 3 x 5 x 6 x 20 = 1800

    1^2 + 8^2 + 12^2 + 21^2 = 3^2 + 4^2 + 7^2 + 24^2 = 650
    1 x 8 x 12 x 21 = 3 x 4 x 7 x 24 = 2016

    1^2 + 3^2 + 26^2 + 30^2 = 2^2 + 5^2 + 6^2 + 39^2 = 1586
    1 x 3 x 26 x 30 = 2 x 5 x 6 x 39 = 2340

    1^2 + 4^2 + 18^2 + 33^2 = 2^2 + 3^2 + 11^2 + 36^2 = 1430
    1 x 4 x 18 x 33 = 2 x 3 x 11 x 36 = 2376
    1^2 + 11^2 + 12^2 + 18^2 = 3^2 + 4^2 + 9^2 + 22^2 = 590
    1 x 11 x 12 x 18 = 3 x 4 x 9 x 22 = 2376

    1^2 + 12^2 + 14^2 + 15^2 = 3^2 + 4^2 + 10^2 + 21^2 = 566
    1 x 12 x 14 x 15 = 3 x 4 x 10 x 21 = 2520

    2^2 + 9^2 + 12^2 + 14^2 = 4^2 + 6^2 + 7^2 + 18^2 = 425
    2 x 9 x 12 x 14 = 4 x 6 x 7 x 18 = 3024

    1^2 + 7^2 + 20^2 + 22^2 = 2^2 + 5^2 + 11^2 + 28^2 = 934
    1 x 7 x 20 x 22 = 2 x 5 x 11 x 28 = 3080

    1^2 + 9^2 + 18^2 + 20^2 = 4^2 + 5^2 + 6^2 + 27^2 = 806
    1 x 9 x 18 x 20 = 4 x 5 x 6 x 27 = 3240

    1^2 + 10^2 + 11^2 + 30^2 = 2^2 + 3^2 + 22^2 + 25^2 = 1122
    1 x 10 x 11 x 30 = 2 x 3 x 22 x 25 = 3300

    1^2 + 6^2 + 18^2 + 32^2 = 3^2 + 4^2 + 8^2 + 36^2 = 1385
    1 x 6 x 18 x 32 = 3 x 4 x 8 x 36 = 3456
    1^2 + 12^2 + 16^2 + 18^2 = 2^2 + 8^2 + 9^2 + 24^2 = 725
    1 x 12 x 16 x 18 = 2 x 8 x 9 x 24 = 3456

    1^2 + 9^2 + 18^2 + 22^2 = 2^2 + 6^2 + 11^2 + 27^2 = 890
    1 x 9 x 18 x 22 = 2 x 6 x 11 x 27 = 3564

    1^2 + 13^2 + 14^2 + 20^2 = 4^2 + 5^2 + 7^2 + 26^2 = 766
    1 x 13 x 14 x 20 = 4 x 5 x 7 x 26 = 3640

    1^2 + 6^2 + 22^2 + 28^2 = 2^2 + 4^2 + 14^2 + 33^2 = 1305
    1 x 6 x 22 x 28 = 2 x 4 x 14 x 33 = 3696

    1^2 + 5^2 + 18^2 + 42^2 = 2^2 + 6^2 + 7^2 + 45^2 = 2114
    1 x 5 x 18 x 42 = 2 x 6 x 7 x 45 = 3780

    1^2 + 4^2 + 30^2 + 33^2 = 3^2 + 5^2 + 6^2 + 44^2 = 2006
    1 x 4 x 30 x 33 = 3 x 5 x 6 x 44 = 3960
    1^2 + 11^2 + 12^2 + 30^2 = 4^2 + 5^2 + 6^2 + 33^2 = 1166
    1 x 11 x 12 x 30 = 4 x 5 x 6 x 33 = 3960

    1^2 + 13^2 + 14^2 + 22^2 = 2^2 + 7^2 + 11^2 + 26^2 = 850
    1 x 13 x 14 x 22 = 2 x 7 x 11 x 26 = 4004

    1^2 + 9^2 + 12^2 + 38^2 = 2^2 + 3^2 + 19^2 + 36^2 = 1670
    1 x 9 x 12 x 38 = 2 x 3 x 19 x 36 = 4104

    1^2 + 7^2 + 20^2 + 30^2 = 3^2 + 4^2 + 10^2 + 35^2 = 1350
    1 x 7 x 20 x 30 = 3 x 4 x 10 x 35 = 4200
    1^2 + 14^2 + 15^2 + 20^2 = 2^2 + 7^2 + 12^2 + 25^2 = 822
    1 x 14 x 15 x 20 = 2 x 7 x 12 x 25 = 4200

    2^2 + 8^2 + 11^2 + 24^2 = 3^2 + 4^2 + 16^2 + 22^2 = 765
    2 x 8 x 11 x 24 = 3 x 4 x 16 x 22 = 4224

    1^2 + 8^2 + 20^2 + 28^2 = 2^2 + 5^2 + 14^2 + 32^2 = 1249
    1 x 8 x 20 x 28 = 2 x 5 x 14 x 32 = 4480

    1^2 + 9^2 + 18^2 + 28^2 = 2^2 + 4^2 + 21^2 + 27^2 = 1190
    1 x 9 x 18 x 28 = 2 x 4 x 21 x 27 = 4536
    1^2 + 12^2 + 18^2 + 21^2 = 3^2 + 6^2 + 9^2 + 28^2 = 910
    1 x 12 x 18 x 21 = 3 x 6 x 9 x 28 = 4536

    1^2 + 10^2 + 14^2 + 33^2 = 2^2 + 6^2 + 11^2 + 35^2 = 1386
    1 x 10 x 14 x 33 = 2 x 6 x 11 x 35 = 4620

    2^2 + 10^2 + 15^2 + 16^2 = 4^2 + 5^2 + 12^2 + 20^2 = 585
    2 x 10 x 15 x 16 = 4 x 5 x 12 x 20 = 4800

    1^2 + 3^2 + 38^2 + 44^2 = 2^2 + 4^2 + 11^2 + 57^2 = 3390
    1 x 3 x 38 x 44 = 2 x 4 x 11 x 57 = 5016

    1^2 + 8^2 + 15^2 + 42^2 = 2^2 + 3^2 + 21^2 + 40^2 = 2054
    1 x 8 x 15 x 42 = 2 x 3 x 21 x 40 = 5040
    2^2 + 8^2 + 15^2 + 21^2 = 3^2 + 7^2 + 10^2 + 24^2 = 734
    2 x 8 x 15 x 21 = 3 x 7 x 10 x 24 = 5040

    3^2 + 5^2 + 18^2 + 20^2 = 4^2 + 6^2 + 9^2 + 25^2 = 758
    3 x 5 x 18 x 20 = 4 x 6 x 9 x 25 = 5400
    3^2 + 9^2 + 10^2 + 20^2 = 4^2 + 5^2 + 15^2 + 18^2 = 590
    3 x 9 x 10 x 20 = 4 x 5 x 15 x 18 = 5400

    1^2 + 6^2 + 28^2 + 33^2 = 3^2 + 4^2 + 11^2 + 42^2 = 1910
    1 x 6 x 28 x 33 = 3 x 4 x 11 x 42 = 5544
    1^2 + 11^2 + 12^2 + 42^2 = 3^2 + 6^2 + 7^2 + 44^2 = 2030
    1 x 11 x 12 x 42 = 3 x 6 x 7 x 44 = 5544

    1^2 + 8^2 + 20^2 + 35^2 = 4^2 + 5^2 + 7^2 + 40^2 = 1690
    1 x 8 x 20 x 35 = 4 x 5 x 7 x 40 = 5600

    1^2 + 5^2 + 22^2 + 52^2 = 2^2 + 4^2 + 13^2 + 55^2 = 3214
    1 x 5 x 22 x 52 = 2 x 4 x 13 x 55 = 5720

    1^2 + 10^2 + 21^2 + 28^2 = 4^2 + 6^2 + 7^2 + 35^2 = 1326
    1 x 10 x 21 x 28 = 4 x 6 x 7 x 35 = 5880

    2^2 + 11^2 + 15^2 + 18^2 = 3^2 + 9^2 + 10^2 + 22^2 = 674
    2 x 11 x 15 x 18 = 3 x 9 x 10 x 22 = 5940

    1^2 + 8^2 + 14^2 + 55^2 = 2^2 + 5^2 + 11^2 + 56^2 = 3286
    1 x 8 x 14 x 55 = 2 x 5 x 11 x 56 = 6160
    1^2 + 8^2 + 22^2 + 35^2 = 2^2 + 7^2 + 11^2 + 40^2 = 1774
    1 x 8 x 22 x 35 = 2 x 7 x 11 x 40 = 6160

    2^2 + 5^2 + 21^2 + 30^2 = 3^2 + 6^2 + 10^2 + 35^2 = 1370
    2 x 5 x 21 x 30 = 3 x 6 x 10 x 35 = 6300

    1^2 + 12^2 + 22^2 + 24^2 = 4^2 + 6^2 + 8^2 + 33^2 = 1205
    1 x 12 x 22 x 24 = 4 x 6 x 8 x 33 = 6336

    1^2 + 8^2 + 14^2 + 57^2 = 2^2 + 3^2 + 19^2 + 56^2 = 3510
    1 x 8 x 14 x 57 = 2 x 3 x 19 x 56 = 6384

    1^2 + 7^2 + 21^2 + 45^2 = 3^2 + 5^2 + 9^2 + 49^2 = 2516
    1 x 7 x 21 x 45 = 3 x 5 x 9 x 49 = 6615

    2^2 + 7^2 + 16^2 + 30^2 = 3^2 + 4^2 + 20^2 + 28^2 = 1209
    2 x 7 x 16 x 30 = 3 x 4 x 20 x 28 = 6720

    1^2 + 5^2 + 30^2 + 46^2 = 2^2 + 3^2 + 23^2 + 50^2 = 3042
    1 x 5 x 30 x 46 = 2 x 3 x 23 x 50 = 6900

    1^2 + 16^2 + 18^2 + 24^2 = 4^2 + 6^2 + 9^2 + 32^2 = 1157
    1 x 16 x 18 x 24 = 4 x 6 x 9 x 32 = 6912

    3^2 + 8^2 + 12^2 + 25^2 = 4^2 + 5^2 + 15^2 + 24^2 = 842
    3 x 8 x 12 x 25 = 4 x 5 x 15 x 24 = 7200

    1^2 + 12^2 + 18^2 + 34^2 = 2^2 + 6^2 + 17^2 + 36^2 = 1625
    1 x 12 x 18 x 34 = 2 x 6 x 17 x 36 = 7344

    1^2 + 7^2 + 30^2 + 36^2 = 3^2 + 4^2 + 14^2 + 45^2 = 2246
    1 x 7 x 30 x 36 = 3 x 4 x 14 x 45 = 7560
    1^2 + 9^2 + 20^2 + 42^2 = 3^2 + 4^2 + 14^2 + 45^2 = 2246
    1 x 9 x 20 x 42 = 3 x 4 x 14 x 45 = 7560
    1^2 + 9^2 + 28^2 + 30^2 = 2^2 + 5^2 + 21^2 + 36^2 = 1766
    1 x 9 x 28 x 30 = 2 x 5 x 21 x 36 = 7560
    2^2 + 7^2 + 15^2 + 36^2 = 3^2 + 4^2 + 18^2 + 35^2 = 1574
    2 x 7 x 15 x 36 = 3 x 4 x 18 x 35 = 7560
    2^2 + 9^2 + 12^2 + 35^2 = 3^2 + 7^2 + 10^2 + 36^2 = 1454
    2 x 9 x 12 x 35 = 3 x 7 x 10 x 36 = 7560

    1^2 + 12^2 + 24^2 + 27^2 = 3^2 + 8^2 + 9^2 + 36^2 = 1450
    1 x 12 x 24 x 27 = 3 x 8 x 9 x 36 = 7776

    2^2 + 12^2 + 13^2 + 25^2 = 4^2 + 5^2 + 15^2 + 26^2 = 942
    2 x 12 x 13 x 25 = 4 x 5 x 15 x 26 = 7800

    1^2 + 10^2 + 24^2 + 33^2 = 3^2 + 6^2 + 11^2 + 40^2 = 1766
    1 x 10 x 24 x 33 = 3 x 6 x 11 x 40 = 7920
    1^2 + 12^2 + 22^2 + 30^2 = 2^2 + 6^2 + 20^2 + 33^2 = 1529
    1 x 12 x 22 x 30 = 2 x 6 x 20 x 33 = 7920

    1^2 + 8^2 + 24^2 + 42^2 = 4^2 + 6^2 + 7^2 + 48^2 = 2405
    1 x 8 x 24 x 42 = 4 x 6 x 7 x 48 = 8064
    1^2 + 12^2 + 24^2 + 28^2 = 2^2 + 6^2 + 21^2 + 32^2 = 1505
    1 x 12 x 24 x 28 = 2 x 6 x 21 x 32 = 8064
    2^2 + 8^2 + 21^2 + 24^2 = 3^2 + 6^2 + 16^2 + 28^2 = 1085
    2 x 8 x 21 x 24 = 3 x 6 x 16 x 28 = 8064
    2^2 + 12^2 + 16^2 + 21^2 = 3^2 + 8^2 + 14^2 + 24^2 = 845
    2 x 12 x 16 x 21 = 3 x 8 x 14 x 24 = 8064

    1^2 + 10^2 + 15^2 + 54^2 = 2^2 + 3^2 + 27^2 + 50^2 = 3242
    1 x 10 x 15 x 54 = 2 x 3 x 27 x 50 = 8100

    1^2 + 6^2 + 22^2 + 63^2 = 2^2 + 7^2 + 9^2 + 66^2 = 4490
    1 x 6 x 22 x 63 = 2 x 7 x 9 x 66 = 8316
    3^2 + 11^2 + 14^2 + 18^2 = 6^2 + 7^2 + 9^2 + 22^2 = 650
    3 x 11 x 14 x 18 = 6 x 7 x 9 x 22 = 8316

    1^2 + 5^2 + 38^2 + 44^2 = 2^2 + 4^2 + 19^2 + 55^2 = 3406
    1 x 5 x 38 x 44 = 2 x 4 x 19 x 55 = 8360

    1^2 + 14^2 + 24^2 + 25^2 = 3^2 + 8^2 + 10^2 + 35^2 = 1398
    1 x 14 x 24 x 25 = 3 x 8 x 10 x 35 = 8400
    1^2 + 15^2 + 20^2 + 28^2 = 4^2 + 5^2 + 12^2 + 35^2 = 1410
    1 x 15 x 20 x 28 = 4 x 5 x 12 x 35 = 8400

    1^2 + 9^2 + 18^2 + 52^2 = 3^2 + 4^2 + 13^2 + 54^2 = 3110
    1 x 9 x 18 x 52 = 3 x 4 x 13 x 54 = 8424
    1^2 + 12^2 + 26^2 + 27^2 = 2^2 + 9^2 + 13^2 + 36^2 = 1550
    1 x 12 x 26 x 27 = 2 x 9 x 13 x 36 = 8424

    1^2 + 4^2 + 44^2 + 48^2 = 2^2 + 6^2 + 11^2 + 64^2 = 4257
    1 x 4 x 44 x 48 = 2 x 6 x 11 x 64 = 8448

    2^2 + 11^2 + 13^2 + 30^2 = 3^2 + 5^2 + 22^2 + 26^2 = 1194
    2 x 11 x 13 x 30 = 3 x 5 x 22 x 26 = 8580

    1^2 + 12^2 + 16^2 + 45^2 = 4^2 + 5^2 + 9^2 + 48^2 = 2426
    1 x 12 x 16 x 45 = 4 x 5 x 9 x 48 = 8640
    1^2 + 16^2 + 18^2 + 30^2 = 4^2 + 5^2 + 12^2 + 36^2 = 1481
    1 x 16 x 18 x 30 = 4 x 5 x 12 x 36 = 8640
    4^2 + 9^2 + 12^2 + 20^2 = 5^2 + 6^2 + 16^2 + 18^2 = 641
    4 x 9 x 12 x 20 = 5 x 6 x 16 x 18 = 8640

    1^2 + 7^2 + 24^2 + 52^2 = 3^2 + 4^2 + 13^2 + 56^2 = 3330
    1 x 7 x 24 x 52 = 3 x 4 x 13 x 56 = 8736

    1^2 + 9^2 + 25^2 + 39^2 = 3^2 + 5^2 + 13^2 + 45^2 = 2228
    1 x 9 x 25 x 39 = 3 x 5 x 13 x 45 = 8775

    1^2 + 17^2 + 20^2 + 26^2 = 4^2 + 5^2 + 13^2 + 34^2 = 1366
    1 x 17 x 20 x 26 = 4 x 5 x 13 x 34 = 8840

    1^2 + 10^2 + 28^2 + 32^2 = 2^2 + 7^2 + 16^2 + 40^2 = 1909
    1 x 10 x 28 x 32 = 2 x 7 x 16 x 40 = 8960

    1^2 + 17^2 + 22^2 + 24^2 = 3^2 + 8^2 + 11^2 + 34^2 = 1350
    1 x 17 x 22 x 24 = 3 x 8 x 11 x 34 = 8976

    1^2 + 13^2 + 21^2 + 33^2 = 3^2 + 7^2 + 11^2 + 39^2 = 1700
    1 x 13 x 21 x 33 = 3 x 7 x 11 x 39 = 9009

    1^2 + 7^2 + 28^2 + 46^2 = 2^2 + 4^2 + 23^2 + 49^2 = 2950
    1 x 7 x 28 x 46 = 2 x 4 x 23 x 49 = 9016

    1^2 + 14^2 + 25^2 + 26^2 = 2^2 + 10^2 + 13^2 + 35^2 = 1498
    1 x 14 x 25 x 26 = 2 x 10 x 13 x 35 = 9100

    1^2 + 15^2 + 22^2 + 28^2 = 2^2 + 11^2 + 12^2 + 35^2 = 1494
    1 x 15 x 22 x 28 = 2 x 11 x 12 x 35 = 9240
    1^2 + 20^2 + 21^2 + 22^2 = 4^2 + 5^2 + 14^2 + 33^2 = 1326
    1 x 20 x 21 x 22 = 4 x 5 x 14 x 33 = 9240

    1^2 + 13^2 + 24^2 + 30^2 = 5^2 + 6^2 + 8^2 + 39^2 = 1646
    1 x 13 x 24 x 30 = 5 x 6 x 8 x 39 = 9360
    2^2 + 8^2 + 15^2 + 39^2 = 3^2 + 6^2 + 13^2 + 40^2 = 1814
    2 x 8 x 15 x 39 = 3 x 6 x 13 x 40 = 9360

    1^2 + 9^2 + 18^2 + 58^2 = 2^2 + 3^2 + 29^2 + 54^2 = 3770
    1 x 9 x 18 x 58 = 2 x 3 x 29 x 54 = 9396

    1^2 + 12^2 + 18^2 + 44^2 = 2^2 + 4^2 + 33^2 + 36^2 = 2405
    1 x 12 x 18 x 44 = 2 x 4 x 33 x 36 = 9504

    1^2 + 10^2 + 30^2 + 32^2 = 3^2 + 4^2 + 20^2 + 40^2 = 2025
    1 x 10 x 30 x 32 = 3 x 4 x 20 x 40 = 9600
    1^2 + 16^2 + 24^2 + 25^2 = 3^2 + 5^2 + 20^2 + 32^2 = 1458
    1 x 16 x 24 x 25 = 3 x 5 x 20 x 32 = 9600

    1^2 + 10^2 + 23^2 + 42^2 = 2^2 + 7^2 + 15^2 + 46^2 = 2394
    1 x 10 x 23 x 42 = 2 x 7 x 15 x 46 = 9660

    1^2 + 17^2 + 22^2 + 26^2 = 2^2 + 11^2 + 13^2 + 34^2 = 1450
    1 x 17 x 22 x 26 = 2 x 11 x 13 x 34 = 9724

    1^2 + 7^2 + 24^2 + 58^2 = 2^2 + 3^2 + 29^2 + 56^2 = 3990
    1 x 7 x 24 x 58 = 2 x 3 x 29 x 56 = 9744

    2^2 + 7^2 + 25^2 + 28^2 = 4^2 + 5^2 + 14^2 + 35^2 = 1462
    2 x 7 x 25 x 28 = 4 x 5 x 14 x 35 = 9800

    2^2 + 10^2 + 15^2 + 33^2 = 3^2 + 5^2 + 22^2 + 30^2 = 1418
    2 x 10 x 15 x 33 = 3 x 5 x 22 x 30 = 9900
    2^2 + 11^2 + 15^2 + 30^2 = 5^2 + 6^2 + 10^2 + 33^2 = 1250
    2 x 11 x 15 x 30 = 5 x 6 x 10 x 33 = 9900

    1^2 + 12^2 + 26^2 + 32^2 = 2^2 + 8^2 + 16^2 + 39^2 = 1845
    1 x 12 x 26 x 32 = 2 x 8 x 16 x 39 = 9984

    Paul.

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