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

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

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

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.

• benvitalis says:

Awesome work. I’ll post your results.