## {a^2 + b^2, a^2 b^2 + c^2 d^2, a^2 d^2 + b^2 c^2}

Let

$X \; = \; a^2 \; + \; b^2$
$Y \; = \; a^2 \, b^2 \; + \; c^2 \, d^2$
$Z \; = \; a^2 \, d^2 \; + \; b^2 \, c^2$

find four positive integers   $(a, \; b, \; c, \; d)$   to make

(1)     $X \cdot Y$,   and   $X \cdot Z$   squares,   or
(2)     $Y \cdot Z$   a square,   or
(3)     $Y$   and   $Z$   squares

math grad - Interest: Number theory
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### One Response to {a^2 + b^2, a^2 b^2 + c^2 d^2, a^2 d^2 + b^2 c^2}

1. Paul says:

Here are solutions for part (1) only with (a, b, c, d) <=50.
Format is
{a, b, c, d}
{X . Y}
{X . Z}

{1,2,31,44}
(1^2 + 2^2 )(1^2 * 2^2 + 31^2 * 44^2) = 3050^2
(1^2 + 2^2 )(1^2 * 44^2 + 2^2 * 31^2) = 170^2

{1,3,9,39}
(1^2 + 3^2 )(1^2 * 3^2 + 9^2 * 39^2) = 1110^2
(1^2 + 3^2 )(1^2 * 39^2 + 3^2 * 9^2) = 150^2

{1,3,13,27}
(1^2 + 3^2 )(1^2 * 3^2 + 13^2 * 27^2) = 1110^2
(1^2 + 3^2 )(1^2 * 27^2 + 3^2 * 13^2) = 150^2

{2,1,44,31}
(2^2 + 1^2 )(2^2 * 1^2 + 44^2 * 31^2) = 3050^2
(2^2 + 1^2 )(2^2 * 31^2 + 1^2 * 44^2) = 170^2

{2,3,23,29}
(2^2 + 3^2 )(2^2 * 3^2 + 23^2 * 29^2) = 2405^2
(2^2 + 3^2 )(2^2 * 29^2 + 3^2 * 23^2) = 325^2

{3,1,27,13}
(3^2 + 1^2 )(3^2 * 1^2 + 27^2 * 13^2) = 1110^2
(3^2 + 1^2 )(3^2 * 13^2 + 1^2 * 27^2) = 150^2

{3,1,39,9}
(3^2 + 1^2 )(3^2 * 1^2 + 39^2 * 9^2) = 1110^2
(3^2 + 1^2 )(3^2 * 9^2 + 1^2 * 39^2) = 150^2

{3,2,29,23}
(3^2 + 2^2 )(3^2 * 2^2 + 29^2 * 23^2) = 2405^2
(3^2 + 2^2 )(3^2 * 23^2 + 2^2 * 29^2) = 325^2

{3,4,5,7}
(3^2 + 4^2 )(3^2 * 4^2 + 5^2 * 7^2) = 185^2
(3^2 + 4^2 )(3^2 * 7^2 + 4^2 * 5^2) = 145^2

{3,9,39,1}
(3^2 + 9^2 )(3^2 * 9^2 + 39^2 * 1^2) = 450^2
(3^2 + 9^2 )(3^2 * 1^2 + 9^2 * 39^2) = 3330^2

{4,3,7,5}
(4^2 + 3^2 )(4^2 * 3^2 + 7^2 * 5^2) = 185^2
(4^2 + 3^2 )(4^2 * 5^2 + 3^2 * 7^2) = 145^2

{4,16,13,47}
(4^2 + 16^2 )(4^2 * 16^2 + 13^2 * 47^2) = 10132^2
(4^2 + 16^2 )(4^2 * 47^2 + 16^2 * 13^2) = 4624^2

{4,22,44,29}
(4^2 + 22^2 )(4^2 * 22^2 + 44^2 * 29^2) = 28600^2
(4^2 + 22^2 )(4^2 * 29^2 + 22^2 * 44^2) = 21800^2

{5,12,3,21}
(5^2 + 12^2 )(5^2 * 12^2 + 3^2 * 21^2) = 1131^2
(5^2 + 12^2 )(5^2 * 21^2 + 12^2 * 3^2) = 1443^2

{5,12,14,32}
(5^2 + 12^2 )(5^2 * 12^2 + 14^2 * 32^2) = 5876^2
(5^2 + 12^2 )(5^2 * 32^2 + 12^2 * 14^2) = 3016^2

{6,7,28,23}
(6^2 + 7^2 )(6^2 * 7^2 + 28^2 * 23^2) = 5950^2
(6^2 + 7^2 )(6^2 * 23^2 + 7^2 * 28^2) = 2210^2

{6,8,10,14}
(6^2 + 8^2 )(6^2 * 8^2 + 10^2 * 14^2) = 1480^2
(6^2 + 8^2 )(6^2 * 14^2 + 8^2 * 10^2) = 1160^2

{6,8,28,5}
(6^2 + 8^2 )(6^2 * 8^2 + 28^2 * 5^2) = 1480^2
(6^2 + 8^2 )(6^2 * 5^2 + 8^2 * 28^2) = 2260^2

{7,6,23,28}
(7^2 + 6^2 )(7^2 * 6^2 + 23^2 * 28^2) = 5950^2
(7^2 + 6^2 )(7^2 * 28^2 + 6^2 * 23^2) = 2210^2

{7,24,4,40}
(7^2 + 24^2 )(7^2 * 24^2 + 4^2 * 40^2) = 5800^2
(7^2 + 24^2 )(7^2 * 40^2 + 24^2 * 4^2) = 7400^2

{7,24,26,45}
(7^2 + 24^2 )(7^2 * 24^2 + 26^2 * 45^2) = 29550^2
(7^2 + 24^2 )(7^2 * 45^2 + 24^2 * 26^2) = 17475^2

{8,6,5,28}
(8^2 + 6^2 )(8^2 * 6^2 + 5^2 * 28^2) = 1480^2
(8^2 + 6^2 )(8^2 * 28^2 + 6^2 * 5^2) = 2260^2

{8,6,14,10}
(8^2 + 6^2 )(8^2 * 6^2 + 14^2 * 10^2) = 1480^2
(8^2 + 6^2 )(8^2 * 10^2 + 6^2 * 14^2) = 1160^2

{8,15,3,42}
(8^2 + 15^2 )(8^2 * 15^2 + 3^2 * 42^2) = 2958^2
(8^2 + 15^2 )(8^2 * 42^2 + 15^2 * 3^2) = 5763^2

{8,15,7,26}
(8^2 + 15^2 )(8^2 * 15^2 + 7^2 * 26^2) = 3706^2
(8^2 + 15^2 )(8^2 * 26^2 + 15^2 * 7^2) = 3961^2

{8,15,14,9}
(8^2 + 15^2 )(8^2 * 15^2 + 14^2 * 9^2) = 2958^2
(8^2 + 15^2 )(8^2 * 9^2 + 15^2 * 14^2) = 3774^2

{8,15,14,25}
(8^2 + 15^2 )(8^2 * 15^2 + 14^2 * 25^2) = 6290^2
(8^2 + 15^2 )(8^2 * 25^2 + 15^2 * 14^2) = 4930^2

{8,15,26,7}
(8^2 + 15^2 )(8^2 * 15^2 + 26^2 * 7^2) = 3706^2
(8^2 + 15^2 )(8^2 * 7^2 + 15^2 * 26^2) = 6698^2

{9,3,1,39}
(9^2 + 3^2 )(9^2 * 3^2 + 1^2 * 39^2) = 450^2
(9^2 + 3^2 )(9^2 * 39^2 + 3^2 * 1^2) = 3330^2

{9,12,15,21}
(9^2 + 12^2 )(9^2 * 12^2 + 15^2 * 21^2) = 4995^2
(9^2 + 12^2 )(9^2 * 21^2 + 12^2 * 15^2) = 3915^2

{9,40,42,25}
(9^2 + 40^2 )(9^2 * 40^2 + 42^2 * 25^2) = 45510^2
(9^2 + 40^2 )(9^2 * 25^2 + 40^2 * 42^2) = 69495^2

{10,24,4,11}
(10^2 + 24^2 )(10^2 * 24^2 + 4^2 * 11^2) = 6344^2
(10^2 + 24^2 )(10^2 * 11^2 + 24^2 * 4^2) = 3796^2

{10,24,6,42}
(10^2 + 24^2 )(10^2 * 24^2 + 6^2 * 42^2) = 9048^2
(10^2 + 24^2 )(10^2 * 42^2 + 24^2 * 6^2) = 11544^2

{10,24,25,11}
(10^2 + 24^2 )(10^2 * 24^2 + 25^2 * 11^2) = 9490^2
(10^2 + 24^2 )(10^2 * 11^2 + 24^2 * 25^2) = 15860^2

{10,24,28,9}
(10^2 + 24^2 )(10^2 * 24^2 + 28^2 * 9^2) = 9048^2
(10^2 + 24^2 )(10^2 * 9^2 + 24^2 * 28^2) = 17628^2

{11,22,1,29}
(11^2 + 22^2 )(11^2 * 22^2 + 1^2 * 29^2) = 5995^2
(11^2 + 22^2 )(11^2 * 29^2 + 22^2 * 1^2) = 7865^2

{12,5,21,3}
(12^2 + 5^2 )(12^2 * 5^2 + 21^2 * 3^2) = 1131^2
(12^2 + 5^2 )(12^2 * 3^2 + 5^2 * 21^2) = 1443^2

{12,5,32,14}
(12^2 + 5^2 )(12^2 * 5^2 + 32^2 * 14^2) = 5876^2
(12^2 + 5^2 )(12^2 * 14^2 + 5^2 * 32^2) = 3016^2

{12,9,21,15}
(12^2 + 9^2 )(12^2 * 9^2 + 21^2 * 15^2) = 4995^2
(12^2 + 9^2 )(12^2 * 15^2 + 9^2 * 21^2) = 3915^2

{12,16,20,28}
(12^2 + 16^2 )(12^2 * 16^2 + 20^2 * 28^2) = 11840^2
(12^2 + 16^2 )(12^2 * 28^2 + 16^2 * 20^2) = 9280^2

{12,35,31,11}
(12^2 + 35^2 )(12^2 * 35^2 + 31^2 * 11^2) = 20017^2
(12^2 + 35^2 )(12^2 * 11^2 + 35^2 * 31^2) = 40441^2

{13,16,4,47}
(13^2 + 16^2 )(13^2 * 16^2 + 4^2 * 47^2) = 5780^2
(13^2 + 16^2 )(13^2 * 47^2 + 16^2 * 4^2) = 12665^2

{13,39,9,31}
(13^2 + 39^2 )(13^2 * 39^2 + 9^2 * 31^2) = 23790^2
(13^2 + 39^2 )(13^2 * 31^2 + 39^2 * 9^2) = 21970^2

{14,48,3,30}
(14^2 + 48^2 )(14^2 * 48^2 + 3^2 * 30^2) = 33900^2
(14^2 + 48^2 )(14^2 * 30^2 + 48^2 * 3^2) = 22200^2

{14,48,5,18}
(14^2 + 48^2 )(14^2 * 48^2 + 5^2 * 18^2) = 33900^2
(14^2 + 48^2 )(14^2 * 18^2 + 48^2 * 5^2) = 17400^2

{14,48,35,22}
(14^2 + 48^2 )(14^2 * 48^2 + 35^2 * 22^2) = 51100^2
(14^2 + 48^2 )(14^2 * 22^2 + 48^2 * 35^2) = 85400^2

{15,8,7,26}
(15^2 + 8^2 )(15^2 * 8^2 + 7^2 * 26^2) = 3706^2
(15^2 + 8^2 )(15^2 * 26^2 + 8^2 * 7^2) = 6698^2

{15,8,9,14}
(15^2 + 8^2 )(15^2 * 8^2 + 9^2 * 14^2) = 2958^2
(15^2 + 8^2 )(15^2 * 14^2 + 8^2 * 9^2) = 3774^2

{15,8,25,14}
(15^2 + 8^2 )(15^2 * 8^2 + 25^2 * 14^2) = 6290^2
(15^2 + 8^2 )(15^2 * 14^2 + 8^2 * 25^2) = 4930^2

{15,8,26,7}
(15^2 + 8^2 )(15^2 * 8^2 + 26^2 * 7^2) = 3706^2
(15^2 + 8^2 )(15^2 * 7^2 + 8^2 * 26^2) = 3961^2

{15,8,42,3}
(15^2 + 8^2 )(15^2 * 8^2 + 42^2 * 3^2) = 2958^2
(15^2 + 8^2 )(15^2 * 3^2 + 8^2 * 42^2) = 5763^2

{15,20,9,35}
(15^2 + 20^2 )(15^2 * 20^2 + 9^2 * 35^2) = 10875^2
(15^2 + 20^2 )(15^2 * 35^2 + 20^2 * 9^2) = 13875^2

{15,20,25,35}
(15^2 + 20^2 )(15^2 * 20^2 + 25^2 * 35^2) = 23125^2
(15^2 + 20^2 )(15^2 * 35^2 + 20^2 * 25^2) = 18125^2

{16,4,47,13}
(16^2 + 4^2 )(16^2 * 4^2 + 47^2 * 13^2) = 10132^2
(16^2 + 4^2 )(16^2 * 13^2 + 4^2 * 47^2) = 4624^2

{16,12,28,20}
(16^2 + 12^2 )(16^2 * 12^2 + 28^2 * 20^2) = 11840^2
(16^2 + 12^2 )(16^2 * 20^2 + 12^2 * 28^2) = 9280^2

{16,13,47,4}
(16^2 + 13^2 )(16^2 * 13^2 + 47^2 * 4^2) = 5780^2
(16^2 + 13^2 )(16^2 * 4^2 + 13^2 * 47^2) = 12665^2

{16,30,21,33}
(16^2 + 30^2 )(16^2 * 30^2 + 21^2 * 33^2) = 28662^2
(16^2 + 30^2 )(16^2 * 33^2 + 30^2 * 21^2) = 27948^2

{16,30,28,18}
(16^2 + 30^2 )(16^2 * 30^2 + 28^2 * 18^2) = 23664^2
(16^2 + 30^2 )(16^2 * 18^2 + 30^2 * 28^2) = 30192^2

{16,30,28,50}
(16^2 + 30^2 )(16^2 * 30^2 + 28^2 * 50^2) = 50320^2
(16^2 + 30^2 )(16^2 * 50^2 + 30^2 * 28^2) = 39440^2

{18,24,30,42}
(18^2 + 24^2 )(18^2 * 24^2 + 30^2 * 42^2) = 39960^2
(18^2 + 24^2 )(18^2 * 42^2 + 24^2 * 30^2) = 31320^2

{18,24,45,11}
(18^2 + 24^2 )(18^2 * 24^2 + 45^2 * 11^2) = 19710^2
(18^2 + 24^2 )(18^2 * 11^2 + 24^2 * 45^2) = 32940^2

{20,15,35,9}
(20^2 + 15^2 )(20^2 * 15^2 + 35^2 * 9^2) = 10875^2
(20^2 + 15^2 )(20^2 * 9^2 + 15^2 * 35^2) = 13875^2

{20,15,35,25}
(20^2 + 15^2 )(20^2 * 15^2 + 35^2 * 25^2) = 23125^2
(20^2 + 15^2 )(20^2 * 25^2 + 15^2 * 35^2) = 18125^2

{20,21,9,17}
(20^2 + 21^2 )(20^2 * 21^2 + 9^2 * 17^2) = 12963^2
(20^2 + 21^2 )(20^2 * 17^2 + 21^2 * 9^2) = 11281^2

{20,21,22,16}
(20^2 + 21^2 )(20^2 * 21^2 + 22^2 * 16^2) = 15892^2
(20^2 + 21^2 )(20^2 * 16^2 + 21^2 * 22^2) = 16298^2

{20,48,8,22}
(20^2 + 48^2 )(20^2 * 48^2 + 8^2 * 22^2) = 50752^2
(20^2 + 48^2 )(20^2 * 22^2 + 48^2 * 8^2) = 30368^2

{20,48,50,22}
(20^2 + 48^2 )(20^2 * 48^2 + 50^2 * 22^2) = 75920^2
(20^2 + 48^2 )(20^2 * 22^2 + 48^2 * 50^2) = 126880^2

{21,20,16,22}
(21^2 + 20^2 )(21^2 * 20^2 + 16^2 * 22^2) = 15892^2
(21^2 + 20^2 )(21^2 * 22^2 + 20^2 * 16^2) = 16298^2

{21,20,17,9}
(21^2 + 20^2 )(21^2 * 20^2 + 17^2 * 9^2) = 12963^2
(21^2 + 20^2 )(21^2 * 9^2 + 20^2 * 17^2) = 11281^2

{21,28,35,16}
(21^2 + 28^2 )(21^2 * 28^2 + 35^2 * 16^2) = 28420^2
(21^2 + 28^2 )(21^2 * 16^2 + 28^2 * 35^2) = 36260^2

{21,28,35,49}
(21^2 + 28^2 )(21^2 * 28^2 + 35^2 * 49^2) = 63455^2
(21^2 + 28^2 )(21^2 * 49^2 + 28^2 * 35^2) = 49735^2

{22,4,29,44}
(22^2 + 4^2 )(22^2 * 4^2 + 29^2 * 44^2) = 28600^2
(22^2 + 4^2 )(22^2 * 44^2 + 4^2 * 29^2) = 21800^2

{22,11,29,1}
(22^2 + 11^2 )(22^2 * 11^2 + 29^2 * 1^2) = 5995^2
(22^2 + 11^2 )(22^2 * 1^2 + 11^2 * 29^2) = 7865^2

{22,44,29,4}
(22^2 + 44^2 )(22^2 * 44^2 + 29^2 * 4^2) = 47960^2
(22^2 + 44^2 )(22^2 * 4^2 + 44^2 * 29^2) = 62920^2

{24,7,40,4}
(24^2 + 7^2 )(24^2 * 7^2 + 40^2 * 4^2) = 5800^2
(24^2 + 7^2 )(24^2 * 4^2 + 7^2 * 40^2) = 7400^2

{24,7,45,26}
(24^2 + 7^2 )(24^2 * 7^2 + 45^2 * 26^2) = 29550^2
(24^2 + 7^2 )(24^2 * 26^2 + 7^2 * 45^2) = 17475^2

{24,10,9,28}
(24^2 + 10^2 )(24^2 * 10^2 + 9^2 * 28^2) = 9048^2
(24^2 + 10^2 )(24^2 * 28^2 + 10^2 * 9^2) = 17628^2

{24,10,11,4}
(24^2 + 10^2 )(24^2 * 10^2 + 11^2 * 4^2) = 6344^2
(24^2 + 10^2 )(24^2 * 4^2 + 10^2 * 11^2) = 3796^2

{24,10,11,25}
(24^2 + 10^2 )(24^2 * 10^2 + 11^2 * 25^2) = 9490^2
(24^2 + 10^2 )(24^2 * 25^2 + 10^2 * 11^2) = 15860^2

{24,10,42,6}
(24^2 + 10^2 )(24^2 * 10^2 + 42^2 * 6^2) = 9048^2
(24^2 + 10^2 )(24^2 * 6^2 + 10^2 * 42^2) = 11544^2

{24,18,11,45}
(24^2 + 18^2 )(24^2 * 18^2 + 11^2 * 45^2) = 19710^2
(24^2 + 18^2 )(24^2 * 45^2 + 18^2 * 11^2) = 32940^2

{24,18,42,30}
(24^2 + 18^2 )(24^2 * 18^2 + 42^2 * 30^2) = 39960^2
(24^2 + 18^2 )(24^2 * 30^2 + 18^2 * 42^2) = 31320^2

{24,45,11,18}
(24^2 + 45^2 )(24^2 * 45^2 + 11^2 * 18^2) = 55998^2
(24^2 + 45^2 )(24^2 * 18^2 + 45^2 * 11^2) = 33507^2

{24,45,42,27}
(24^2 + 45^2 )(24^2 * 45^2 + 42^2 * 27^2) = 79866^2
(24^2 + 45^2 )(24^2 * 27^2 + 45^2 * 42^2) = 101898^2

{26,32,19,2}
(26^2 + 32^2 )(26^2 * 32^2 + 19^2 * 2^2) = 34340^2
(26^2 + 32^2 )(26^2 * 2^2 + 32^2 * 19^2) = 25160^2

{28,21,16,35}
(28^2 + 21^2 )(28^2 * 21^2 + 16^2 * 35^2) = 28420^2
(28^2 + 21^2 )(28^2 * 35^2 + 21^2 * 16^2) = 36260^2

{28,21,49,35}
(28^2 + 21^2 )(28^2 * 21^2 + 49^2 * 35^2) = 63455^2
(28^2 + 21^2 )(28^2 * 35^2 + 21^2 * 49^2) = 49735^2

{28,45,16,27}
(28^2 + 45^2 )(28^2 * 45^2 + 16^2 * 27^2) = 70596^2
(28^2 + 45^2 )(28^2 * 27^2 + 45^2 * 16^2) = 55332^2

{28,45,49,27}
(28^2 + 45^2 )(28^2 * 45^2 + 49^2 * 27^2) = 96831^2
(28^2 + 45^2 )(28^2 * 27^2 + 45^2 * 49^2) = 123543^2

{30,16,18,28}
(30^2 + 16^2 )(30^2 * 16^2 + 18^2 * 28^2) = 23664^2
(30^2 + 16^2 )(30^2 * 28^2 + 16^2 * 18^2) = 30192^2

{30,16,33,21}
(30^2 + 16^2 )(30^2 * 16^2 + 33^2 * 21^2) = 28662^2
(30^2 + 16^2 )(30^2 * 21^2 + 16^2 * 33^2) = 27948^2

{30,16,50,28}
(30^2 + 16^2 )(30^2 * 16^2 + 50^2 * 28^2) = 50320^2
(30^2 + 16^2 )(30^2 * 28^2 + 16^2 * 50^2) = 39440^2

{32,26,2,19}
(32^2 + 26^2 )(32^2 * 26^2 + 2^2 * 19^2) = 34340^2
(32^2 + 26^2 )(32^2 * 19^2 + 26^2 * 2^2) = 25160^2

{35,12,11,31}
(35^2 + 12^2 )(35^2 * 12^2 + 11^2 * 31^2) = 20017^2
(35^2 + 12^2 )(35^2 * 31^2 + 12^2 * 11^2) = 40441^2

{39,13,31,9}
(39^2 + 13^2 )(39^2 * 13^2 + 31^2 * 9^2) = 23790^2
(39^2 + 13^2 )(39^2 * 9^2 + 13^2 * 31^2) = 21970^2

{40,9,25,42}
(40^2 + 9^2 )(40^2 * 9^2 + 25^2 * 42^2) = 45510^2
(40^2 + 9^2 )(40^2 * 42^2 + 9^2 * 25^2) = 69495^2

{40,42,18,34}
(40^2 + 42^2 )(40^2 * 42^2 + 18^2 * 34^2) = 103704^2
(40^2 + 42^2 )(40^2 * 34^2 + 42^2 * 18^2) = 90248^2

{40,42,25,9}
(40^2 + 42^2 )(40^2 * 42^2 + 25^2 * 9^2) = 98310^2
(40^2 + 42^2 )(40^2 * 9^2 + 42^2 * 25^2) = 64380^2

{40,42,44,32}
(40^2 + 42^2 )(40^2 * 42^2 + 44^2 * 32^2) = 127136^2
(40^2 + 42^2 )(40^2 * 32^2 + 42^2 * 44^2) = 130384^2

{42,40,9,25}
(42^2 + 40^2 )(42^2 * 40^2 + 9^2 * 25^2) = 98310^2
(42^2 + 40^2 )(42^2 * 25^2 + 40^2 * 9^2) = 64380^2

{42,40,32,44}
(42^2 + 40^2 )(42^2 * 40^2 + 32^2 * 44^2) = 127136^2
(42^2 + 40^2 )(42^2 * 44^2 + 40^2 * 32^2) = 130384^2

{42,40,34,18}
(42^2 + 40^2 )(42^2 * 40^2 + 34^2 * 18^2) = 103704^2
(42^2 + 40^2 )(42^2 * 18^2 + 40^2 * 34^2) = 90248^2

{44,22,4,29}
(44^2 + 22^2 )(44^2 * 22^2 + 4^2 * 29^2) = 47960^2
(44^2 + 22^2 )(44^2 * 29^2 + 22^2 * 4^2) = 62920^2

{45,24,18,11}
(45^2 + 24^2 )(45^2 * 24^2 + 18^2 * 11^2) = 55998^2
(45^2 + 24^2 )(45^2 * 11^2 + 24^2 * 18^2) = 33507^2

{45,24,27,42}
(45^2 + 24^2 )(45^2 * 24^2 + 27^2 * 42^2) = 79866^2
(45^2 + 24^2 )(45^2 * 42^2 + 24^2 * 27^2) = 101898^2

{45,28,27,16}
(45^2 + 28^2 )(45^2 * 28^2 + 27^2 * 16^2) = 70596^2
(45^2 + 28^2 )(45^2 * 16^2 + 28^2 * 27^2) = 55332^2

{45,28,27,49}
(45^2 + 28^2 )(45^2 * 28^2 + 27^2 * 49^2) = 96831^2
(45^2 + 28^2 )(45^2 * 49^2 + 28^2 * 27^2) = 123543^2

{48,14,18,5}
(48^2 + 14^2 )(48^2 * 14^2 + 18^2 * 5^2) = 33900^2
(48^2 + 14^2 )(48^2 * 5^2 + 14^2 * 18^2) = 17400^2

{48,14,22,35}
(48^2 + 14^2 )(48^2 * 14^2 + 22^2 * 35^2) = 51100^2
(48^2 + 14^2 )(48^2 * 35^2 + 14^2 * 22^2) = 85400^2

{48,14,30,3}
(48^2 + 14^2 )(48^2 * 14^2 + 30^2 * 3^2) = 33900^2
(48^2 + 14^2 )(48^2 * 3^2 + 14^2 * 30^2) = 22200^2

{48,20,22,8}
(48^2 + 20^2 )(48^2 * 20^2 + 22^2 * 8^2) = 50752^2
(48^2 + 20^2 )(48^2 * 8^2 + 20^2 * 22^2) = 30368^2

{48,20,22,50}
(48^2 + 20^2 )(48^2 * 20^2 + 22^2 * 50^2) = 75920^2
(48^2 + 20^2 )(48^2 * 50^2 + 20^2 * 22^2) = 126880^2

Paul.