{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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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.

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