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

Find positive integers   $a, \; b, \; c, \; d$   such that

$a^2 \, b^2 \; + \; c^2 \, d^2$
$a^2 \, c^2 \; + \; b^2 \, d^2$
$a^2 \, d^2 \; + \; b^2 \, c^2$

$(a, b, c, d) = 1$

for   $a \; \leq \; 100$

(a,   b,   c,   d)   =   (3,   14,   8,   5)

$(3^2) \,(14^2) \; + \; (8^2) \,(5^2) \; = \; 58^2$
$(3^2) \,(8^2) \; + \; (14^2) \,(5^2) \; = \; 74^2$
$(3^2) \,(5^2) \; + \; (14^2) \,(8^2) \; = \; 113^2$

(a,   b,   c,   d)   =   (5,   38,   21,   16)

$(5^2) \,(38^2) \; + \; (21^2) \,(16^2) \; = \; 386^2$
$(5^2) \,(21^2) \; + \; (38^2) \,(16^2) \; = \; 617^2$
$(5^2) \,(16^2) \; + \; (38^2) \,(21^2) \; = \; 802^2$

(a,   b,   c,   d)   =   (8,   14,   5,   3)

$(8^2) \,(14^2) \; + \; (5^2) \,(3^2) \; = \; 113^2$
$(8^2) \,(5^2) \; + \; (14^2) \,(3^2) \; = \; 58^2$
$(8^2) \,(3^2) \; + \; (14^2) \,(5^2) \; = \; 74^2$

(a,   b,   c,   d)   =   (8,   26,   15,   7)

$(8^2) \,(26^2) \; + \; (15^2) \,(7^2) \; = \; 233^2$
$(8^2) \,(15^2) \; + \; (26^2) \,(7^2) \; = \; 218^2$
$(8^2) \,(7^2) \; + \; (26^2) \,(15^2) \; = \; 394^2$

(a,   b,   c,   d)   =   (11,   62,   35,   24)

$(11^2) \,(62^2) \; + \; (35^2) \,(24^2) \; = \; 1082^2$
$(11^2) \,(35^2) \; + \; (62^2) \,(24^2) \; = \; 1537^2$
$(11^2) \,(24^2) \; + \; (62^2) \,(35^2) \; = \; 2186^2$

(a,   b,   c,   d)   =   (13,   86,   48,   35)

$(13^2) \,(86^2) \; + \; (48^2) \,(35^2) \; = \; 2018^2$
$(13^2) \,(48^2) \; + \; (86^2) \,(35^2) \; = \; 3074^2$
$(13^2) \,(35^2) \; + \; (86^2) \,(48^2) \; = \; 4153^2$

(a,   b,   c,   d)   =   (15,   26,   7,   8)

$(15^2) \,(26^2) \; + \; (7^2) \,(8^2) \; = \; 394^2$
$(15^2) \,(7^2) \; + \; (26^2) \,(8^2) \; = \; 233^2$
$(15^2) \,(8^2) \; + \; (26^2) \,(7^2) \; = \; 218^2$

(a,   b,   c,   d)   =   (15,   26,   8,   7)

$(15^2) \,(26^2) \; + \; (8^2) \,(7^2) \; = \; 394^2$
$(15^2) \,(8^2) \; + \; (26^2) \,(7^2) \; = \; 218^2$
$(15^2) \,(7^2) \; + \; (26^2) \,(8^2) \; = \; 233^2$

(a,   b,   c,   d)   =   (16,   38,   21,   5)

$(16^2) \,(38^2) \; + \; (21^2) \,(5^2) \; = \; 617^2$
$(16^2) \,(21^2) \; + \; (38^2) \,(5^2) \; = \; 386^2$
$(16^2) \,(5^2) \; + \; (38^2) \,(21^2) \; = \; 802^2$

(a,   b,   c,   d)   =   (21,   38,   5,   16)

$(21^2) \,(38^2) \; + \; (5^2)(16^2) \; = \; 802^2$
$(21^2) \,(5^2) \; + \; (38^2)(16^2) \; = \; 617^2$
$(21^2) \,(16^2) \; + \; (38^2)(5^2) \; = \; 386^2$

(a,   b,   c,   d)   =   (24,   62,   35,   11)

$(24^2) \,(62^2) \; + \; (35^2) \,(11^2) \; = \; 1537^2$
$(24^2) \,(35^2) \; + \; (62^2) \,(11^2) \; = \; 1082^2$
$(24^2) \,(11^2) \; + \; (62^2) \,(35^2) \; = \; 2186^2$

(a,   b,   c,   d)   =   (32,   134,   77,   45)

$(32^2) \,(134^2) \; + \; (77^2) \,(45^2) \; = \; 5513^2$
$(32^2) \,(77^2) \; + \; (134^2) \,(45^2) \; = \; 6514^2$
$(32^2) \,(45^2) \; + \; (134^2) \,(77^2) \; = \; 10418^2$

(a,   b,   c,   d)   =   (33,   74,   40,   7)

$(33^2) \,(74^2) \; + \; (40^2) \,(7^2) \; = \; 2458^2$
$(33^2) \,(40^2) \; + \; (74^2) \,(7^2) \; = \; 1418^2$
$(33^2) \,(7^2) \; + \; (74^2) \,(40^2) \; = \; 2969^2$

(a,   b,   c,   d)   =   (35,   62,   11,   24)

$(35^2) \,(62^2) \; + \; (11^2) \,(24^2) \; = \; 2186^2$
$(35^2) \,(11^2) \; + \; (62^2) \,(24^2) \; = \; 1537^2$
$(35^2) \,(24^2) \; + \; (62^2) \,(11^2) \; = \; 1082^2$

(a,   b,   c,   d)   =   (39,   98,   55,   16)

$(39^2) \,(98^2) \; + \; (55^2) \,(16^2) \; = \; 3922^2$
$(39^2) \,(55^2) \; + \; (98^2) \,(16^2) \; = \; 2657^2$
$(39^2) \,(16^2) \; + \; (98^2) \,(55^2) \; = \; 5426^2$

(a,   b,   c,   d)   =   (40,   74,   7,   33)

$(40^2) \,(74^2) \; + \; (7^2) \,(33^2) \; = \; 2969^2$
$(40^2) \,(7^2) \; + \; (74^2) \,(33^2) \; = \; 2458^2$
$(40^2) \,(33^2) \; + \; (74^2) \,(7^2) \; = \; 1418^2$

(a,   b,   c,   d)   =   (40,   74,   33,   7)

$(40^2) \,(74^2) \; + \; (33^2) \,(7^2) \; = \; 2969^2$
$(40^2) \,(33^2) \; + \; (74^2) \,(7^2) \; = \; 1418^2$
$(40^2) \,(7^2) \; + \; (74^2) \,(33^2) \; = \; 2458^2$

(a,   b,   c,   d)   =   (48,   86,   13,   35)

$(48^2) \,(86^2) \; + \; (13^2) \,(35^2) \; = \; 4153^2$
$(48^2) \,(13^2) \; + \; (86^2) \,(35^2) \; = \; 3074^2$
$(48^2) \,(35^2) \; + \; (86^2) \,(13^2) \; = \; 2018^2$

(a,   b,   c,   d)   =   (48,   86,   35,   13)

$(48^2) \,(86^2) \; + \; (35^2) \,(13^2) \; = \; 4153^2$
$(48^2) \,(35^2) \; + \; (86^2) \,(13^2) \; = \; 2018^2$
$(48^2) \,(13^2) \; + \; (86^2) \,(35^2) \; = \; 3074^2$

(a,   b,   c,   d)   =   (55,   98,   39,   16)

$(55^2) \,(98^2) \; + \; (39^2) \,(16^2) \; = \; 5426^2$
$(55^2) \,(39^2) \; + \; (98^2) \,(16^2) \; = \; 2657^2$
$(55^2) \,(16^2) \; + \; (98^2) \,(39^2) \; = \; 3922^2$

(a,   b,   c,   d)   =   (56,   122,   65,   9)

$(56^2) \,(122^2) \; + \; (65^2) \,(9^2) \; = \; 6857^2$
$(56^2) \,(65^2) \; + \; (122^2) \,(9^2) \; = \; 3802^2$
$(56^2) \,(9^2) \; + \; (122^2) \,(65^2) \; = \; 7946^2$

(a,   b,   c,   d)   =   (65,   122,   56,   9)

$(65^2) \,(122^2) \; + \; (56^2) \,(9^2) \; = \; 7946^2$
$(65^2) \,(56^2) \; + \; (122^2) \,(9^2) \; = \; 3802^2$
$(65^2) \,(9^2) \; + \; (122^2) \,(56^2) \; = \; 6857^2$

(a,   b,   c,   d)   =   (77,   134,   32,   45)

$(77^2) \,(134^2) \; + \; (32^2) \,(45^2) \; = \; 10418^2$
$(77^2) \,(32^2) \; + \; (134^2) \,(45^2) \; = \; 6514^2$
$(77^2) \,(45^2) \; + \; (134^2) \,(32^2) \; = \; 5513^2$

(a,   b,   c,   d)   =   (80,   146,   17,   63)

$(80^2) \,(146^2) \; + \; (17^2) \,(63^2) \; = \; 11729^2$
$(80^2) \,(17^2) \; + \; (146^2) \,(63^2) \; = \; 9298^2$
$(80^2) \,(63^2) \; + \; (146^2) \,(17^2) \; = \; 5618^2$

(a,   b,   c,   d)   =   (80,   146,   63,   17)

$(80^2) \,(146^2) \; + \; (63^2) \,(17^2) \; = \; 11729^2$
$(80^2) \,(63^2) \; + \; (146^2) \,(17^2) \; = \; 5618^2$
$(80^2) \,(17^2) \; + \; (146^2) \,(63^2) \; = \; 9298^2$

(a,   b,   c,   d)   =   (80,   182,   99,   19)

$(80^2) \,(182^2) \; + \; (99^2) \,(19^2) \; = \; 14681^2$
$(80^2) \,(99^2) \; + \; (182^2) \,(19^2) \; = \; 8642^2$
$(80^2) \,(19^2) \; + \; (182^2) \,(99^2) \; = \; 18082^2$

(a,   b,   c,   d)   =   (88,   266,   153,   65)

$(88^2) \,(266^2) \; + \; (153^2) \,(65^2) \; = \; 25433^2$
$(88^2) \,(153^2) \; + \; (266^2) \,(65^2) \; = \; 21914^2$
$(88^2) \,(65^2) \; + \; (266^2) \,(153^2) \; = \; 41098^2$

(a,   b,   c,   d)   =   (91,   158,   51,   40)

$(91^2) \,(158^2) \; + \; (51^2) \,(40^2) \; = \; 14522^2$
$(91^2) \,(51^2) \; + \; (158^2) \,(40^2) \; = \; 7841^2$
$(91^2) \,(40^2) \; + \; (158^2) \,(51^2) \; = \; 8842^2$

(a,   b,   c,   d)   =   (96,   182,   11,   85)

$(96^2) \,(182^2) \; + \; (11^2) \,(85^2) \; = \; 17497^2$
$(96^2) \,(11^2) \; + \; (182^2) \,(85^2) \; = \; 15506^2$
$(96^2) \,(85^2) \; + \; (182^2) \,(11^2) \; = \; 8402^2$

(a,   b,   c,   d)   =   (96,   182,   85,   11)

$(96^2) \,(182^2) \; + \; (85^2) \,(11^2) \; = \; 17497^2$
$(96^2) \,(85^2) \; + \; (182^2) \,(11^2) \; = \; 8402^2$
$(96^2) \,(11^2) \; + \; (182^2) \,(85^2) \; = \; 15506^2$

(a,   b,   c,   d)   =   (99,   182,   19,   80)

$(99^2) \,(182^2) \; + \; (19^2) \,(80^2) \; = \; 18082^2$
$(99^2) \,(19^2) \; + \; (182^2) \,(80^2) \; = \; 14681^2$
$(99^2) \,(80^2) \; + \; (182^2) \,(19^2) \; = \; 8642^2$

Find other solutions.

math grad - Interest: Number theory
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### 2 Responses to Make {a^2 b^2 + c^2 d^2, a^2 c^2 + b^2 d^2, a^2 d^2 + b^2 c^2} squares

1. Paul says:

It seems it wont let me post anything in a list atm, so here a link to some more

https://www.dropbox.com/s/pqhmhiw4wm5vui9/3%20squares%20from%204.txt?dl=0

Paul.