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

are made to be squares

 

(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.
 
 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s