{(a^2 b^2) + n^2(a^2 + b^2), (a^2 b^2) + n^2 c^2}

 
 
 
Find three squares such that the product of any two added to the product of a given
square   n^2   by either the sum of those two or the remaining one gives a square

(a^2 \cdot b^2) \; + \; n^2 \,(a^2 + b^2)
(a^2 \cdot b^2) \; + \; n^2 \cdot c^2

(a^2 \cdot c^2) \; + \; n^2 \,(a^2 + c^2)
(a^2 \cdot c^2) \; + \; n^2 \cdot b^2

(b^2 \cdot c^2) \; + \; n^2 \,(b^2 + c^2)
(b^2 \cdot c^2) \; + \; n^2 \cdot a^2

are all squares.

 

Here’s one example,
  

a^2 = 5^2,     b^2 = 8^2,     c^2 = 14^2,     n = 3

(5^2\times 8^2) \; + \; 3^2 \,(5^2 + 8^2) \; = \; 49^2
(5^2\times 8^2) \; + \; (3^2\times 14^2) \; = \; 58^2

(5^2\times 14^2) \; + \; 3^2 \,(5^2 + 14^2) \; = \; 83^2
(5^2\times 14^2) \; + \; (3^2\times 8^2) \; = \; 74^2

(8^2\times 14^2) \; + \; 3^2 \,(8^2 + 14^2) \; = \; 122^2
(8^2\times 14^2) \; + \; (3^2\times 5^2) \; = \; 113^2

 

 
 

 
 

 
 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

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

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