{(ab+1), (ac+1), (bc+1)} are squares

 
 
If   (a \,b + 1)   is a square, then there exists a positive integer   c   such that

(a \,c + 1)   and   (b \,c + 1)   are both squares.

 
if   a \,b \; + \; 1 \; = \; n^2
then   c \; = \; 2 \,n \; + a \; + \; b

a \,c \; + \; 1 \; = \; a \,(2 \,n + a + b) \; + \; 1
a \,c \; + \; 1 \; = \; 2 \,a \,n \; + \; a^2 \; + \; a \,b \; + \; 1
a \,c \; + \; 1 \; = \; 2 \,a \,n \; + \; a^2 \; + \; (a \,b + 1)
a \,c \; + \; 1 \; = \; 2 \,a \,n \; + \; a^2 \; + \; n^2
a \,c \; + \; 1 \; = \; (n + a)^2

b \,c \; + \; 1 \; = \; b \,(2 \,n + a + b) \; + \; 1
b \,c \; + \; 1 \; = \; 2 \,b \,n \; + \; a \,b \; + \; b^2 \; + \; 1
b \,c \; + \; 1 \; = \; 2 \,b \,n \; + \; b^2 + (a \,b + 1)
b \,c \; + \; 1 \; = \; 2 \,b \,n \; + \; b^2 \; + \; n^2
b \,c \; + \; 1 \; = \; (n + b)^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