Diophantine equation: (x – y – z)(x – y + z)(x + y – z) = 8 xyz

 
 
Does the Diophantine equation

(x - y - z) \,(x - y + z) \,(x + y - z) \; = \; 8 \, x \, y \, z

have an infinite number of relatively prime solutions?

 
 
Solution:

 
we have the trivial solutions   (x, \; y, \; z) \; = \; (\pm 1, \; \pm 1, \; 0)   and permutations thereof.

For other solutions, note that each of the three equations

x \; - \; y \; - \; z \; = \; 2 \, \sqrt {y z}
x \; - \; y \; + \; z \; = \; 2 \, \sqrt {x z}
x \; + \; y \; - \; z \; = \; 2 \, \sqrt {x y}

is satisfied by   \sqrt {x} \; = \; \sqrt {y} \; + \; \sqrt {z}

consequently,   we have the infinite set of solutions

y \; = \; m^2
z \; = \; n^2
x \; = \; (m + n)^2

where   (m, \; n) \; = \; 1

I don’t know whether or not there are any other infinite sets of relatively prime solutions.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

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

2 Responses to Diophantine equation: (x – y – z)(x – y + z)(x + y – z) = 8 xyz

  1. Paul says:

    Yes, if you look at these few solutions you can see the pattern.

    {1,4,9}
    {1,9,16}
    {1,16,25}
    {1,25,36}
    {1,36,49}
    {1,49,64}
    {1,64,81}
    {4,9,25}
    {4,25,49}
    {4,49,81}
    {9,16,49}
    {9,25,64}
    {16,25,81}

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