Diophantine Equations

Date: 06/29/2001 at 13:45:36
From: Deva Mishra
Subject: Diophantine Equations

I have recently become interested in number theory. I am reading a 
book by Andrew Weil that discusses Diophantine Equations, and he 
proposed a question:

Find rational x and y such that x^2+x^2*y^2 and y^2+x^2*y^2 are 
perfect squares. Or, more simply, x^2+x^2*y^2 = m^2 and y^2+x^2*y^2 = 
n^2, where n and m are rational numbers. 

I tried applying the typical form to the two equations, but I got 
stuck. The solution would be greatly appreciated. 

Thank you very much.
Deva Mishra

Date: 06/29/2001 at 16:10:39
From: Doctor Jaffee
Subject: Re: Diophantine Equations

Hi Deva,

Here is how I would approach this problem.

x^2+x^2*y^2 = x^2(1 + y^2)

Since x^2 is a perfect square for any rational value of x, we need to 
concentrate on 1 + y^2.

Consider any Pythagorean triple (a,b,c) and let y = a/b.  

                              a^2 + b^2            c^2
Then 1 + y^2 = 1 + a^2/b^2 = ------------------ = ------, so m = c/b
                                   b^2             b^2

Likewise, you can find a value for x.

Besides Pythagorean triples, you can let x or y = 0. Also, integers 
such as -a and -b can be used where (a,b,c) is a Pythagorean triple.

I hope my answer has helped you understand the problem better. If you 
want to discuss it more, or if you have other questions, write back.  
Thanks for writing to Ask Dr. Math.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/