# Math Help - Number Theory

1. ## Number Theory

There do not exist positive integers x,y,z satisfying 2xz=y^2 & x+z=997

2. ## Re: Number Theory

Is there a...question?

3. ## Re: Number Theory

Proceed by using these two equations to form a quadratic equation. Since we have two equations and three variables we cannot have a unique solution. we will get infinitely many solutions.
from first one we have z = (y^2)/(2x) plug in this in equation 2, we get
x + (y^2)/(2x) = 997
Form this quadratic and then try and reason out

4. ## Re: Number Theory

Originally Posted by swarna
There do not exist positive integers x,y,z satisfying 2xz=y^2 & x+z=997
Suppose x is even
$x=2t$
$4\text{tz}=y^2$
$t=u^2 , z=v^2$ since t and z are relatively prime
$2u^2+v^2=997$ which is impossible (Use congruence mod 8)