What Are the Solutions to the Diophantine Equation x^2 = 2y^2 = 3z^2?

[lei123]

Homework Statement

x^2 = 2y^2 = 3z^2
Find all the solutions.

Homework Equations

There is a method once I substitute the above formula into the form ap^2 + q^2 = z^2, to get all the solutions.

The Attempt at a Solution

I'm having a little trouble with the substitution.
I was thinking, at first that one could argue that x and y are both congruent to 0, mod 3. Then x-y = 3p and x+ 2y = 3q; one could solve for x, and y, and then backsubstitute, when I get the formulas for x and y in terms of integers, they don't check in the first equation

 

[chwala]

this problem did not get any feedback?

 

[nikilKRP]

Homework Statement

x^2 = 2y^2 = 3z^2

Are you sure this is correct? Is it supposed to be $x^2 + 2y^2 =3z^2$ or something similar?
$x^2 = 2y^2 = 3z^2$ does not seem correct, maybe it's a typo.