Prove that the equation x²+2y²=3z², with gcd(x,y,z) = 1 has an infinite number of positive integer solutions.

Set x=(u^2-6v^2),
Set y=(u^2+6uv+6v^2) = (u+3v)^2-3v^2
Set z=(u^2+4uv+6v^2) = (u+2v)^2+2v^2

Since u and v can (subject to x being greater than zero) be any numbers we choose, it follows that there are an infinite number of positive integer solutions with gcd(x,y,z) = 1.

If there are any other methods, I'd be interested to see them.

Edited on May 7, 2011, 2:39 pm

Posted by broll on 2011-05-07 14:32:35