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

(In reply to Possible solution by broll)

I'm not following you.

I agree your very neat parametric solution provides an infinite number of solutions (x,y,z), but since not all of those solutions gives gcd(x,y,z)=1, how can you be sure an infinity of them does?

For example, for even u, (x,y,z) are all even.

For u=multiple of 3, (x,y,z) are all multiples of by 3.

For u=multiple of v, (x,y,z) are all multiples of v^2.

Posted by xdog on 2011-05-10 00:33:18