Is this true? I can prove that there doesn't exist $(x, y, z) \in \mathbb{Z}_{+}$ such that $x^4 - y^4 = z^2$.

trungnghia215 wrote: Is this true? I can prove that there doesn't exist $(x, y, z) \in \mathbb{Z}_{+}$ such that $x^4 - y^4 = z^2$. I agree with you, for the proof See here http://math.stackexchange.com/questions/153546/solving-x4-y4-z2

@jlammy Sorry but I do not know how to keep the latex.

^At least keep the LaTeX of the stackexchange post in the link...

huricane wrote: Prove that there exist an infinite number of odd natural numbers $m_1<m_2<...$ and an infinity of natural numbers $n_1<n_2<...$ ,such that $(m_k,n_k)=1$ and $m_k^4-2n_k^4$ is a perfect square,for all $k\in\mathbb{N}$.

First, we know $(m,n)=(3,2)$ is a solution. We can construct infinite solution as follows: Assume that $(m,n)$ is a solution.(So $gcd(m,2n)=1$) We can take positive integer $r$ such that $m^4-2n^4=r^2$. Set $(x,y)$$=$($m^4+2n^4, 2mnr$). Then, you can easily confirm $gcd(x,2y)=1$ and $x^4-2y^4$ is a perfect square. You are done. (example: (x,y)=(113, 84), (262621633, 151245528))