Solving $x^4-y^4=z^2$ I have a question 
show that $x^4- y^4 = z^2$ has no nontrivial solution where $x$, $y$ and $z$ are nonzero integers
I tried infinite descent to find solution but I could not find it. square of a number in mod 4 is 1 or 0 I also tried to use but got nothing.
Can you help? 
thanks
 A: Suppose $z^2=y^4-x^4$ with $xyz\not=0$ for the smallest possible value of $y^4$. First we rewrite the
equation as $y^4=x^4+z^2$ so that $\{z,x^2,y^2\}$ is a Pythagorean triple.
It must be primitive, since if some prime $p$ divides $\gcd(x^2,y^2)$, then
$p\,|\,y^2$ implies $p\,|\,y$ which gives $p^4\,|\,y^4$.
Similarly, $p^4\,|\,x^4$, so $p^4\,|\,z^2$. This implies $p^2\,|\,z$, so that
$\left({z/p^2}\right)^2=\left({y/p}\right)^4-\left({x/p}\right)^4$ is a smaller solution. 
The Pythagorean triple $z,x^2,y^2$ is primitive and there are two cases:
If $x$ is even, then for some $m>n$, $(m,n)=1$,
 and $m\not\equiv n \pmod2$ we have
 $$ z=m^2-n^2,\quad x^2=2mn,\quad  y^2=m^2+n^2.$$
  Since $m,n$ have opposite parity, we can let $o$ denote the odd number and $e$ the even number among $\{m,n\}$.
  The primitive Pythagorean triple $\{n,m,y\}$ gives $$o=t^2-s^2,\quad e=2st,\quad y=t^2+s^2,$$ for some
 $t>s$, $(t,s)=1$,  and $t\not\equiv s\pmod2$. The formula for $x^2$ now
 gives $$(x/2)^2=ts(t^2-s^2)$$ which expresses the product of three relatively prime numbers as a square.
 That means all   three of them are squares: $s=u^2$, $t=v^2$, and $t^2-s^2=w^2$.
 In other words, $v^4-u^4=w^2$  is another
  solution to our equation, and it is smaller, since $v^4<t^2+s^2=y\leq y^4$.
If $x$ is odd, then for some $m>n$, $(m,n)=1$,
 and $m\not\equiv n\pmod2$ we have
 $$ x^2=m^2-n^2,\quad z=2mn,\quad  y^2=m^2+n^2.$$
 In this case $m^4-n^4=(xy)^2$ is a smaller solution, since $m^4<(m^2+n^2)^2=y^4$.
