Rational points on $4x^5 + y^2 = z^2$

Question

Does the title curve have any nonzero rational points ?

I have to admit that i didn't find any significant insight to this problem.

Answer 1

I might be missing something simple here, but:

$$z^2-y^2=4x^5 \Rightarrow (z-y)(z+y)=4x^5$$

Now, let $s,r$ be any rational numbers with $r \neq 0$. Then the system $$ z-y= r\\ z+y = \frac{4s^5}{r} $$ has unique solution $$z=\frac{r^2+4s^5}{2r} \\ y=\frac{4s^5-r^2}{2r}$$

Which leads to the rational points: $$(s, \frac{4s^5-r^2}{2r}, \frac{r^2+4s^5}{2r})$$