<s>[INST] Solutions to diophantinte equation $x^4+2y^4=z^2$ It is well known that the equation $x^4+y^4=z^2$ has no non-trivial solutions. The same holds also for the equation $x^4+2y^4=z^4$. What about the equation 
$x^4+2y^4=z^2$? 
 [/INST] The classic book Diophantine Analysis by R. D. Carmichael says on page 17 that the equation $x^4+2y^4=z^2$ has no non-trivial solutions. Here is the original text, slightly edited:


  
*The equation $x^4+2y^4=z^2$ is impossible in integers $x$, $y$, $z$, all of which
  are different from zero. 
  Suggestion. This may be proved by the method of infinite descent.  Begin by writing $z$ in the form
  $
z = x^2 + \frac{2py^2}{q},
$
  where $p$ and $q$ are relatively prime integers, and thence show that $x^2=q^2-2p^2$,
  $y^2=2pq$, provided that $x$, $y$, $z$ are prime each to each.
  

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