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I'm told that Ellenberg and Bruin proved that the Diophantine equation $x^{4} + y^{4} = z^n$ has no primitive solutions in which $xyz\neq 0$ and $n\geq 4$. Does the proof use the same methods that Wiles-Taylor used to prove FLT ? If yes, to what extent is the proof reliant on those methods ? I mean, is the argument heavily based on the techniques that Wiles-Taylor used ?

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    $\begingroup$ Sorry to say this, but I have the feeling that your question is not focused enough for this forum. Can you make it more to the point? $\endgroup$
    – András Bátkai
    Commented Jul 6, 2019 at 21:42
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    $\begingroup$ @AndrásBátkai, i have no control over what you can ''feel''. But seriously, is there anything unclear about this question ?? $\endgroup$
    – Merton
    Commented Jul 6, 2019 at 21:50

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I think that this result was first published by Bennett-Ellenberg-Ng in Int. J. Number Theory 6 (2010), 311-338. According to the MathSciNet review, the argument is heavily based on the Taylor-Wiles method: "The proof follows from an artful strengthening of the methods of Ellenberg's previous article. It is based on a careful study of the properties of certain elliptic curves associated to each primitive solution (analogous to the Frey elliptic curves that eventually led to the proof of Fermat's Last Theorem)."

Looking at this paper and "Ellenberg's previous article", it is clear that many inventive ideas were needed in addition to the Taylor-Wiles method.

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