A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,z\in \mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two different elliptic curves. To transform the Fermat Curve of degree $3$, the substitutions $$ a=\frac{12z}{x+y},\quad b=\frac{36(x-y)}{x+y} $$ produce $b^2=a^3-432$, an elliptic curve. Similarly for the Fermat Curve of degree $4$, the substitutions $$ a=\frac{2(y^2+z^2)}{x^2},\quad b=\frac{4y(y^2+z^2)}{x^3} $$ give $b^2=a^3-4a$. However, the substitutions used are not at all obvious, which leads me to wonder,

Is there a similar substitution which can relate a Fermat curve of arbitrary degree to an elliptic curve?

How can we even begin to prove this? I suspect the proof or disproof of this statement will be way above my level; I truly have no clue where to begin. Can someone help out? If this is somehow an open problem, then any links to literature is also appreciated!

Edit: This question has been asked and answered on MO. (Yay!)

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    $\begingroup$ In the language of algebraic geometry, you're asking whether every Fermat curve has a nonconstant morphism to an elliptic curve. $\endgroup$ – Eric Wofsey Dec 17 '18 at 1:31

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