Non-number theoretic formulation of Fermat's last theorem?

Question

We have dozens of non-number theoretic formulations of Riemann hypothesis. I was wondering if there are any non-number theoretic formulations of Fermat's last theorem? I am in particular curious about some analytic formulation?

Answer 1

Depends on what you call "non-number theoretic". Wikipedia says,

Equivalent statement 4 – connection to elliptic curves: If $a, b, c$ is a non-trivial solution to $x^p + y^p = z^p$, $p$ odd prime, then $y^2 = x(x − a^p)(x + b^p)$ (Frey curve) will be an elliptic curve.

Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form. However, the proof by Andrew Wiles proves that any equation of the form $y^2 = x(x − a^n)(x + b^n)$ does have a modular form. Any non-trivial solution to $x^p + y^p = z^p$ (with $p$ an odd prime) would therefore create a contradiction, which in turn proves that no non-trivial solutions exist.

In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be "attacked" for all numbers at once.

Answer 2

There are several [non-trivially-different] combinatorial formulations. In fact, there are many elementary reformulations — Ribenboim’s Fermat’s Last Theorem For Amateurs has a whole chapter of them, but it doesn’t include any analytic formulations (because of the scope of the book).