Has Fermat's Last Theorem per se been used?

Question

There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current proofs of FLT is their use of the modularity thesis which is just the opposite: arcane, and richly connected to a lot of results.

But have there been uses of FLT itself? Beyond implying simple variants of itself, are there any more serious uses yet?

I notice the discussion in Fermat's Last Theorem and Computability Theory concludes that one purported use is not seriously using FLT.

Answer 1

Corollary 3.17 in this paper of Stefan Keil uses FLT for exponent 7 to show that if $E/\mathbb{Q}$ is an elliptic curve with a rational 7-torsion point $P$, and $E\rightarrow E'$ is the 7-isogeny with kernel $\langle P\rangle$, then $E'(\mathbb{Q})[7]=0$. There are of course lots of ways of proving this, but the paper does it by writing down a parametrisation of all elliptic curves over $\mathbb{Q}$ with 7-torsion and of their rational 7-isogenies, and then playing with parameters to get a contradiction to FLT.

Answer 2

It is perhaps an indication of the average age of today's MOers that nobody remembers the work of Hellegouarch who introduced around 1970 what is nowadays called the Frey curve precisely in order to deduce information about elliptic curves from (the then) known results about Fermat's Last Theorem.

Answer 3

Recall that around 1977 Mazur has completely classified the possible torsion groups of elliptic curves over $\mathbb Q$. A few years prior, Kubert has worked on this problem and has established a number of partial results, including, in the paper "Universal bounds on the torsion of elliptic curves", the following statement (Main result 1, second part):

If $\ell>3$ is a prime for which Fermat's last theorem is valid, then $\ell^2\nmid |E_\mathrm{tor}(\mathbb Q)|$.

(let me remark that the proof splits into cases $\ell>5$, which substantially uses the assumption, and $\ell=5$ which doesn't and uses a complicated descent argument)

With this theorem we can, relatively easily, prove that if FLT holds, then $|E_\mathrm{tor}(\mathbb Q)|$ is a product of a squarefree number and a factor of $12$.

Of course, this result predates FLT by a long shot, and was quickly superseded by Mazur's theorem, but it is still noteworthy because it relies on full FLT and not just a single case.

Answer 4

Do applications to physics count?

Supersymmetry Breakings and Fermat's Last Theorem, Hitoshi Nishino, Mod.Phys.Lett. A 10 (1995) 149-158.

In this paper, we give the first application of Fermat's last theorem (FLT) to physical models, in particular to supersymmetric models in two or four dimensions. It is shown that FLT implies that supersymmetry is exact at some irrational number points in parameter space, while it is broken at all rational number points except for the origin. This mechanism presents a peculiar link between the FLT in number theory and the vacuum structure of supersymmetry. Previously, the only well-known connection between number theory and supersymmetry has been via topological effects, such as instantons and monopoles in supersymmetric models.