Has Fermat's Last Theorem per se been used?

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.

• mathteacherctk.com/blog/2010/09/… Aug 26, 2013 at 15:11
• @Koundinya Vajjha: so, FLT would be just as useful if Wiles had finally disproved it! Aug 26, 2013 at 16:00
• It has been used to good effect to prove the irrationality of $2^{1/n}$ here: mathoverflow.net/a/42519/8430 (sorry, couldn't resist this joke) Aug 26, 2013 at 16:19
• Alex B.'s answer is certainly a great application of the $p=7$ case, proved by Lamé in 1839. I would be interested to know if there is a substantial application for which pre-Wiles work did not suffice. Aug 26, 2013 at 23:16
• @suvrit, that application is the same as in Carlo's comment. Aug 26, 2013 at 23:44

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.

• In that corollary he needs to show certain rational numbers $q$ and $q-1$ cannot both be 7th powers, and he gets that immediately from degree 7 FLT. Looks like a serious use to me. Aug 26, 2013 at 17:33
• @ColinMcLarty: it is an absolutely serious use. There are other ways of doing this (rational points on the corresponding modular curves), but possibly not easier or more elementary than this one. Aug 26, 2013 at 17:40
• On the one hand this is, of course, awesome. On the other, this only uses FLT for n=7 so it is possible to argue it doesn't count as a real application. Still very satisfying though! Aug 27, 2013 at 10:52
• FLT for exponent 7 can be proved using a standard 2-descent. It is arguably simpler than FLT for exponent 3, which requires 3-descent or a 2-descent over number fields. Aug 27, 2013 at 11:58
• This does not refute the centuries old consensus that FLT is a rather isolated fact in number theory. But it is a serious application. Keil wanted to complete a calculation and the easiest way he could see to do it was FLT. Aug 27, 2013 at 22:14

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.

• See for example math.unicaen.fr/~nitaj/hellegouarch.html. Aug 27, 2013 at 10:23
• I Have been reading Hellegouarch but he says much less about the Fermat equations than I expected. He cites and uses many results on elliptic curves but can you point me to some place where he uses a then known case of FLT? Thanks. Aug 27, 2013 at 11:59
• In what follows $A$ is an elliptic curve defined over the field of fractions $K$ of a Dedekind ring $R$, and all the points of order $2$ on $A$ are defined over $K$. A rational prime $p$ is said to be "idoine'' if $(−2)^m\equiv−1\mod p$ for some integer $m$. When $R=\mathbf{Z}$, the rational integers, it is shown that there is no point of order $p^2$ defined over $K$, where $p$ is an idoine prime, provided that Fermat's equation $x^p+y^p=z^p$ is insoluble. (ams.org/mathscinet-getitem?mr=274451) Aug 27, 2013 at 12:20
• Thanks. This led me to a cleaner more general paper by Hellegouarch, Points d’ordre 2p sur les courbes elliptiques. Acta Arith. 26 (1974/75), no. 3, 253–263. MR0379507 (52 #412). Aug 27, 2013 at 12:56
• It is mentioned in the link in my first comment. Aug 27, 2013 at 12:58

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.

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.

• FLT comes into this via their specification of a specific superpotential $W$ which includes a term $\Phi_4[\Phi_1^p+\Phi_2^p+ (t\Phi_3)^p]$ (p.~3). Is there something in the theory that constrains them to that, or do they simply have the freedom to pick a superpotential that will relate to FLT this way? Aug 27, 2013 at 12:26
• @Colin --- more general non-polynomial superpotentials are considered in section 4. Aug 27, 2013 at 13:11
• I do not see that Section 4 uses FLT. It looks like in this paper FLT is just a nice source of integer polynomials without rational solutions. Aug 27, 2013 at 13:34