TITLE: Has Fermat's Last Theorem per se been used?
QUESTION [57 upvotes]: 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.

REPLY [16 votes]: 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Of course, this result predates FLT by a long shot, and was quickly superseded by Mazur's theorem, but it is still noteworthy because it relies on full FLT and not just a single case.