Recall that around 1977 Mazur has completely classified the possible torsion groups of elliptic curvers 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 exceeding $2$ for which Fermat's last theorem is valid, then $\ell^2\nmid |E_{tor}(\mathbb Q)|$.

(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 $|E_{tor}(\mathbb Q)|$ 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Of course, this result predates FLT by a long shot, and was quickly superseded by Mazur's theorem, but it is still noteworthy because it relies on full FLT and not just a single case.

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Of course, this result predates FLT by a long shot, and was quickly superseded by Mazur's theorem, but it is still noteworthy because it relies on full FLT and not just a single case.