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](https://mathoverflow.net/questions/50183/fermats-last-theorem-and-computability-theory) concludes that one purported use is not seriously using FLT.



Has Fermat's Last Theorem per se been used?
Corollary 3.17 in [this paper of Stefan Keil](http://arxiv.org/abs/1206.1822) 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.


