In our discourse FLT is Fermat's Last Theorem. I am unaware of any theorems or conjectures that begin assuming FLT is true, or otherwise use FLT as a starting point or tool. The small amount of literature review I've done on this question reveals nothing.
My question is: Where can I find a work requiring FLT, or some useful implication of FLT? Even an implication of a polynomial inequality, that may not be FLT, may be a good answer to this question, as I'll likely try to use it to find something regarding FLT.
The following is not acceptable as an answer to this question:
$$ (a^{x_{1}}_{1} + b^{x_{1}}_{1} - c^{x_{1}}_{1}) \ldots (a^{x_{n}}_{n} + b^{x_{n}}_{n} - c^{x_{n}}_{n}) \not= 0 : x_{i} > 2 $$
and it's expansions imply (something trivial)
Another acceptable answer to this question would be a proof requiring FLT to be false.
Thanks and please let me know if I can ask this question in a way more fitting math.se (new user).
I don't know if this helps. But you can consider this for fun in the meantime while you search for something significant. More of a "nuking the mosquito"
To prove $2^\frac{1}{n}$ is an irrational number when $n\ge3$.
$$2^\frac{1}{n}=\frac{p}{q}$$
$$p^n=q^n+q^n$$ which contradicts the $FLT$, therefore proving $2^\frac{1}{n}$ is indeed irrational.
On a side note, $FLT$ is not strong enough to prove the irrationality of $2^\frac{1}{n}$ for case $n=2$