Remembering that a STEP mathematics question which guided us through showing $n^3 + (n-3)^3 = (n+3)^3$ has no solutions in the integers could be sledgehammered with Fermat's Last Theorem, I wondered if there are other fun ways to apply Fermat's Last Theorem.
Another example:
Claim $ \sqrt[n]{2}$ is irrational for all integers n > 3
Proof Suppose to the contrary, that $ \sqrt[n]{2} = \frac{p}{q}$ for integers p and q. Then $q^n + q^n = p^n $. A contradiction (Wiles, 95).
Any other examples? I'd be happy to see applications of other big theorems (such as the Catalan conjecture/Mihăilescu's theorem). Fermat fits naturally because of the gulf in the elementariness of the statement and the difficulty of the proof.