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.
Best Answer
This so called application of Fermat's great theorem was given at a Romanian olympiad. It is not suitable for such a purpose, but I'm not sure why this was accepted as a contest problem.
Where $P(a,b,c)$ is a statement about $a,b,c$ which can be deduced from $a=b=-c$.
Proof: $$ (a^n)^3+(b^n)^3+(-c^n)^3-3(a^n\cdot b^n\cdot (-c^n))=0$$
and then we use $$ x^3+y^3+z^3-3xyz=\frac{1}{2}(x+y+z)((x-y)^2+(y-z)^2+(z-x)^2)$$