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I am currently taking a course on optimisation and the lecturer has said that it is possible to pose Fermat's Last Theorem ($a^n + b^n = c^n,\ n > 2$ has no solutions in the integers) as an optimisation problem, however he said no more on it.

How exactly can this be done?

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I don't see how to do this in a non-silly way. However, if you really stretch the meaning of "optimization" you can sort of get it to work: asking whether FLT holds for a given $n$ is the same as asking whether the minimum value of the function $$f:(\mathbb{N}_{>0})^3\rightarrow\mathbb{N}_{\ge 0}:(a,b,c)\mapsto\vert a^n+b^n-c^n\vert$$ is positive. (My notation above is due to the fact that different texts disagree over whether $0\in\mathbb{N}$.)

Of course, this is utterly useless since we can't bring any optimization techniques to bear, the domain of $f$ not having any good geometry. Optimization problems over $\mathbb{R}$ (or similar) work pretty well since we have calculus; with a discrete domain like the natural numbers, we're out of luck in this regard.

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