# Effective bounds for Fermat's Last Theorem

Suppose $$n>2$$. By Fermat's Last Theorem, we know that $$a^{n}+b^{n}=c^{n}$$ has no non-trivial solutions. Can we quantify it more?

More specifically, given $$a,b,c,n\in\mathbb{N}$$ with $$n>2$$ and $$a^{n}+b^{n}\neq c^{n}$$, can we prove lower bounds on $$\lvert a^{n}+b^{n}-c^{n}\rvert$$ in terms of $$a,b,c,n$$? Also assume that $$a,b,c$$ are distinct.

• I think for $n=3$ the difference $a^3+b^3-c^3$ is equal to $1$ infinitely often, so no nontrivial lower bounds probably exist. Nov 26, 2019 at 11:49
• Not duplicate but closely related: mathoverflow.net/q/214422/30186 Nov 26, 2019 at 11:50
• I think the $a^n+b^n\neq c^n$ condition is unnecessary... Nov 26, 2019 at 12:06
• @Wojowu, yeah I had looked at that related question before posting this. Here we are more interested in the asymptotics of any lower bounds we can prove. Nov 26, 2019 at 13:40
• @GoravJindal You probably won't have much luck proving any lower bounds if we can't even (unconditionally) prove that this value tends to infinity! Nov 26, 2019 at 13:42