Is $\;\big\{(a,b,c)\in\mathbb{N}^3: \big(\exists m,n,\ell \in (\mathbb{N}\setminus\{0,1,2\})\big): a^m + b^n= c^\ell\big\}\;$ computable?
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asked Dec 6 at 7:48
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Yes. If there are solutions for $1\in\{a,b,c\}$, the equation has the form $a^m\pm b^n=1$, where $\log a/\log b$ is irrational, and decidability follows from strong enough effective lower bound on $|m\log a-n\log b|$ (e.g., Gel'fond-Baker bound), or from Størmer theorem, which is completely elementary. If $a^m+b^n=c^\ell$ and $a,b,c>1$, then $\max\{m,n,\ell\}\le6500(\log\max\{a,b,c\})^3$, according to [1].
[1] Y.-Z. Hu, M.-H. Le. An upper bound for the number of solutions of ternary purely exponential diophantine equations. J. Number Theory, 183:62–73, 2018.
