For c = 7, 13, and 97, the equation a^x+b^y=c^z has at most one positive solution except (a,b)=(3,10)/(10,3) with c=13, and primes c=2^r·3+1 are covered up to effectively computable exceptions.
Bennett,On some exponential equations of S
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Handling some Diophantine equation via Euclidean algorithm and its application to purely exponential equations
For c = 7, 13, and 97, the equation a^x+b^y=c^z has at most one positive solution except (a,b)=(3,10)/(10,3) with c=13, and primes c=2^r·3+1 are covered up to effectively computable exceptions.