The Scott-Styer conjecture on at most one solution to a^x + b^y = c^z is confirmed for c=7,13,97 and primes c of form 2^r*3+1 except finitely many effectively determinable cases.
Luca,On the system of Diophantine equationsa2 +b 2 = (m 2 + 1)r anda x +b y = (m 2 + 1)z, Acta Arith.153(2012), 373–392
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Handling some Diophantine equation via Euclidean algorithm and its application to purely exponential equations
The Scott-Styer conjecture on at most one solution to a^x + b^y = c^z is confirmed for c=7,13,97 and primes c of form 2^r*3+1 except finitely many effectively determinable cases.