On a conjecture on exponential Diophantine equations

We study the solutions of a Diophantine equation of the form a^x+b^y=c^z where a\equiv 2 \pmod 4 b\equiv 3 \pmod 4and \gcd (a,b,c)=1 The main result is that if there exists a solution (x,y,z)=(2,2,r)with r>1odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values (c,r) We also prove the uniqueness of such a solution if any of a b cis a prime power. In a...