On the exponential Diophantine equation \(a^x+b^y=c^z\) (Q1607592)

Let \(a\), \(b\), \(c\) be fixed positive integers such that \(\min(a,b,c)>1\) and \(\gcd(a,b,c)=1\). In this paper the author proves that if \(a,b,c\) satisfy some congruence conditions, then the equation \((*)\) \(a^x+b^y=c^z\) has only one positive integer solution \((x,y,z)\). For example, he proves that if \(a\equiv -1\pmod{b^2}\), \(b\) is an odd prime with \(b\equiv 3\pmod 4\) and \(a^2+b=c\), then \((*)\) has only the positive integer solution \((x,y,z)= (2,1,1)\).