At most one solution to \(a^x + b^y = c^z\) of \(a\), \(b\), \(c\)

For fixed coprime positive integers \(a\), \(b\), and \(c\) with \(\min(a, b, c) > 1\), we consider the number of solutions in positive integers \((x, y, z)\) for the purely exponential Diophantine equation \(a^x + b^y = c^z\). Apart from a list of known exceptions, a conjecture published in 2016 claims that this equation has at most one solution in positive integers \(x\), \(y\), and \(z\). We show that this is true for some ranges of \(a\), \(b\), \(c\), for instance, when \(1 \lt a,b \lt 3600\) and \(c \lt 10^{10}\). The conjecture also holds for small pairs \((a,b)\) independent of \(c\), where \(2 \le a,b \le 10\) with \(\gcd(a,b)=1\). We show that the Pillai equation \(a^x - b^y = r \gt 0\) has at most one solution (with a known list of exceptions) when \(2 \le a,b \le 3600\) (with \(\gcd(a,b)=1\)). Finally, the primitive case of the Jeśmanowicz conjecture holds when \(a \le 10^6\) or when \(b \le 10^6\). This work highlights the power of some ideas of Miyazaki and Pink and the usefulness of a theorem by Scott.