Number of solutions to a special type of unit equations in two unknowns, III

It is conjectured that for any fixed relatively prime positive integers $a,b$ and c all greater than 1 there is at most one solution to the equation aˣ+bʸ=cᶻ in positive integers $x,y$ and z, except for specific cases. We develop the methods in our previous work which rely on a variety from Baker's theory and thoroughly study the conjecture for cases where c is small relative to a or b. Using restrictions derived under which there is more than one solution to the equation, we obtain a number of finiteness results on the conjecture, which in particular enables us to find some new values of c being presumably infinitely many such that for each such c the conjecture holds true except for only finitely many pairs of a and b. Most importantly we prove that if $c=13$ then the equation has at most one solution, except for $(a,b)=(3,10)$ or $(10,3)$ which exactly gives two solutions. Further our study with the help of Schmidt Subspace Theorem among others brings strong contributions to the study of Pillai's type Diophantine equations, which includes a general and satisfactory result on a well-known conjecture of M. Bennett on the equation aˣ-bʸ=c for any fixed positive integers $a,b$ and c with both a and b greater than 1. Some conditional results are presented under the $abc$-conjecture as well.