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

Abstract 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 from the hypothesis that there is more than one solution to the equation, we obtain a number of finiteness results on the conjecture. In particular, we find some, presumably infinitely many, new values of c with the property 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) each of which gives exactly two solutions. Further, our study with the help of the Schmidt Subspace Theorem among others more, brings strong contributions to the study of Pillai’s type Diophantine equations, notably 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.