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

Abstract

This paper contributes to the conjecture of R. Scott and R. Styer which asserts 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 $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. The fundamental result proves the conjecture under some congruence condition modulo $c$ on $a$ and $b$. As applications the conjecture is confirmed to be true if $c$ takes some small values including the Fermat primes found so far, and in particular this provides an analytic proof of the celebrated theorem of Scott [R. Scott, On the equations $p^x-b^y=c$ and $a^x+b^y=c^z$, J. Number Theory 44(1993), no.2, 153-165] solving the conjecture for $c=2$ in a purely algebraic manner. The method can be generalized for smaller modulus cases, and it turns out that the conjecture holds true for infinitely many specific values of $c$ not being perfect powers. The main novelty is to apply a special type of the $p$-adic analogue to Baker's theory on linear forms in logarithms via a certain divisibility relation arising from the existence of two hypothetical solutions to the equation. The other tools include Baker's theory in the complex case and its non-Archimedean analogue for number fields together with various elementary arguments through rational and quadratic numbers, and extensive computation.