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

TL;DR

This paper proves a conjecture about the uniqueness of solutions to a specific exponential equation involving three variables, using advanced number theory techniques and extending known results for certain cases.

Contribution

It introduces a novel application of p-adic analogues of Baker's theory to confirm the conjecture for infinitely many values of c and under certain congruence conditions.

Findings

Confirmed the conjecture for small values of c including Fermat primes.

Provided an algebraic proof for the case c=2.

Extended the conjecture's validity to infinitely many specific c values.

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.