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This paper contributes to the conjecture of R. Scott and R. Styer whichasserts that for any fixed relatively prime positive integers $a,b$ and $c$greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ inpositive integers $x,y$ and $z$, except for specific cases. The fundamentalresult proves the conjecture under some congruence condition modulo $c$ on $a$and $b$. As applications the conjecture is confirmed to be true if $c$ takessome small values including the Fermat primes found so far, and thisparticularly 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 Theory44(1993), 153-165] solving the conjecture for $c=2$ in a purely algebraicmanner. The method can be generalized for smaller modulus cases, and it turnsout 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 certaindivisibility relation arising from the existence of two hypothetical solutionsto the equation. The other tools include Baker's theory in the complex case andits non-Archimedean analogue for number fields together with various elementaryarguments through rational and quadratic numbers, and extensive computation.

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