Scott–Styer's prime-base conjecture on solutions to a ternary purely exponential Diophantine equation

Let $a$, $b$, and $c$ be distinct primes with $a<b$, and let $N(a,b,c)$ denote the number of solutions $(x,y,z)\in\mathbb{N}^3$ to

$$a^x+b^y=c^z.$$

Scott–Styer's prime-base conjecture. We have $N(a,b,c)\leq1$, apart from the six explicitly listed exceptional cases: $(2,3,5)$, $(2,3,11)$, $(2,5,3)$, $(2,7,3)$, $(3,5,2)$, and $(3,13,2)$, with the solution tuples given in the source statement. This restricted form is used in the paper because the prime-base setting makes the classification problem more tractable; the preceding theorems give substantial restrictions on any non-exceptional counterexample, but the conjecture itself remains open.