The following two decision problems capture the complexity of comparing integers or rationals that are succinctly represented in product-of-exponentials notation, or equivalently, via arithmetic circuits using only multiplication and division gates, and integer inputs. Input instance: Four lists of positive integers: a 1,..., an ∈N+ n ; b 1,..., bn ∈N+ n ; c 1,..., cm ∈N+ m ; d 1, ..., dm ∈N+ m ; where each of the integers is represented in binary. Problem 1 (equality testing): Decide whether a 1 b 1 a 2 b 2⋯ anbn = c 1 d 1 c 2 d 2⋯ cmdm . Problem 2 (inequality testing): Decide whether a 1 b 1 a 2 b 2⋯ anbn ≥ c 1 d 1 c 2 d 2⋯ cmdm . Problem 1 is easily decidable in polynomial time using a simple iterative algorithm. Problem 2 is much harder. We observe that the complexity of Problem 2 is intimately connected to deep conjectures and results in number theory. In particular, if a refined form of the ABC conjecture formulated by Baker in 1998 holds, or if the older Lang-Waldschmidt conjecture (formulated in 1978) on linear forms in logarithms holds, then Problem 2 is decidable in P-time (in the standard Turing model of computation). Moreover, it follows from the best available quantitative bounds on linear forms in logarithms, namely, by Baker and Wüstholz [1993] or Matveev [2000], that if m and n are fixed universal constants then Problem 2 is decidable in P-time (without relying on any conjectures). This latter fact was observed earlier by Shub [1993]. We describe one application: P-time maximum probability parsing for arbitrary stochastic context-free grammars (where ε -rules are allowed).