Refined abc Conjecture (Robert–Stewart–Tenenbaum)

Establish the refined abc conjecture proposed by Robert, Stewart, and Tenenbaum, which asserts that for coprime integers a, b, and c with a + b = c, the exceptional inequality in the abc conjecture can be strengthened to rad(abc) < c^{1−o(1/(log c log log c))}, where the o(·) term tends to 0 as c → ∞.

Background

Remark 2.4 connects sharper bounds on the distribution of integers by their radicals with quantitative formulations of the abc conjecture. Building on de Bruijn’s more precise estimates, Robert, Stewart, and Tenenbaum proposed a refined abc conjecture that replaces the fixed ε in the exceptional inequality rad(abc) < c^{1−ε} with a slowly varying term o(1/(log c log log c)).

This refinement aims to capture more precisely how close rad(abc) must be to c for coprime triples solving a + b = c. The present expository note provides elementary bounds on the number of exceptions but does not address the truth of this refined conjecture.