A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the *quality* or *power* of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$, where $\text{rad}(k)$ denotes the product of distinct prime divisors of $k$.

One version of the $abc$-Conjecture is that for each $\varepsilon > 0$, there are finitely many $abc$-triples such that $P(a,b,c) > 1 + \varepsilon$.

There are finitely many known triples satisfying $P > 1.4$, the so called *good triples*, and the largest (quality) is $P(2,3^{10} \cdot 109, 23^{5}) = 1.629911684 \dots$ (discovered by E. Reyssat).

**Question:** Are there any known upper bounds for $P(a,b,c)$ sharper than $\log_{p^{n}} c$, where $p$ is the minimum prime dividing $abc$ and $n$ is the number of distinct prime divisors of $abc$?

**Question:** Is there an absolute upper bound for $P(a,b,c)$ so that no triple has higher quality?

**Best Answer:** If there were such a bound, asymptotic FLT would be in hand. (Thanks Ace of Base!)