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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: Is there an upper bound for $P(a,b,c)$ simply in terms of $c$ and absolute constants (sharper than $\log_{2} c$)?

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, then asymptotic FLT would be in hand. (Thanks Ace of Base on MO!)

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  • $\begingroup$ The same question was asked as mathoverflow.net/questions/44308 on mathoverflow. $\endgroup$ – azimut Aug 22 '13 at 9:50
  • $\begingroup$ Actually, I asked it on MO when I got no answers here. $\endgroup$ – user02138 Aug 22 '13 at 12:52

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