Editing Abc conjecture (section)

==Refined forms, generalizations and related statements==
The ''abc'' conjecture is an integer analogue of the [[Mason–Stothers theorem]] for polynomials.

A strengthening, proposed by {{Harvtxt|Baker|1998}}, states that in the ''abc'' conjecture one can replace rad(''abc'') by

{{block indent|''ε''<sup>−''ω''</sup> rad(''abc''),}}

where ''ω'' is the total number of distinct primes dividing ''a'', ''b'' and ''c''.{{sfnp|Bombieri|Gubler|2006|p=404}}

[[Andrew Granville]] noticed that the minimum of the function <math>\big(\varepsilon^{-\omega}\operatorname{rad}(abc)\big)^{1+\varepsilon}</math> over <math>\varepsilon > 0</math> occurs when <math>\varepsilon = \frac{\omega}{\log\big(\operatorname{rad}(abc)\big)}.</math>

This inspired {{Harvtxt|Baker|2004}} to propose a sharper form of the ''abc'' conjecture, namely:
{{block indent|<math>c < \kappa \operatorname{rad}(abc) \frac{\Big(\log\big(\operatorname{rad}(abc)\big)\Big)^\omega}{\omega!}</math>}}
with ''κ'' an absolute constant. After some computational experiments he found that a value of <math>6/5</math> was admissible for ''κ''. This version is called the "explicit ''abc'' conjecture".

{{Harvtxt|Baker|1998}} also describes related conjectures of [[Andrew Granville]] that would give upper bounds on ''c'' of the form

{{block indent|<math>K^{\Omega(a b c)} \operatorname{rad}(a b c),</math>}}

where Ω(''n'') is the total number of prime factors of ''n'', and

{{block indent|<math>O\big(\operatorname{rad}(a b c) \Theta(a b c)\big),</math>}}

where Θ(''n'') is the number of integers up to ''n'' divisible only by primes dividing ''n''.

{{Harvtxt|Robert|Stewart|Tenenbaum|2014}} proposed a more precise inequality based on {{Harvtxt|Robert|Tenenbaum|2013}}.
Let ''k'' = rad(''abc''). They conjectured there is a constant ''C''<sub>1</sub> such that

{{block indent|<math>c < k \exp\left(4\sqrt{\frac{3\log k}{\log\log k}}\left(1+\frac{\log\log\log k}{2\log\log k}+\frac{C_{1}}{\log\log k}\right)\right)</math>}}

holds whereas there is a constant ''C''<sub>2</sub> such that

{{block indent|<math>c > k \exp\left(4\sqrt{\frac{3\log k}{\log\log k}}\left(1+\frac{\log\log\log k}{2\log\log k}+\frac{C_{2}}{\log\log k}\right)\right)</math>}}

holds infinitely often.

{{harvtxt|Browkin|Brzeziński|1994}} formulated the [[n conjecture]]—a version of the ''abc'' conjecture involving ''n'' > 2 integers.