Editing Abc conjecture (section)

==Formulations==
Before stating the conjecture, the notion of the  [[radical of an integer]] must be introduced: for a [[positive integer]] ''<math>n</math>'', the radical of ''<math>n</math>'', denoted ''<math>\text{rad}(n)</math>'', is the product of the distinct [[prime factor]]s of ''<math>n</math>''. For example,

<math>\text{rad}(16)=\text{rad}(2^4)=\text{rad}(2)=2</math>

<math>\text{rad}(17)=17</math>

''<math>\text{rad}(18)=\text{rad}(2\cdot 3^2)=2\cdot3 =6</math>''

''<math>\text{rad}(1000000)=\text{rad}(2^6 \cdot 5^6)=2\cdot5=10</math>''

If ''a'', ''b'', and ''c'' are [[coprime]]<ref group="notes">When ''a'' + ''b'' = ''c'', any common factor of two of the values is necessarily shared by the third.  Thus, coprimality of ''a'', ''b'', ''c'' implies [[Pairwise coprime|pairwise coprimality]] of ''a'', ''b'', ''c''. So in this case, it does not matter which concept we use.</ref> positive integers such that ''a'' + ''b'' = ''c'', it turns out that "usually" ''<math>c<\text{rad}(abc)</math>''. The ''abc conjecture'' deals with the exceptions. Specifically, it states that:

{{block indent|1=For every positive [[real number]] ''ε'', there exist only finitely many triples (''a'', ''b'', ''c'') of coprime positive integers, with ''a'' + ''b'' = ''c'', such that{{sfn|Waldschmidt|2015}}
{{block indent|<math>c > \operatorname{rad}(abc)^{1+\varepsilon}.</math>}}}}

An equivalent formulation is:

{{block indent|1=For every positive real number ''ε'', there exists a constant ''K<sub>ε</sub>'' such that for all triples (''a'', ''b'', ''c'') of coprime positive integers, with ''a'' + ''b'' = ''c'':{{sfn|Waldschmidt|2015}}
{{block indent|<math>c < K_{\varepsilon} \cdot \operatorname{rad}(abc)^{1+\varepsilon}.</math>}}}}

Equivalently (using the [[little o notation]]):

{{block indent|1=For all triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'', rad(''abc'') is at least ''c''<sup>1-''o''(1)</sup>.}}

A fourth equivalent formulation of the conjecture involves the ''quality'' ''q''(''a'', ''b'', ''c'') of the triple (''a'', ''b'', ''c''), which is defined as

{{block indent|<math> q(a, b, c) = \frac{\log(c)}{\log\big(\textrm{rad}(abc)\big)}.</math>}}

For example:

{{block indent|1=''q''(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
:''q''(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...}}

A typical triple (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' will have ''c'' < rad(''abc''), i.e. ''q''(''a'', ''b'', ''c'') < 1. Triples with ''q'' > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small [[prime number]]s. The fourth formulation is:

{{block indent|1=For every positive real number ''ε'', there exist only finitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1 + ''ε''.}}

Whereas it is known that there are infinitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1, the conjecture predicts that only finitely many of those have ''q'' > 1.01 or ''q'' > 1.001 or even ''q'' > 1.0001, etc. In particular, if the conjecture is true, then there must exist a triple (''a'', ''b'', ''c'') that achieves the maximal possible quality ''q''(''a'', ''b'', ''c'').