Editing Abc conjecture (section)

==Theoretical results==
The ''abc'' conjecture implies that ''c'' can be [[upper bound|bounded above]] by a near-linear function of the radical of ''abc''. Bounds are known that are [[exponential function|exponential]]. Specifically, the following bounds have been proven:

{{block indent|<math>c < \exp{ \left(K_1 \operatorname{rad}(abc)^{15}\right) } </math> {{harv|Stewart|Tijdeman|1986}},}}
{{block indent|<math>c < \exp{ \left(K_2 \operatorname{rad}(abc)^{\frac{2}{3} + \varepsilon}\right) } </math> {{harv|Stewart|Yu|1991}}, and}}
{{block indent|<math>c < \exp{ \left(K_3 \operatorname{rad}(abc)^{\frac{1}{3}}\left(\log(\operatorname{rad}(abc)\right)^3\right) } </math> {{harv|Stewart|Yu|2001}}.}}

In these bounds, ''K''<sub>1</sub> and ''K''<sub>3</sub> are [[Constant (mathematics)|constants]] that do not depend on ''a'', ''b'', or ''c'', and ''K''<sub>2</sub> is a constant that depends on ''ε'' (in an [[effectively computable]] way) but not on ''a'', ''b'', or ''c''. The bounds apply to any triple for which ''c'' > 2.

There are also theoretical results that provide a lower bound on the best possible form of the ''abc'' conjecture.  In particular, {{Harvtxt|Stewart|Tijdeman|1986}} showed that there are infinitely many triples (''a'', ''b'', ''c'') of coprime integers with ''a'' + ''b'' = ''c'' and

{{block indent|<math>c > \operatorname{rad}(abc) \exp{ \left(k \sqrt{\log c}/\log\log c \right) } </math>}}

for all ''k'' < 4.  The constant ''k'' was improved to ''k'' = 6.068 by {{Harvtxt|van Frankenhuysen|2000}}.