structural:scholar_edition
Theoretical results

The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. Bounds are known that are exponential. Specifically, the following bounds have been proven:

In these bounds, K and K are constants that do not depend on a, b, or c, and K 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, Stewart & Tijdeman (1986) showed that there are infinitely many triples (a, b, c) of coprime integers with a + b = c and

for all k < 4. The constant k was improved to k = 6.068 by van Frankenhuysen (2000).