Theoretical Results
The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. However, exponential bounds are known. Specifically, the following bounds have been proven:
- (Stewart & Tijdeman 1986),
- (Stewart & Yu 1991), and
- (Stewart & Yu 2001).
In these bounds, K1 is a constant that does not depend on a, b, or c, and K2 and K3 are constants that depend on ε (in an effectively computable way) but not on a, b, or c. The bounds apply to any triple for which c > 2.
Read more about this topic: abc Conjecture
Famous quotes containing the words theoretical and/or results:
“The desire to serve the common good must without fail be a requisite of the soul, a necessity for personal happiness; if it issues not from there, but from theoretical or other considerations, it is not at all the same thing.”
—Anton Pavlovich Chekhov (1860–1904)
“The peace conference must not adjourn without the establishment of some ordered system of international government, backed by power enough to give authority to its decrees. ... Unless a league something like this results at our peace conference, we shall merely drop back into armed hostility and international anarchy. The war will have been fought in vain ...”
—Virginia Crocheron Gildersleeve (1877–1965)