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 (18601904)
“A distinction of property results from that very protection which a free Government gives to unequal faculties of acquiring it.”
—James Madison (17511836)