Catalan's Conjecture
The conjecture made by Belgian mathematician Eugène Charles Catalan in 1844 that 8 and 9 (
and 
) are the only
consecutive powers (excluding 0 and 1). In other words,
 |
(1)
|
is the only nontrivial solution to Catalan's Diophantine problem
 |
(2)
|
The special case 
and 
is the 
case of a Mordell
curve.
Interestingly, more than 500 years before Catalan formulated his conjecture, Levi ben Gerson (1288-1344) had already noted that the only powers of 2 and 3 that apparently
differed by 1 were 
and 
(Peterson 2000).
This conjecture had defied all attempts to prove it for more than 150 years, although Hyyrő and Makowski proved that no three consecutive powers exist (Ribenboim 1996), and it was also known that 8 and 9 are the only consecutive cubic and square numbers (in either order). Finally, on April 18, 2002, Mihăilescu sent a manuscript proving the entire conjecture to several mathematicians (van der Poorten 2002). The proof has now appeared in print (Mihăilescu 2004) and is widely accepted as being correct and valid (Daems 2003, Metsänkylä 2003).
Tijdeman (1976) showed that there can be only a finite number of exceptions should the conjecture not hold.
More recent progress showed the problem to be decidable in a finite
(but more than astronomical) number of steps and that, in particular, if 
and 
are powers,
then 
(Guy 1994, p. 155).
In 1999, M. Mignotte showed that if a nontrivial solution exists, then 
, 
(Peterson 2000).
It had also been known that if additional solutions to the equation exist, either the exponents 
must be double
Wieferich prime pairs, or 
and 
must satisfy a
class number divisibility condition (Steiner 1998). Constraints on this class number
condition were continuously improved starting with Inkeri (1964) and continuing through
the work of Steiner (1998). Then, in the spring of 1999, Bugeaud and Hanrot proved
the weakest possible class number condition holds unconditionally (i.e., irrespective
of whether 
and 
are a double
Wieferich prime pair or not). Subsequently, in Autumn 2000, Mihailescu proved
that the double Wieferich prime pair
condition also must hold unconditionally (Peterson 2000).
A generalization to Gaussian integers that differ by a unit is given by
 |
(3)
|
Contact the MathWorld Team
catalan's conjecture