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,
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).
Bilu, Y. F. "Catalan's Conjecture (After Mihăilescu)." Astérisque, No. 294, 1-26, 2004.Bilu, Y. F. "Catalan
Without Logarithmic Forms (after Bugeaud, Hanrot and Mihăilescu)." J.
Théor. Nombres Bordeaux17, 69-85, 2005.Catalan, E.
"Note extraite d'une lettre adressée à l'éditeur."
J. reine angew. Math.27, 192, 1844.Daems, J. "A
Cyclotomic Proof of Catalan's Conjecture." Sept. 29, 2003. http://www.math.leidenuniv.nl/~jdaems/scriptie/Catalan.pdf.Guy,
R. K. "Difference of Two Power." §D9 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 155-157,
1994.Inkeri, K. "On Catalan's Problem." Acta Arith.9,
285-290, 1964.Metsänkylä, T. "Catalan's Conjecture: Another
Old Diophantine Problem Solved." Bull. Amer. Math. Soc.41, 43-57,
2003.Mihăilescu, P. "A Class Number Free Criterion for Catalan's
Conjecture." J. Number Th.99, 225-231, 2003.Mihăilescu,
P. "Primary Cyclotomic Units and a Proof of Catalan's Conjecture." J.
reine angew. Math.572, 167-195, 2004.Peterson, I. "MathTrek:
Zeroing In on Catalan's Conjecture." Dec. 4, 2000. http://www.sciencenews.org/20001202/mathtrek.asp.Ribenboim,
P. "Consecutive Powers." Expositiones Mathematicae2, 193-221,
1984.Ribenboim, P. Catalan's
Conjecture: Are 8 and 9 the only Consecutive Powers? Boston, MA: Academic
Press, 1994.Ribenboim, P. "Catalan's Conjecture." Amer.
Math. Monthly103, 529-538, 1996.Steiner, R. "Class
Number Bounds and Catalan's Equation." Math. Comput.67, 1317-1322,
1998.Tijdeman, R. "On the Equation of Catalan." Acta Arith.29,
197-209, 1976.van der Poorten, A. "Concerning: Catalan's Conjecture
Proved?." 5 May 2002. http://listserv.nodak.edu/scripts/wa.exe?A2=ind0205&L=nmbrthry&P=269.Weisstein,
E. W. "Draft Proof of Catalan's Conjecture Circulated." MathWorld
Headline News, May 5, 2002. http://mathworld.wolfram.com/news/2002-05-05/catalan/.Wells,
D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, pp. 71 and 73, 1986.
and 
)
are the only consecutive powers (excluding 0 and 1). In
other words,
and 
is the 
case of a Mordell curve.
and 
(Peterson 2000).
and 
are powers, then 
(Guy 1994, p. 155). In 1999, M. Mignotte
showed that if a nontrivial solution exists, then 
, 
(Peterson 2000).
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).
