MathWorld Headline News

Draft Proof of Catalan's Conjecture Circulated

By Eric W. Weisstein

May 5, 2002--Today, in an email sent to the NMBRTHRY mailing list, number theorist Alf van der Poorten confirmed that an apparent proof of the long-outstanding Catalan's conjecture has been circulated to a group of mathematicians by number theorist Preda Mihailescu.

The conjecture in question was made by Belgian mathematician Eugène Charles Catalan in 1844, and it states that 8 and 9 (2³ and 3²) are the only consecutive powers--excluding 0 and 1. In other words, Catalan conjectured that

is the only nontrivial solution to the so-called Catalan's Diophantine problem,

Although Hyyrö and Makowski had previously proved that no three consecutive powers existed (Ribenboim 1996), Catalan's conjecture itself has stubbornly refused attack for more than a century and a half. The first groundbreaking result was that of Robert Tijdeman (1976), who showed that there can be at most a finite number of exceptions should the conjecture not hold. This led to considerable computational efforts, and in 1999, Maurice Mignotte showed that if a nontrivial solution exists, then 10⁷ < p < 7.15 x 10¹¹ and 10⁷ < q < 7.78 x 10¹⁶ (Peterson 2000).

The most significant recent progress towards settling the conjecture had been achieved by pursuing the known result that if additional solutions to the equation exist, either the exponents p and q are a so-called double Wieferich prime pairs (Steiner 1998), or they satisfy a so-called class number condition. Contraints on this class number condition were continuously improved starting with Inkeri 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 p and q 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).

Then, on April 18, 2002, Mihailescu reportedly sent a manuscript to several mathematicians purporting to prove the entire conjecture. The paper was apparently also accompanied by an expository analysis written by colleague Yuri Bilu (van der Poorten 2002).

Until Mihailescu's results are analyzed by the mathematical community and made public, Catalan's conjecture must remain unsettled. However, like the famous Fermat's last theorem recently proved by mathematician Andrew Wiles, Catalan's own famous problem in Diophantine number theory looks likely to fall within the very near future as a result of Mihailescu's work.