Catalan's conjecture states that 8 and 9 are the only consecutive integer

powers. More formally, it claims that the only solution to

*x*^{p}-

*y*^{q}=1 (where

*x, y, p* and

*q* are

integers greater than 1) is 3

^{2}-2

^{3}=1.

Whilst most number theorists believe Catalan's conjecture to be true, as of 2001 no-one has yet found a proof. Thanks to Robert Tijdeman, it has been known since 1976 that the equation above has only a finite number of solutions. Preda Mihailescu has also proven that if a second solution to the equation exists, then *p* and *q* must be double Wieferich primes. This result, in combination with some bounds on *p* and *q* proven by Maurice Mignotte, has brought the hunt for counterexamples to Catalan's conjecture within the capabilities of distributed brute force searching. (See `http://catalan.ensor.org` if you want to contribute computing time to this effort.) However, the amount of computation required is still huge, and it may turn out that a theoretical advance resolves Catalan's conjecture before an exhaustive computer search is complete. (In fact, Mihailescu claimed in May 2002 to have proved the conjecture; the proof is currently being verified.)