Catalan's conjecture (idea)

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 3²-2³=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.)