Catalan's conjecture states that 8 and 9 are the only consecutive integer powers. More formally, it claims that the only solution to xp-yq=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 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.)

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