Catalan's conjecture is a simple conjecture in number theory that was proposed by the mathematician Eugène Charles Catalan.
To understand the conjecture notice that 23 = 8 and 32 = 9 are two consecutive powerss of natural numbers. Catalan's conjecture states that this is the only case of two consecutive powers.
That is to say, Catalan's conjecture states that the only solution in the natural numbers of
In particular, notice that it's unimportant that the same numbers 2 and 3 are repeated in the equation 32 − 23 = 1. Even a case where the numbers were not repeated would still be a counterexample to Catalan's conjecture.
A proof of Catalan's conjecture, which would make it a theorem, was claimed by the mathematician Preda Mihailescu in April 2002. The proof is still being checked.