**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 2^{3} = 8 and 3^{2} = 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

`x`^{a}−`y`^{b}= 1

In particular, notice that it's unimportant that the same numbers 2 and 3 are repeated in the equation 3^{2} − 2^{3} = 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.