To understand the conjecture notice that 2^{3} = 8 and 3^{2} = 9 are two consecutive powers 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 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.
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