To understand the conjecture, notice that 23 and 32 are two consecutive powers of natural numbers, whose values 8 and 9 respectively, are also consecutive. Mihăilescu's theorem states that this is the only case of two consecutive powers.
That is to say, Mihăilescu's theorem states that the only solution in the natural numbers of
xa − yb = 1
for x, a, y, b > 1 is x = 3, a = 2, y = 2, b = 3.
In 1974, Robert Tijdeman applied methods from the theory of transcendental numbers to show that there is a computable constant C so that the exponents of all consecutive powers are less than C. As the results of a number of other mathematicians collectively established a bound for the base dependent only on the exponents, this resolved Catalan's conjecture (as Mihăilescu's theorem was then known) for all but a finite number of cases. However, the finite calculation required to complete the proof of the theorem was nonetheless too time-consuming to perform.
Catalan's conjecture was proved by Preda Mihăilescu in April 2002, so it is now sometimes called Mihăilescu's theorem. The proof was published in the Journal für die reine und angewandte Mathematik, 2004. It makes extensive use of the theory of cyclotomic fields and Galois modules. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki.
Pillai's conjecture concerns a general difference of perfect powers. It states that the differences in the sequence of all perfect powers tend to infinity, so that each given difference occurs only finitely many times. It is an open problem (though Chudnowski has claimed to prove it) and is named for S. S. Pillai. Paul Erdős conjectured that there is some positive constant c such that if d is the difference of a perfect power n, then d>nc for sufficiently large n.
* Størmer's theorem