Mihăilescu's theorem

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Mihăilescu's theorem (formerly "Catalan's conjecture") is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proved in 2002 by Preda Mihăilescu.

To understand the conjecture, notice that 2³ = 8 and 3² = 9 are two consecutive powers of natural numbers. 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

x^a − y^b = 1

for x, a, y, b > 1 is x = 3, a = 2, y = 2, b = 3.

In particular, notice that it is 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 Mihăilescu's theorem.

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 as of 2004 (though Chudnowski has claimed to prove it) and is named for S. S. Pillai. Erdős conjectured that there is some positive constant c such that if d is the difference of a perfect power n, then d>n^c for sufficiently large n.

[edit] See also

Tijdeman's theorem

[edit] Reference

[1] Preda Mihăilescu, "Primary Cyclotomic Units and a Proof of Catalan's Conjecture", J. reine angew. Math. 572 (2004), 167–195. MathSciNet Featured Review MR 2076124