The Catalan's conjecture reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Catalan's conjecture

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

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'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.

Catalan's conjecture was proved by Preda Mihailescu in April 2002, so it is now a theorem. The proof was checked by Yuri Bilu and make extensive use of the theory of cyclotomic fields and Galois modules

External links

• http://www.maa.org/mathland/mathtrek_06_24_02.html