Catalan conjecture

The Catalan conjecture (also known as Mihăilescu's theorem) is a number theory theorem proposed by the mathematician Eugène Charles Catalan in 1884 and proved for the first time by Preda Mihăilescu in April 2002.

To understand this conjecture, note that 2³ = 8 and 3² = 9 are two numbers that are consecutive powers of natural numbers. The Catalan conjecture says that this is the only case of two consecutive powers.

That is, the Catalan conjecture affirms that the only solution in the set of natural numbers of

x^a− and^b= 1

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

In particular, note that it doesn't matter that the same numbers 2 and 3 are repeated in the equation 3² − 2³ = 1.

Catalan's conjecture was proved by Preda Mihăilescu in 2002. The proof was published in the Journal für die reine und angewandte Mathematik, 2004. It makes extensive use of cyclotomic field theory and modulus de Galois. An exposition of said test was given by Yuri Bilu at the Bourbaki Seminar.