# 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 2^{3} = 8 and 3^{2} = 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

`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 3^{2} − 2^{3} = 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 Mihăilescu; in April 2002, so it is now a theorem. The proof was checked by Yuri Bilu and makes extensive use of the theory of cyclotomic fields and Galois modules.

**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 and is named for S. S. Pillai.

## External links

- Ivars Peterson's MathTrek
- Metsänkylä, Tauno (2003). Catalan's conjecture: another old Diophantine problem solved,
*Bull. (New Ser.) Amer. Math. Soc.***41**(1), 43–57.