Catalan's conjecture is a simple conjecture in number theory that was proposed by the mathematician Eugène Charles Catalan. Mathmatical and Non-Mathamatical Definitions In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Eugène Charles Catalan (May 30, 1814 - February 14, 1894) was a Belgian mathematician. ...

To understand the conjecture notice that 2³ = 8 and 3² = 9 are two consecutive powers of natural numbers. Catalan's conjecture states that this is the only case of two consecutive powers. In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

That is to say, Catalan's conjecture states that the only solution in the natural numbers of In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ...

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. In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ...

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. Preda Mihăilescu (1955 —) is a Romanian mathematician who received his education at the ETH Zürich and later did research at the University of Paderborn, Germany. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense...

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. Subbayya Sivasankaranarayana Pillai (1901-1950) was an Indian mathematician, well known for his work in number theory. ...

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.