Catalan's conjecture

Catalan's conjecture states that 8 and 9 are the only consecutive positive perfect powers. Preda Mihăilescu proved it in 2002.

Mihăilescu's theorem: The theorem that 8 and 9 are the only consecutive positive perfect powers. This is the proved form of Catalan's conjecture.
Eugène Charles Catalan: A Belgian mathematician who formulated Catalan's conjecture in 1844. He asked whether any consecutive perfect powers besides 8 and 9 exist.
Integer: A number in the set of positive whole numbers, zero, and their negatives. The conjecture concerns positive integers and their powers.
Number theory: The branch of mathematics that studies integers and their properties. The conjecture is a landmark problem about integer powers.
Perfect power: An integer equal to an integer base raised to an integer exponent greater than one. The conjecture compares two members of this broader class.
Catalan's 1844 letter: A letter in which Eugène Charles Catalan posed the problem of consecutive perfect powers. It records the conjecture's original formulation.
Exponentiation: The operation of raising a number to a specified power. Perfect powers are defined through repeated multiplication.
Diophantine equation: A polynomial equation for which integer or rational solutions are sought. Its proof advances methods for equations requiring integer solutions.
Exponential Diophantine equation: A Diophantine equation in which unknowns occur in exponents. The conjecture asks when two exponential expressions differ by one.
Pierre de Fermat: A seventeenth-century French mathematician known for work in number theory. His equation involving consecutive squares is an early special case of the problem.

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