Catalan's conjecture

In 1844, Eugène Charles Catalan proposed in a letter to the editor of Crelle's journal that 8 and 9 are the only consecutive natural numbers that are both perfect powers. The question remained open for 158 years, until Preda Mihailescu produced a proof.

Schoof's book presents this result at a level suited to advanced undergraduates. The early chapters need little beyond basic mathematics and elementary number theory, while later chapters draw on Galois theory, algebraic number theory and a small amount of commutative algebra. Background such as the arithmetic of cyclotomic fields is explained within the text.

The author examines Mihailescu's proof along with the earlier results it depends on, choosing streamlined and transparent versions of each argument and keeping the material self-contained. Class field theory appears only in the proof of Thaine's theorem, included in the hope that it will encourage readers to study that theory further.

The book is aimed at number theory specialists and at anyone curious about how the ideas of algebraic number theory apply to a famous problem.