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Eugene Charles Catalan made his famous conjecture that 8 and 9 are the only two consecutive perfect powers of natural numbers in 1844 in a letter to the editor of Crelle s mathematical journal. One hundred and fifty-eight years later, Preda Mihailescu proved it.

Catalan s Conjecture presents this spectacular result in a way that is accessible to the advanced undergraduate. The author dissects both Mihailescu s proof and the earlier work it made use of, taking great care to select streamlined and transparent versions of the arguments and to keep the text self-contained. Only in the proof...

Invited articles by leading researchers explore various aspects of the parallel worlds of function fields and number fields

Topics range from Arakelov geometry, the search for a theory of varieties over the field with one element, via Eisenstein series to Drinfeld modules, and t-motives

Aimed at graduate students, mathematicians, and researchers interested in geometry and arithmetic and their connections