Author Description

<p>In 1842 the Belgian mathematician Eugène Charles Catalan asked whether 8 and 9 are the only consecutive pure powers of non-zero integers. 160 years after, the question was answered affirmatively by the Swiss mathematician of Romanian origin Preda Mihăilescu. In other words, 32 – 23 = 1 is the only solution of the equation <i>xp</i> – <i>yq</i> = 1 in integers <i>x, y, p, q</i> with <i>xy</i> ≠ 0 and <i>p, q </i>≥<i> </i>2.</p><p></p><p>In this book we give a complete and (almost) self-contained exposition of Mihăilescu’s work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume a very modest background:a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.</p>