"Numbers with altitude": Review of Mathematical Mountaintops, by John Casti. Reviewed by Robert Matthews. New Scientist, 5 January 2002, page 34.
This book focuses on five famous mathematical problems whose solutions are considered to count among the major achievements in the field in the past century. The first problem Casti discusses is whether there exists a decision process for solving a certain type of polynomial equation, called a Diophantine equation, that involves only rational numbers. This is problem 10 in David Hilbert's legendary list of 23 problems presented at the International Congress of Mathematicians in Paris in 1900. The reviewer says this is an "awful" way to start the book: "an abstruse question with an even more abstruse solution." However, he says the book improves as Casti moves on to the other four problems. One of them is the first problem on Hilbert's list, Cantor's Continuum Hypothesis. The other three are the four-color theorem, Kepler's Conjecture about the most efficient way to pack spheres, and, of course, Fermat's Last Theorem. The reviewer concludes: "Apart from the Diophantine misjudgment, Casti has given us something rare here: a book on higher mathematics that challenges and entertains in equal measure."
--- Allyn Jackson