foreword by Martin Davis and Hilary Putnam In 1900, the German mathematician David Hilbert put forth a list of 23 unsolved problems that he saw as being the greatest challenges for twentieth-century mathematics. Hilbert's 10th problem, to find a method for deciding whether a Diophantine equation has an integral solution, was solved by Yuri Matiyasevich in 1970. Proving the undecidability of Hilbert's 10th problem is clearly one of the great mathematical results of the century. This book presents the full, self-contained negative solution of Hilbert's 10th problem. In addition it contains a number of diverse, often striking applications of the technique developed for that solution, describes the many improvements and modifications of the original proof since the problem was "unsolved" 20 years ago, and adds several new, previously unpublished proofs. More on this book...