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The Ultimate Challenge
The $3x+1$ problem, or Collatz problem, concerns the following seemingly innocent arithmetic procedure applied to integers: If an integer $x$ is odd then “multiply by three and add one”, while if it is even then “divide by two”. The $3x+1$ problem asks whether, starting from any positive integer, repeating this procedure over and over will eventually reach the number 1. Despite its simple appearance, this problem is unsolved. Generalizations of the problem are known to be undecidable, and the problem itself is believed to be extraordinarily difficult. This book reports on what is known on this problem. It consists of a collection of papers, which can be read independently of each oth...
Erds on Graphs
This book is a tribute to Paul Erd\H{o}s, the wandering mathematician once described as the "prince of problem solvers and the absolute monarch of problem posers." It examines -- within the context of his unique personality and lifestyle -- the legacy of open problems he left to the world after his death in 1996. Unwilling to succumb to the temptat
The Art of Computer Programming
The bible of all fundamental algorithms and the work that taught many of today's software developers most of what they know about computer programming. –Byte, September 1995 I can't begin to tell you how many pleasurable hours of study and recreation they have afforded me! I have pored over them in cars, restaurants, at work, at home... and even at a Little League game when my son wasn't in the line-up. –Charles Long If you think you're a really good programmer... read [Knuth's] Art of Computer Programming... You should definitely send me a resume if you can read the whole thing. –Bill Gates It's always a pleasure when a problem is hard enough that you have to get the Knuths off the sh...
CryptoSchool
This book offers an introduction to cryptology, the science that makes secure communications possible, and addresses its two complementary aspects: cryptography—--the art of making secure building blocks—--and cryptanalysis—--the art of breaking them. The text describes some of the most important systems in detail, including AES, RSA, group-based and lattice-based cryptography, signatures, hash functions, random generation, and more, providing detailed underpinnings for most of them. With regard to cryptanalysis, it presents a number of basic tools such as the differential and linear methods and lattice attacks. This text, based on lecture notes from the author’s many courses on the ...
The Art of Computer Programming, Volume 4A
The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1 Knuth’s multivolume analysis of algorithms is widely recognized as the definitive description of classical computer science. The first three volumes of this work have long comprised a unique and invaluable resource in programming theory and practice. Scientists have marveled at the beauty and elegance of Knuth’s analysis, while practicing programmers have successfully applied his “cookbook” solutions to their day-to-day problems. The level of these first three volumes has remained so high, and they have displayed so wide and deep a familiarity with the art of computer programming, that a sufficient “review...
Concrete Mathematics
This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas,...
Conversational Problem Solving
This book features mathematical problems and results that would be of interest to all mathematicians, but especially undergraduates (and even high school students) who participate in mathematical competitions such as the International Math Olympiads and Putnam Competition. The format is a dialogue between a professor and eight students in a summer problem solving camp and allows for a conversational approach to the problems as well as some mathematical humor and a few nonmathematical digressions. The problems have been selected for their entertainment value, elegance, trickiness, and unexpectedness, and have a wide range of difficulty, from trivial to horrendous. They range over a wide variety of topics including combinatorics, algebra, probability, geometry, and set theory. Most of the problems have not appeared before in a problem or expository format. A Notes section at the end of the book gives historical information and references.
Modern Computer Algebra
Now in its third edition, this highly successful textbook is widely regarded as the 'bible of computer algebra'.
Modern Computer Arithmetic
Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favourite language, while keeping a high-level description and avoiding too low-level or machine-dependent details. The book is intended for anyone interested in the design and implementation of efficient high-precision algorithms for computer arithmetic, and more generally efficient multiple-precision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions to selected exercises are available from the authors.
