This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) open-ended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems. The mathematical prerequisites for this book are few. Early chapters contain topics such as integer divisibility, modular arithmetic, and applications to cryptography, while later chapters contain more specialized topics, such as Diophantine approximation, number theory of dynamical systems, and number theory with polynomials. Students of all levels will be drawn in by the patterns and relationships of number theory uncovered through data driven exploration.

One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

Mathematics is not, and never will be, an empirical science, but mathematicians are finding that the use of computers and specialized software allows the generation of mathematical insight in the form of conjectures and examples, which pave the way for theorems and their proofs. In this way, the experimental approach to pure mathematics is revolutionizing the way research mathematicians work. As the first of its kind, this book provides material for a one-semester course in experimental mathematics that will give students the tools and training needed to systematically investigate and develop mathematical theory using computer programs written in Maple. Accessible to readers without prior programming experience, and using examples of concrete mathematical problems to illustrate a wide range of techniques, the book gives a thorough introduction to the field of experimental mathematics, which will prepare students for the challenge posed by open mathematical problems.

New mathematical insights and rigorous results are often gained through extensive experimentation using numerical examples or graphical images and analyzing them. Today computer experiments are an integral part of doing mathematics. This allows for a more systematic approach to conducting and replicating experiments. The authors address the role of

With the continued advance of computing power and accessibility, the view that "real mathematicians don't compute" no longer has any traction for a newer generation of mathematicians. The goal in this book is to present a coherent variety of accessible examples of modern mathematics where intelligent computing plays a significant role and in so doi

This revised and updated second edition maintains the content and spirit of the first edition and includes a new chapter, "Recent Experiences", that provides examples of experimental mathematics that have come to light since the publication of the first edition in 2003. For more examples and insights, Experimentation in Mathematics: Computational P

Keith Devlin and Jonathan Borwein, two well-known mathematicians with expertise in different mathematical specialties but with a common interest in experimentation in mathematics, have joined forces to create this introduction to experimental mathematics. They cover a variety of topics and examples to give the reader a good sense of the current sta

As discrete mathematics rapidly becomes a required element of undergraduate mathematics programs, algebraic software systems replace compiled languages and are now most often the computational tool of choice. Newcomers to university level mathematics, therefore, must not only grasp the fundamentals of discrete mathematics, they must also learn to use an algebraic manipulator and develop skills in abstract reasoning. Experimental Mathematics with MAPLE uniquely responds to these needs. Following an emerging trend in research, it places abstraction and axiomatization at the end of a learning process that begins with computer experimentation. It introduces the foundations of discrete mathematics and, assuming no previous knowledge of computing, gradually develops basic computational skills using the latest version of the powerful MAPLE® software. The author's approach is to expose readers to a large number of concrete computational examples and encourage them to isolate the general from the particular, to synthesize computational results, formulate conjectures, and attempt rigorous proofs. Using this approach, Experimental Mathematics with MAPLE enables readers to build a foundation in discrete mathematics, gain valuable experience with algebraic computing, and develop a familiarity with basic abstract concepts, notation, and jargon. Its engaging style, numerous exercises and examples, and Internet posting of selected solutions and MAPLE worksheets make this text ideal for use both in the classroom and for self-study.

Thirty years ago mathematical, as opposed to applied numerical, computation was difficult to perform and so relatively little used. Three threads changed that: the emergence of the personal computer; the discovery of fiber-optics and the consequent development of the modern internet; and the building of the Three “M’s” Maple, Mathematica and Matlab. We intend to persuade that Mathematica and other similar tools are worth knowing, assuming only that one wishes to be a mathematician, a mathematics educator, a computer scientist, an engineer or scientist, or anyone else who wishes/needs to use mathematics better. We also hope to explain how to become an "experimental mathematician" while learning to be better at proving things. To accomplish this our material is divided into three main chapters followed by a postscript. These cover elementary number theory, calculus of one and several variables, introductory linear algebra, and visualization and interactive geometric computation.

Classic, lively explanation of one of the byways of mathematics. Klein bottles, Moebius strips, projective planes, map coloring, problem of the Koenigsberg bridges, much more, described with clarity and wit.