**MSRI Mathematical Circles Library**

Volume: 16;
2015;
158 pp;
Softcover

MSC: Primary 00;
Secondary 34; 68; 20; 11

**Print ISBN: 978-0-8218-9416-3
Product Code: MCL/16**

List Price: $29.00

Individual Price: $21.75

**Electronic ISBN: 978-1-4704-2550-0
Product Code: MCL/16.E**

List Price: $29.00

Individual Price: $21.75

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#### Supplemental Materials

# Experimental Mathematics

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*V. I. Arnold*

Translated by Dmitry Fuchs and Mark Saul.

A co-publication of the AMS and the Mathematical Sciences Research Institute

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).

#### Readership

Undergraduate and graduate students and research mathematicians interested in mathematics.

#### Table of Contents

# Table of Contents

## Experimental Mathematics

- Cover 11
- Title page 22
- Preface to the English Translation 66
- Introduction 1010
- Lecture 1. The Statistics of Topology and Algebra 1212
- Lecture 2. Combinatorial Complexity and Randomness 6868
- Lecture 3. Random Permutations and Young Diagrams of Their Cycles 9292
- 1. Statistics of Young Diagrams of Permutations of Small Numbers of Objects 9494
- 2. Experimentation with Random Permutations of Larger Numbers of Elements 101101
- 3. Random Permutations of 𝑝² Elements Generated by Galois Fields 105105
- 4. Statistics of Cycles of Fibonacci Automorphisms 106106
- Editor’s notes 115115

- Lecture 4. The Geometry of Frobenius Numbers for Additive Semigroups 120120
- 1. Sylvester’s Theorem and the Frobenius Numbers 121121
- 2. Trees Blocked by Others in a Forest 124124
- 3. The Geometry of Numbers 126126
- 4. Upper Bound Estimate of the Frobenius Number 130130
- 5. Average Values of the Frobenius Numbers 141141
- 6. Proof of Sylvester’s Theorem 144144
- 7. The Geometry of Continued Fractions of Frobenius Numbers 146146
- 8. The Distribution of Points of an Additive Semigroup on the Segment Preceding the Frobenius Number 157157
- Editor’s notes 163163

- Bibliography 166166
- Back Cover 169169