100 YEARS OF MATH MILESTONES The Pi Mu Epsilon Centennial Collection Stephan Ramon Garcia Steven J. Miller 100 YEARS OF MATH MILESTONES The Pi Mu Epsilon Centennial Collection 10.1090/mbk/121 100 YEARS OF MATH MILESTONES The Pi Mu Epsilon Centennial Collection Stephan Ramon Garcia Steven J. Miller 2010 Mathematics Subject Classification. Primary 00A08, 00A30, 00A35, 05-01, 11-01, 30-01, 54-01, 60-01. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-121 Library of Congress Cataloging-in-Publication Data Names: Garcia, Stephan Ramon, author. | Miller, Steven J., 1974- author. Title: 100 years of math milestones : the Pi Mu Epsilon centennial collection / Stephan Ramon Garcia, Steven J. Miller. Other titles: One hundred years of math milestones | Pi Mu Epsilon centennial collection Description: Providence, Rhode Island : American Mathematical Society, [2019] | Includes bibli- ographical references and indexes. Identifiers: LCCN 2019000982 | ISBN 9781470436520 (alk. paper) Subjects: LCSH: Mathematics–United States–History. | Pi Mu Epsilon. | AMS: General – General and miscellaneous specific topics – Philosophy of mathematics. msc | General – General and miscellaneous specific topics – Methodology of mathematics, didactics. msc | Combinatorics – Instructional exposition (textbooks, tutorial papers, etc.). msc | Number theory – Instruc- tional exposition (textbooks, tutorial papers, etc.). msc | Functions of a complex variable – Instructional exposition (textbooks, tutorial papers, etc.). msc | General topology – Instruc- tional exposition (textbooks, tutorial papers, etc.). msc | Probability theory and stochastic processes – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA27.U5 G37 2019 | DDC 510.9–dc23 LC record available at https://lccn.loc.gov/2019000982 Copying and reprinting. 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Visit the AMS home page at https://www.ams.org/ 10987654321 242322212019 Stephan Ramon Garcia dedicates this book to his wife, Gizem Karaali, and their children, Reyhan and Altay. Thanks also go to his parents for their constant support and affection. Steven Miller dedicates this book with thanks to his many colleagues and stu- dents who assisted in writing this book, to his in-laws Jeffrey and Judy Gelfand for providing a hospitable environment where many of these entries were written and edited, and to his friends at Pi Mu Epsilon (especially Harold Reiter, a previous editor of the Problem Section) for their support of this project. Contents Preface xi Notation xiii 1913. Paul Erd˝os 1 1914. Martin Gardner 7 1915. General Relativity and the Absolute Differential Calculus 11 1916. Ostrowski’s Theorem 17 1917. Morse Theory, but Really Cantor 21 1918. Georg Cantor 27 1919. Brun’s Theorem 33 1920. Waring’s Problem 39 1921. Mordell’s Theorem 45 1922. Lindeberg Condition 51 1923. The Circle Method 57 1924. The Banach–Tarski Paradox 61 1925. The Schr¨odinger Equation 67 1926. Ackermann’s Function 71 1927. William Lowell Putnam Mathematical Competition 75 1928. Random Matrix Theory 79 1929. G¨odel’s Incompleteness Theorems 85 1930. Ramsey Theory 89 1931. The Ergodic Theorem 95 1932. The 3x + 1 Problem 101 1933. Skewes’s Number 107 1934. Khinchin’s Constant 113 vii viii CONTENTS 1935. Hilbert’s Seventh Problem 117 1936. Alan Turing 121 1937. Vinogradov’s Theorem 127 1938. Benford’s Law 131 1939. The Power of Positive Thinking 137 1940. A Mathematician’s Apology 141 1941. The Foundation Trilogy 145 1942. Zeros of ζ(s) 151 1943. Breaking Enigma 157 1944. Theory of Games and Economic Behavior 163 1945. The Riemann Hypothesis in Function Fields 169 1946. Monte Carlo Method 175 1947. The Simplex Method 181 1948. Elementary Proof of the Prime Number Theorem 187 1949. Beurling’s Theorem 193 1950. Arrow’s Impossibility Theorem 199 √ 1951. Tennenbaum’s Proof of the Irrationality of 2 205 1952. NSA Founded 209 1953. The Metropolis Algorithm 215 1954. Kolmogorov–Arnold–Moser Theorem 221 1955. Roth’s Theorem 227 1956. The GAGA Principle 233 1957. The Ross Program 235 1958. Smale’s Paradox 241 1959. QR Decomposition 247 1960. The Unreasonable Effectiveness of Mathematics 251 1961. Lorenz’s Nonperiodic Flow 257 1962. The Gale–Shapley Algorithm and the Stable Marriage Problem 263 1963. Continuum Hypothesis 269 CONTENTS ix 1964. Principles of Mathematical Analysis 275 1965. Fast Fourier Transform 281 1966. Class Number One Problem 287 1967. The Langlands Program 293 1968. Atiyah–Singer Index Theorem 299 1969. Erd˝os Numbers 305 1970. Hilbert’s Tenth Problem 311 1971. Society for American Baseball Research 317 1972. Zaremba’s Conjecture 323 1973. Transcendence of e Centennial 329 1974. Rubik’s Cube 333 1975. Szemer´edi’s Theorem 339 1976. Four Color Theorem 345 1977. RSA Encryption 351 1978. Mandelbrot Set 357 1979. TEX 363 1980. Hilbert’s Third Problem 369 1981. The Mason–Stothers Theorem 375 1982. Two Envelopes Problem 381 1983. Julia Robinson 385 1984. 1984 391 1985. The Jones Polynomial 395 1986. Sudokus and Look and Say 401 1987. Primes, the Zeta Function, Randomness, and Physics 407 1988. Mathematica 413 1989. PROMYS 421 1990. The Monty Hall Problem 427 1991. arXiv 433 1992. Monstrous Moonshine 439 xCONTENTS 1993. The 15-Theorem 445 1994. AIM 451 1995. Fermat’s Last Theorem 457 1996. Great Internet Mersenne Prime Search (GIMPS) 463 1997. The Nobel Prize of Merton and Scholes 469 1998. The Kepler Conjecture 475 1999. Baire Category Theorem 481 2000. R 487 2001. Colin Hughes Founds Project Euler 493 2002. PRIMES in P 499 2003. Poincar´e Conjecture 505 2004. Primes in Arithmetic Progression 511 2005. William Stein Developed Sage 519 2006. The Strong Perfect Graph Theorem 525 2007. Flatland 531 2008. 100th Anniversary of the t-Test 537 2009. 100th Anniversary of Brouwer’s Fixed-Point Theorem 543 2010. Carmichael Numbers 549 2011. 100th Anniversary of Egorov’s Theorem 555 2012. National Museum of Mathematics 561 Index of People 565 Index 571 Preface In 2013, the second named author had the honor of succeeding Ashley Ahlin and Harold Reiter as the editor of the Problem Department of the ΠME Journal. This event essentially coincided with the 100th anniversary of Pi Mu Epsilon, so Miller thought it would be fun and appropriate to recognize this milestone in some way. Many others agreed. For example, Mike Pinter, from Belmont University in Nashville, Tennessee, proposed the base-16 celebratory equation PMEMATH +SOCIETY HUNDRED (which was used in the Spring 2014 issue). Many readers submitted correct solu- tions, the first being Jessica Lehr of Elizabethtown College. We leave the task of determining all possible solutions as a fun exercise for you. Being still somewhat young, energetic, and new to the job, while also gravely worried about finding enough good problems for issue after issue (not yet aware of the excellent submissions that would consistently arrive), Miller decided to celebrate with one hundred problems related to important mathematical milestones of the past century. Since one hundred is a large number of problems relative to the normal operation of the Problem Department (there are typically five or six problems per issue), he asked many colleagues for contributions. This resulted in four centennial articles, which appeared in The Pi Mu Epsilon Journal in 2013–2014 (13 (2013), no. 9, 513–534; 13 (2014), no. 10, 577–608; 14 (2014), no. 1, 65–99; and 14 (2014), no. 2, 100–134). The four articles were well received and there was strong interest in converting them into a book. The first named author came on board early in the process as a collaborator. Every entry was either expanded jointly by us from the four centennial articles or simply written anew. The second option was an essential step in converting the collection from a series of disjointed problems into a unified whole. We have used the original descriptions as springboards to introduce a variety of mathematical ideas, techniques, and applications. Whenever possible, we have quoted primary sources. Concepts are often introduced early on and then threaded through and expanded upon in later entries. The final result is a tour through much of mathematics, with an emphasis on beauty, big ideas, and interesting problems. There are several influential collections of problems that have motivated and guided mathematics. Hilbert’s problems and the Clay Millennium Problems are notable examples. We have a different emphasis here. Pi Mu Epsilon is an un- dergraduate mathematics honor society and thus, in addition to being important, the problems must be accessible to students. Although some of them do require analysis or algebra, number theory or probability, as a whole we hope they will be xi xii PREFACE appealing to energetic and enthusiastic math majors of all stripes. We wanted to create a collection that would motivate people who are still trying to decide what to do with their lives, as well as those who already have. No list can be complete and there are far too many items to celebrate. This book necessarily misses many old favorites. It is largely a reflection of the personal tastes and inclinations of the two authors.