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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 | 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 | 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. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2019 by the authors. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 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. 7 1915. General Relativity and the Absolute Differential 11

1916. Ostrowski’s Theorem 17 1917. Morse Theory, but Really Cantor 21 1918. 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 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 in Function Fields 169

1946. Monte Carlo Method 175

1947. The Method 181

1948. Elementary Proof of the 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 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. 345

1977. RSA Encryption 351

1978. 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, , and Physics 407

1988. Mathematica 413

1989. PROMYS 421

1990. The 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 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 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 to being important, the problems must be accessible to students. Although some of them do require analysis or algebra, 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. Accessibility counted far more than importance in breaking the many ties, and thus the collection below is well represented with problems that are somewhat recreational but also serve as springboards to great mathematics. We thank all the people who have helped us over the last several years. This in- cludes the problem proposers, James M. Andrews and Avery T. Carr, who helped edit some of the original collection of problems; Miles C. Fippinger, who helped with some of the initial organization; and Ben Logsdon, who carefully read an early draft. We owe particular gratitude to Zachary Glassman, who made numerous Tikz drawings for some of the earlier entries. We learned many Tikz tricks and techniques from him, without which many of the remaining illustrations would not have been possible. In addition, we are greatly indebted to Yo Akiyama, Kather- ine Blake, Paula Burkhardt-Guim, Max Chao-Haft, Amina Diop, Alexandre Gue- ganic, Mark Hay, Bjørn Kjos-Hanssen, Forest Kobayashi, Scott Duke Kominers, Jeffrey Lagarias, David Lee, Clayton Mizgerd, Jos´eMu˜noz-L´opez, Giebien Na, Carl Pomerance, Harald Schilly, Zachary Siegel, Lily Shao, William A. Stein, Hong Suh, Alexander Summers, James Tener, Gabe Udell, and Hunter Wieman for spotting numerous mistakes, typos, and errors throughout the book or suggesting various improvements to the text. The first named author also thanks Kathy Sheldon for her considerable logistical support. We were fortunate to work with a terrific staff at AMS (Marcia Almeida, Brian Bartling, John Brady, Sergei Gelfand, Eriko Hironaka, Arlene O’Sean, and Court- ney Rose), whose tireless efforts from the start of this project years ago to the careful reading of the final draft greatly enhanced the book before you. Although we are no longer young or energetic, it has been a fun and enlighten- ing experience working on so many diverse topics and with so many distinguished people. Read on, enjoy, and for those of you who someday aspire to be the Problem Editor for PME, here is some useful advice: start assembling the next hundred problems today!

Stephan Ramon Garcia Steven J. Miller Claremont, CA Williams College May 2, 2019 Williamstown, MA 01267 Carnegie Mellon University Pittsburgh, PA 15213 May 2, 2019 Notation

• ∅ ...... emptyset •|A| ...... cardinalityofa set A

• (...)b ...... numberinbase-b • log x ...... base-e logarithm of x • logb x ...... base-b logarithm of x • a|b ...... a divides b •x ...... greatestintegerfunction • gcd...... greatestcommondivisor • lcm...... leastcommonmultiple • a ≡ b (mod m) ...... congruencemodulo m • n i=1 ai ...... productof a1,a2,...,an • N ...... theset {1, 2, 3,...} of natural numbers • Z ...... theset {...,−2, −1, 0, 1, 2,...} of integers • Q ...... thesetofrationalnumbers • R ...... thesetofrealnumbers • C ...... thesetofcomplexnumbers • Re z ...... realpartofthecomplexnumber z • Im z ...... imaginarypartofthecomplexnumber z ∼ • = ...... equinumerosity(p.28) • f ∼ g ...... asymptoticequivalence(p.33) • π(x) ...... thenumberofprimesatmost x (p. 33) • Li(x) ...... (offset)logarithmic integral of x (p. 107)

xiii

Index of People

Abbott, Derek, 252 Bellaso, Giovan Battista, 212 Caesar, Julius, 125, 212 Abbott, Edwin Abbott, 531 Berge, Claude, 525 Caldwell, Chris K., 388 Ackermann, Wilhelm, 71 Bergelson, Vitaly, 341, 342 Campbell, John, 145, 533 Adams, Douglas, 27, 392 Bernoulli, Jacob, 381 Cantor, Georg, 21, 27, 71, Adams, John Couch, 221 Bernstein, Sergei, 196 118, 269, 329 Adleman, Leonard, 351, 501 Bertrand, Joseph, 129 Caplan, Seth, 531 Agmon, Shmuel, 197 Beukers, Frits, 365 Carlitz, Leonard, 460 Agrawal, Manindra, 501 Beurling, Arne, 194 Carmichael, Robert Daniel, Aguayo, Daniel, 489 Bhargava, Manjul, 47, 287, 549 Ahlin, Ashley, xi 446, 562 Carr, Avery T., xii, 222, 242, Aigner, Martin, 3, 189 Bieberbach, Ludwig, 393 383, 386 Akhmedov, Azer, 63 Bigelow, Stephen, 63 Catalan, Eug`ene Charles, Akiyama, Yo, xii Binet, Jacques Philippe 254 al-Din al-Tusi, Sharaf, 557 Marie, 495 Cauchy, Augustin-Louis, 448 Alcuin of York, 473 Birkar, Caucher, 562 Cellarosi, Francesco, 556 Alexander, James Waddell, Birkenmajer, Ludwik An- Chang, Alan, 334 396 toni, 370 Chang, Paul, 559 Alford, William, 549 Birkhoff, George, 21, 96 Chao-Haft, Max, xii Anderson, Randy L., 435 Birman, Joan, 399 Chˆau, Ngˆo Bao, 294 Andrade, Julio, 170 Black, Fischer, 469 Chebyshev, Pafnuty, 129, Andrews, James M., xii, 91, Blake, Katherine, xii 381, 515 231 Bohr, Neils, 14 Chen, Hang, 306 Ap´ery, Roger, 171, 364 Boltzmann, Ludwig, 95 Cheng, Christine, 265 Appel, Kenneth, 346 Bolyai, J´anos, 369 Cheng, Yuanyou, 388 Arnold, Vladimir, 221 Bolyai, Wolfgang Farkas, 369 Chioniadis, Gregory, 557 Arrow, Kenneth J., 199 Bombieri, Enrico, 170 Chowla, Sarvadaman, 552 Artin, Emil, 170 Borcherds, Richard, 441 Chudnovsky, Maria, 526 Aschbacher, Michael, 514 Bott, Raoul, 300 Church, Alonzo, 122 Asimov, Isaac, 14, 145, 533 Bourbaki, Nicholas, 234 Churchill, Winston, 158 Atiyah, Michael, 299, 490 Bourgain, Jean, 97, 326 Cipolla, Michele, 410 Augustine of Hippo, 473 Boyer, Carl Benjamin, 552 Cipra, Barry, 247 Axelsson, Ake,˚ 489 Brassau, Pierre, 489 Civario, Gilles, 402 Brauer, Richard, 515 Clausen, Thomas, 459 Babbage, Charles, 212 Broad, Steven, 23 Clay, Landon T., 490 Bach, Johann Sebastian, 88 Bronstein, Manuel, 303 Cocks, Clifford, 351 Bacon, Kevin, 1, 306 Brooks, Robert W., 357 Cohen, Paul, 269, 307, 484 Baez, John, 559 Brouwer, Luitzen Egbertus Cole, Frank Nelson, 464 Baire, Ren´e-Louis, 481 Jan, 543 Collatz, Lothar, 101 Baker, Alan, 288 Brown, Gordon, 123 Condorcet, Nicolas de, 199 Baker, Roger C., 388 Brun, Viggo, 33, 58 Connes, Alain, 163 Banach, Stefan, 62 Buffett, Warren, 470 Conrad, Brian, 233 Banks, William D., 552 Bunyakovsky, Viktor Conway, John Horton, 402, Banzhaf III, John F., 201 Yakovlevich, 521 441, 446 Barlow, William, 476 Burkhardt-Guim, Paula, xii Cooley, James William, 281 Bateman, Paul T., 521 Burt, David, 18 Cooper, Curtis, 306

565 566 INDEX OF PEOPLE

Corsi, Craig, 2, 258 Figalli, Alessio, 562 Grossman, Jerrold, 72, 306 Cram´er, Harald, 410, 552 Fippinger, Miles C., xii Grothendieck, Alexander, Firk, Frank W. K., 13 233, 376 Dantzig, George, 137, 182 Focardi, Sergio M., 252 Gueganic, Alexandre, xii Davids, Bob, 317 Fraenkel, Aviezri, 306 Guthrie, Francis, 345 Davis, Martin, 312, 385, 386 Francis, John G. F., 247 Gy´arf´as, Andr´as, 526 de Branges, Louis, 394 Franklin, Benjamin, 14 de Grey, Aubrey, 527 Franklin, Philip, 347 Hadamard, Jacques, 21, 187, de Moivre, Abraham, 495 Freedman, Michael, 505 190, 249 Debrunner, Hans, 369 Freeman, Jesse, 118 Hadwiger, Hugo, 527 Dehn, Max, 369, 396 Frege, Gottlob, 85 H¨aggstr¨om, Olle, 383 Delaunay, Charles-Eug´ene, Frenkel, Igor, 441 Haken, Wolfgang, 346 221 Frey, Gerhard, 234 Hales, Thomas C., 476 Diaconis, Persi, 381 Fried, David, 421 Hall, Monty, 427 Dickson, Leonard Eugene, Fry, John, 451 Halparin, Monte, 427 521 Fry, Roger, 39 Hamel, Georg, 278 Diop, Amina, xii Furstenberg, Hillel, 3, 230, Hamilton, Richard S., 505 Diophantus of Alexandria, 340 Hamming, Richard W., 252 457, 458 Hammond, Christopher N. Dirichlet, Peter Gustav Leje- Gale, David, 263 B., 306 une, 3, 172, 458 Galilei, Galileo, 27 Hanke, Jonathan P., 447 Duren, Peter, 308 Gallian, Joseph, 75 Hardy, Godfrey Harold, 14, Dyson, Freeman, 81, 228, Galois, Evariste,´ 145, 514 39, 57, 59, 141, 153, 187, 408 Gardner, Martin, 7, 342, 383, 195, 381, 521, 529, 551, 442, 546 555 Eder, Maciej, 487 Garfield, James A., 315 Harman, Glyn, 388 Egorov, Dmitri, 556 Harriot, Thomas, 476 Einstein, Albert, 11 Gauss, Carl Friedrich, 11, 97, Haselgrove, C. Brian, 93 Ekhad, Shalosh B., 403 143, 281, 287, 381, 395, 423, 445, 448 Hasse, Helmut, 48, 170 Elga, Adam, 428 Hawkins, David, 408 Elkies, Noam, 39 Gelfand, Israel, 148, 197, 252 Hay, Mark, xii Eppstein, David, 527 Gelfond, Alexandr, 117 Heath-Brown, Roger, 552 Erd˝os, Paul, 1, 90, 102, 129, Gentleman, Robert, 487 Heaviside, Oliver, 11 187, 189, 305–307, 340 Germain, Sophie, 460 Heawood, Percy J., 346 Escher, Maurits Cornelis, 88, Gerwien, Paul, 369 Heegner, Kurt, 288, 454 145 Geyer, Lukas, 362 Eskew, Monroe, 559 Gibbon, Edward, 145 Heeringa, Brent, 122 Eubulides of Miletus, 85 Gibbs, Josiah Willard, 11, 95 Helfgott, Harald Andr´es, 127 Euclid of Alexandria, 4, 272, Ginsparg, Paul, 433 Hellegouarch, Yves, 234 275, 369, 474 Gladwell, Malcolm, 138 Hensel, Kurt, 17 Euler, Leonhard, 2, 39, 91, Glassman, Zachary, xii Hermite, Charles, 329, 342 117, 118, 171, 295, 378, G¨odel, Kurt, 86, 122, 141, Hilbert, David, 40, 71, 86, 422, 423, 458, 463, 474, 269, 484 117, 269, 311, 369, 385, 513, 533 Goffman, Casper, 275, 305 457, 490, 564 Evans, Jonny, 399 Goldbach, Christian, 127 Hirzebruch, Friedrich, 300 Goldfeld, Dorian, 187 Hoover, Colleen, 23 Fabozzi, Frank J., 252 Goldston, Daniel, 528 Horn, Roger A., 521, 523 Faltings, Gerd, 378 Golomb, Solomon, 527 Horner, William George, 285 Fedi, Zolt, 306 Gomory, Ralph E., 509 Householder, Alston Scott, Ferguson, Samuel P., 476 Gorenstein, Daniel, 514 247 Fermat, Cl´ement-Samuel, Gosset, William Sealy, 537 Huang, Ming-Deh A., 501 458 Gowers, Timothy, 90, 490 Hughes, Colin, 493 Fermat, Pierre de, 145, 208, Graham, Ronald, 90, 431, 234, 311, 421, 448, 457, 442 Ihaka, Ross, 487 512 Granville, Andrew, 549 Ingham, Albert, 172, 388 Feynman, Richard, 76 Green, Ben, 4, 58, 511 Irons, Jeremy, 144 INDEX OF PEOPLE 567

Jacobi, Carl Gustav Jacob, Lagarias, Jeffrey, xii, 103, Manasse, Mark, 355 445 108, 370, 477 Marchal, Christian, 476 James, Bill, 319 Lagrange, Joseph-Louis, 39, Markov, Andrey, 220, 381, Jensen, Alexandra, 346 445, 475 487 Jensen, Johan Ludwig, 460 Lambert, Joel, 306 Mason, Richard C., 375 Johnson, Charles R., 523 Lam´e, Gabriel, 458, 499 Masser, David, 376 Johnson, Dano, 531 Landau, Edmund, 172, 521 Matelski, J. Peter, 357 Jones, James P., 386 Lander, Leon J., 39 Mathey, Steven, 491 Jones, Michael, 200 Langlands, Robert, 293 Matiyasevich, Yuri, 312, 385, Jones, Peter, 277 Laplace, Pierre-Simon, 381 386 Jones, Toby, 144 Le Gall, Fran¸cois, 283 Maynard, James, 33, 528 Jones, Vaughan F. R., 163, Le Verrier, Urbain, 221 Mazur, Barry, 47 396 Lebesgue, Henri, 458 McGarvey, Joey, 158 Lebesgue, Victor-Am´ed´ee, McGuire, Gary, 402 Kahoro, Elvis, 416 458 McKay, John, 440 Kanigel, Robert, 143 Leclerc, Georges-Louis, 176 Mercer, Idris D., 230 Kant, Immanuel, 273 Lee, David, xii Mersenne, Marin, 463 Karp, Richard, 402 Lee, Harper, 487 Mertens, Franz, 110, 189 Kasiski, Friedrich, 212 Lefschetz, Solomon, 300 Merton, Robert C., 469 Katz, Nick, 82 Legendre, Adrien-Marie, Merton, Robert K., 552 Kayal, Neeraj, 501 445, 458 Metropolis, Nicholas, 219 Kehle, Paul, 265 Lehmer, Derrick Henry, 465 Meurman, Arne, 441 Kempe, Alfred, 346 Lehr, Jessica, xi Mili´cevi´c, Djordje, 77 Kepler, Johannes, 476 Leibniz, Gottfried Wilhelm, Miller, Gary Lee, 501 Kestemont, Mike, 487 147, 561 Miller, Stephen D., 365 Khavinson, Dmitry, 362 Lemke Oliver, Robert, 415 Mills, William H., 388 Khinchin, Aleksandr, 97, 113 Lenstra, Arjen K., 355 Milnor, John, 76 Khovanov, Mikhail, 400 Leontief, Wassily, 472 Mirzakhani, Maryam, 562 Kjos-Hanssen, Bjørn, xii Lepowsky, James, 441 Mishkin, Pamela, 346, 465 Klamkin, Murray S., 103 Levi-Civita, Tullio, 11 Mizgerd, Clayton, xii Kleene, Stephen, 122 Levinson, Norman, 153 M¨obius, August Ferdinand, Klein, Felix, 243, 440 Lewis, Michael, 317 242 Klyachko, Alexander, 68 Lie, Sophus, 514 Mochizuki, Shinichi, 376 Knuth, Donald, 363, 443 Lindeberg, Jarl, 54 Molchanov, Stanislav, 253 Knutson, Allen, 68 Lindemann, Ferdinand von, Monaco, Jane J., 435 Kobayashi, Forest, xii 329 Montague, David, 207 Kodaira, Kunihiko, 234 Linnik, Yuri Vladimirovich, Montgomery, Hugh, 81, 408 Koebe, Paul, 394 553 Mordell, Louis, 46 Kolmogorov, Andrey, 221, Liouville, Joseph, 118, 228, Moreno, Samuel G., 35 381 329 Morgan, Frank, 306, 460, 507 Kominers, Scott Duke, xii, Listing, Johann Benedict, Morgenstern, Oskar, 163 447 242 Morin, Bernard, 241 Kontorovich, Alex, 306, 323, Littlewood, John Edensor, Morse, Marston, 21 326, 402 39, 57, 59, 108, 143, 383, Moser, J¨urgen, 221 Korselt, Alwin, 552 521, 529, 555 Moser, Leo, 91, 527 Kraitchek, Maurice, 383 Logsdon, Ben, xii Moser, William, 527 Krantz, Steven G., 141 Lorenz, Edward, 257 Mumford, David, 76 Krohn, Maxwell, 489 Lov´asz, L´aszl´o, 525 Mu˜noz-L´opez, Jos´e, xii Kublanovskaya, Vera, 247 Lowell, Percival, 75 Munroe, M. E., 275 Kummer, Ernst, 459 Lowry, John, 369 M¨untz, Herman, 197 Kurschak, Josef, 17 Luca, Florian, 306, 417 Murray, Francis, 398 Kuzmin, Rodion, 117 Lucas, Edouard,´ 423, 464 Murty, M. Ram, 306 Lyons, Richard, 515 Myerson, Gerry, 505 Labb´e, Cyril, 490 Lacan, Jacques, 436, 466 Mackall, Blake, 441 Na, Giebien, xii 568 INDEX OF PEOPLE

Nash Jr., John Forbes, 163, Quillen, Daniel, 76 Scott, Alex, 526 543 Segal, Irving, 559 Navier, Claude-Louis, 491 Rabin, Michael Oser, 501 Selberg, Atle, 153, 187 Nelson, Edward, 527, 559 Rainich, Georg Yuri, 534 Seldon, Hari, 145 Neuenschwander, Dwight E., Raleigh, Walter, 476 Selhorst-Jones, Vincent, 306, 251 Ramanujan, Srinivasa, 40, 307 , Isaac, 67, 435, 561 59, 143, 172, 342, 365, Selvin, Steve, 427 Neyman, Jerzy, 137 381, 446, 448, 551 Serre, Jean-Pierre, 233, 299 Nicely, Thomas, 33 Ramsey, Frank Plumpton, 89 Severini, Carlo, 556 Nimitz, Chester W., 160 Reid, Constance, 385 Seymour, Paul, 526 Nishikado, Tomohiro, 359 Reidemeister, Kurt, 396 Shakespeare, William, 487 Nobel, Alfred, 469 Reiter, Harold, xi, 253 Shamir, Adi, 351 Norton, Simon P., 441 R´enyi, Alfr´ed, 306 Shao, Lily, xii Norwich, John Julius, 201 Reznick, Bruce, 76 Shapiro, Arnold, 241 Ribet, Ken, 234 Shapiro, Daniel, 235 O’Brien, Miles, 392 Ricci-Curbastro, Gregorio, Shapiro, Harold S., 306 O’Neill, Cathy, 267 11 Shapley, Lloyd, 263 Odlyzko, Andrew, 81, 407, Riemann, Bernhard, 11, 151, Shavgulidze, E. T., 63 408 188 Sheil-Small, Terence, 362 Oesterl´e, Joseph, 376 Ringel, Gerhard, 347 Sheldon, Kathy, xii Olbers, Wilhelm, 284 Risch, Robert Henry, 302 Sherman, David, 306 Ono, Ken, 60, 448 Rivest, Ronald, 351 Shiing-Shen, Chern, 293 Orwell, George, 391 Robertson, Neil, 526 Shor, Peter, 352 Ostrowski, Alexander, 17 Robinson, Julia, 312, 385, Siegel, Carl Ludwig, 228, 460 386 Siegel, Zachary, xii Palka, Bruce, 534 Robinson, Raphael, 61, 71 Sierpi´nski, Waclaw, 271 Pan, Chengdong, 553 Rochefort, Joseph J., 160 Silva, Cesar E., 97 Parkin, Thomas R., 39 Rosenthal, Jeffrey, 219 Silverman, Joseph H., 376, Patel, Dev, 144 Ross, Arnold, 235, 421 377 Penrose, Lionel, 201 Ross, W. Bruce, 91 Simerka,ˇ V´aclav, 552 Perelman, Grigori, 433, 505 Ross, William T., 306 Simpson, Homer, 241 Perichon, Benoˆıt, 511 Roth, Alvin, 263 Singer, Isador, 299 Perpetua, Byron, 9 Roth, Klaus Friedrich, 228, Skewes, Stanley, 108 Picard, Charles Emile,´ 166 340 Smale, Stephen, 241, 505 Picard, Jean-Luc, 145, 166, Rowling, J. K., 487 Smith, Stephen D., 514 392 Royden, Halsey, 307 Smith, Winston, 391 Pinch, Richard G. E., 549 Rubik, Ern˝o, 333 Snow, Joanne, 23 Pinter, Mike, xi Rudin, Walter, 275 Snyder, Noah, 376 Pintz, J´anos, 388, 528 Russell, Bertrand, 85, 86 Sokal, Alan, 436, 466 Poincar´e, Henri, 21, 221, 257 Rybicki, Jan, 487 S´os, Vera, 306 P´olya, George, 91, 407 Soundararajan, Kannan, Pomerance, Carl, xii, 503, Sally, Paul, 371 415, 448, 528 504, 549, 551 Sarason, Donald, 197 Spencer, Joel, 91, 93 Post, Emil Leon, 122 Sarnak, Peter, 82 Sperner, Emanuel, 494 Pratt, Kyle, 288 Sato, Daihachiro, 386 Spirkl, Sophie, 526 Punnett, Reginald Crundall, Savant, Marilyn vos, 427 Stark, Eberhard L., 35 142 Saxena, Nitin, 501 Stark, Harold, 288 Pushkin, Alexander, 220, Schilly, Harald, xii Stein, William A., xii, 519 487 Schinzel, Andrzej, 305, 521 Stepanov, Sergei Aleksan- Putinar, Mihai, 306 Schneeberger, William, 446 drovich, 170 Putnam, Elizabeth Lowell, Schneider, Theodor, 117 Stevens, Glenn H., 421 75 Scholes, Myron S., 469 Stigler, Stephen, 552 Putnam, Hilary, 312, 385, Scholze, Peter, 562 Stoiciu, Mihai, 482 386 Schr¨odinger, Erwin, 67, 383 Stokes, George Gabriel, 491 Putnam, William Lowell, 75 Schultz, William Henry, 208 Stone, Daniel F., 164 INDEX OF PEOPLE 569

Stone, Harlan F., 69 Trinh, Minh-Tam, 460 Wellens, Jake, 114 Stone, Marshall H., 69 Tripp, Samuel, 403 Weston, J. D., 556 Stothers, Walter Wilson, 375 Truman, Harry S., 209 Wetzel, John E., 307 Strassen, Volker, 283 Tugemann, Bastian, 402 Wheeler, Jeffery Paul, 376 Stribling, Jeremy, 489 Tukey, John, 281 Whitehead, Alfred North, 86 Su, Francis, 558 Tunnell, Jerrold Bates, 454 Whitehead, Ian, 158, 294 Suh, Hong, xii Tur´an, P´al, 340 Wieman, Hunter, xii Summers, Alexander, xii Turing, Alan, 121, 157 Wien, Douglas, 386 Swi¸´ atek, Grzegorz, 362 Wiener, Norbert, 146, 148 Sylvester, James Joseph, 474 Udell, Gabe, xii Wigner, Eugene, 79, 82, 251 Sz´asz, Otto, 197 Ulam, Stanislaw, 101, 164, Wilcox, James, 364 Szekeres, George, 340 175, 546 Wiles, Andrew, 145, 208, Szemer´edi, Endre, 58, 340 234, 294, 457 Vall´ee-Poussin, Charles Jean Wiley, Chad, 397 de la, 187, 190 Tai, Mary M., 434 Wilf, Herbert, 276, 416 van der Waerden, Bartel Tait, Peter Guthrie, 346 Wilmshurst, A. S., 362 Leendert, 90 Takagi, Teiji, 360 Wilson, Edward Osborne, Vazsonyi, Andrew, 430 252 Tao, Terence, 4, 33, 58, 68, Velupillai, K. Vela, 251 511, 512, 559 Wilson, Kenneth, 76 Venkatesh, Akshay, 562 Wishart, John, 79 Tarjan, Robert, 71 Vigen`ere, Blaise de, 212 Witten, Edward, 400 Tarski, Alfred, 62 Vinogradov, Ivan Matveye- Wolfram, Stephen, 413 Tate, John, 490 vich, 127 Woltman, George, 464 Tausk, Daniel, 559 Vitali, Giuseppe, 65 Taussky-Todd, Olga, 407 von Dyck, Walther, 254, 509 Xylouris, Triantafyllos, 553 Taylor, Richard, 234, 457 von Mangoldt, Hans Carl Tener, James, xii, 397 Friedrich, 409 Tennenbaum, Stanley, 205 von Mises, Richard, 138 Yeates, Bree, 282 Thai, Minh, 333 von Neumann, John, 63, 67, Yıldırım, Cem, 528 Thielman, H. P., 275 96, 163, 179, 398 Youngs, John W. T., 347 Thomas, Robin, 526 von Staudt, Karl, 459 Yule, Udny, 142 Thompson, John, 63 Vongsathorn, Xan, 334 Thue, Axel, 228, 442, 475 Zaremba, Stanislaw, 323 Thurston, William, 241 Wada, Hideo, 386 Zeckendorf, Edouard, 312 Tijdeman, Robert, 442 Wallace, William, 369 Zeilberger, Doron, 403, 416 Titchmarsh, Edward Wallis, John, 539 Zemdegs, Feliks, 333 Charles, 142 Wantzel, Pierre, 424 Zhang, Scott Sicong, 334 Tits, Jacques, 514 Waring, Edward, 39 Zhang, Yitang, 33, 128, 528 Tonelli, Leonida, 556 Weil, Andr´e, 76, 170, 293 Ziegler, G¨unter M., 3, 189 T´oth, L´aszl´o Fejes, 475 Weinberg, Wilhelm, 142 Zong, Chuanming, 371 Travis, Jeffrey, 531 Weiss, Gary, 306 Zuboff, Arnold, 428

Index

3x + 1 conjecture, 101 , 117, 169, Ap´ery’s constant, 365 3x + 1 problem, 101, 563 295, 329, 332 Archimedean, 18 QR-decomposition, 473 algorithm, 335; 196-, 104; di- Aristotle, 370 γ, 172 vision, 461; Euclidean, arithmetic progression, 1, φ, 223, 313 207, 499; fast Fourier 228, 339; arbitrarily long, π, 227, 228 transform, 281; Gale– 2, 339; as a Diophantine π, 98, 113, 115, 134, 222, 230, Shapley, 264; greedy, 312, set, 386; forming a topol- 290, 294, 329, 332, 478 364; Horner’s method, ogy, 230; monochromatic, 2 284; integer factoriza- π√ , 189 90; of perfect , 3, 2, 205 tion (naive), 181, 499, 512; of primes, 2, 3, 58, abc-conjecture, 376, 378, 521, 500; Markov chain Monte 511 563 Carlo, 219; matrix mul- Arrow’s theorem, 199 e, 98, 115, 134, 227, 258, 324, tiplication, 282; Me- Artin–Whaples product for- 329, 331, 332, 460 tropolis, 219; Newton’s mula, 17 e + π, 118, 330 method, 259; PageRank, arXiv, 433 − eπ, 118, 330 465; Pollard’s p 1, 353; ASCII, 123 j-invariant, 343, 440 polynomial-time, 501; asymptotically equivalent, t-test, 538 Risch, 302, 414; RSA, 33 352; Rubik’s Cube, 334; 15-theorem, 446 Atiyah–Singer index theo- Shor’s, 352; simplex, 182, 196-algorithm, 104 rem, 299 185; Strassen, 283; to 290-theorem, 447 automorphic forms, 293 computesquareroots, , 62, 65, 86, 277; Wilf–Zeilberger, 416 108, 141, 269, 277, 278, Abel Prize, 76, 163, 299, 300, almost everywhere, 555 482, 557 340 alternating group, 439, 514 axiom of foundation, 85 Abel–Ruffini theorem, 329 alternating harmonic series, axiom of infinity, 85, 87 absolute differential calculus, 110 11 alternating series test, 110 axiom of pairing, 85 absolute value, 17; p-adic, amenable, 63 axiomatic set theory, 85 17; standard, 17; trivial, American Institute of Math- 17 ematics, 451 Bacon number, 306 absolutely convergent, 110, American Mathematical So- badness, 364 146, 148 ciety, 385 Baire category theorem, 481 abstract nonsense, 295 American Standard Code for Ackermann’s function, 443 Information Interchange, Baker–Heegner–Stark theo- additively large, 340 123 rem, 288, 534 Advanced Encryption Stan- analysis situs, 242 ballotino, 201 dard, 124 analytic continuation, 151, Banach algebra, 148 AES, 124 365, 407 Banach–Tarski paradox, 61, AIM, 451 analytic function, 151, 393 241, 277, 369, 381 AKS primality test, 501 analytic index, 302 base for a topology, 230 Alexander polynomial, 396 analytic rank, 47, 48 baseball, 317–319 Alexander’s theorem, 399 Ann Arbor Problem Book, Basel problem, 35, 140, 294 algebraic conjugate, 495 307 basic construction, 398 algebraic irrational, 114 AP-rich, 339 basic solution, 182

571 572 INDEX

Bateman–Horn conjecture, , 22, 270, 271, 482 Collatz sequence, 101 34, 57, 129, 464, 512, 521, Cantor surjection theorem, companion matrix, 296 528, 533, 546, 550, 563 24 complete graph, 442 Bateman–Horn constant, Cantor’s powerset theorem, completeness theorem, 86 522, 534 31 conditionally convergent, Battle of Midway, 160 cardinality, 27, 28, 269 110 bell curve, 52 Carmichael number, 501, Condorcet cycle, 200 Benford’s law, 54, 102, 131, 504, 549 Condorcet winner, 200 223 Catalan number, 253, 254, Condorcet winner criterion, Bernoulli numbers, 459 540 200 Bernoulli random variable, category theory, 481 congruence obstruction, 57 55, 179 Cauchy functional equation, congruent number, 452, 453 Bernstein polynomial, 196 278 congruent number problem, Bertrand’s postulate, 129 Cauchy product, 109 452 Beurling’s theorem, 195 Cauchy random variable, 53 conjecture; 3x + 1, 101; Bieberbach conjecture, 393 Cauchy–Riemann equations, abc-, 376, 378, 521, bijection, 28 8 563; Bateman–Horn, 34, billiards, 258 central limit theorem, 51, 52, 57, 129, 464, 512, 521, Binet’s formula, 313, 495, 54, 55, 79, 176, 179, 303, 528, 533, 546, 550, 563; 519 411, 537, 539 Bieberbach, 393; Birch binomial random variable, 55 chain of subsets, 31 and Swinnerton-Dyer, Birch and Swinnerton-Dyer chaos, 221 47, 454, 455; Conway– conjecture, 47, 454, 455, character, 148 Norton, 441; epsilon, 234; 491 characteristic function, 54, Erd˝os–Tur´an, 340, 343; Birkhoff ergodic theorem, 96 95 Erd˝os, 2, 343; Euler’s on birthday attack, 139 characteristic polynomial, sums of powers, 39; Fer- birthday paradox, 139 77, 247, 296, 496 mat’s, 422; Gauss’s class birthday problem, 138 Chebyshev’s bias, 415 number, 288; Goldbach, Black–Scholes model, 470 checksum, 124 57; Goldbach binary, blancmange function, 360 , 493 127; Goldbach ternary, Blaschke condition, 195, 197 Chinese remainder theorem, 127; Hardy–Littlewood Bletchley Park, 157, 210 238, 535 k-tuple, 57, 58, 128; Boneyard Book, 308 chromatic number, 525 Hardy–Littlewood (twin Borsuk–Ulam theorem, 165 ciphertext, 124 primes), 34; Heawood, Boston Red Sox, 317 circle method, 40, 57, 127, 346; Hilbert–P´olya, 407; braid group, 399 441 Kepler, 346, 476; Lan- Brouncker’s formula, 115 class field theory, 294 dau’s, 528; Mordell’s, 378; Brouwer’s fixed-point theo- class number, 287, 446, 459 Poincar´e, 433, 505, 507; rem, 164, 494, 543 class number one problem, Polignac’s, 33; P´olya’s, 91; , 470 287 Ramanujan, 294; Sato– Brun’s constant, 33, 37 classification of finite simple Tate, 294; Taniyama– Brunn–Minkowski theorem, groups, 300, 439, 513 Shimura, 48; Thwaites, 23 classification of surfaces, 506 101; twin prime, 33, 57, Buffon’s needle problem, 176 Clausen–von Staudt theo- 408, 433, 522; Ulam’s, Burali–Forti paradox, 88 rem, 459–461 101; Zaremba’s, 323, 326 busy beaver, 122 Clay Millennium Problems, connected sum, 508 busy beaver function, 122 xi, 47, 108, 153, 487, 505 consistent, 86, 270, 484 butterfly effect, 221, 257 clique number, 525 constant; Ap´ery’s, 365; closed graph theorem, 481 Bateman–Horn, 522, C∗-algebra, 302 closed set, 230 534; Brun’s, 33, 37; Caesar cipher, 124, 212 CoCalc, 521 Conway, 402; Euler’s, Calkin–Wilf sequence, 29 Cole Prize, 464 see also e;Euler– canonical linear program- collaboration graph, 305 Mascheroni, 134, 152, 172; ming problem, 183 Collatz function, 101 Gelfond–Schneider, 117; Cantor dust, 24 Collatz graph, 101 Khinchin’s, 113–115, 134; INDEX 573

Liouville’s, 119, 227, 329; Diophantine equation, 311, 378, 461; functional (zeta Meissel–Mertens, 189; 385, 386 function), 152; Orwell’s, Mills’s, 388; Planck’s, 67; Diophantine set, 386 391; Schr¨odinger, 67 Ramanujan’s, 342, 534; Dirac delta functional, 80 equidecomposable, 369 twin primes, 34 Dirichlet divisor problem, equidistributed modulo 1, constraint matrix, 181 172 96, 131, 133 constructible, 424 Dirichlet’s approximation equinumerous, 27, 270, 545 constructible , 423 theorem, 223, 227 equivalence relation, 62, 64 consumption matrix, 472 Dirichlet’s box principle, 223 Eratosthenes of Cyrene, 408 continued fraction, 73, 97, Dirichlet’s theorem on Erd˝os–Tur´an conjecture, 113, 323, 324, 326 primes in arithmetic pro- 340, 343 continuum hypothesis, 86, gressions, 3, 58, 291, 354, Erd˝os conjecture, 2, 343 269, 272, 277, 307 415, 522, 528, 552, 553 Erd˝os number, 1 contraction mapping princi- discrete dynamical system, Erd˝os–Bacon number, 306 ple, 165 95 ergodic hypothesis, 95 Conway’s constant, 402 discrete Fourier transform, ergodicity, 96 Conway–Norton conjecture, 281 error function, 470 441 discriminant, 287; of an ellip- Euclid’s theorem, 3, 87, 111, cookie problem, 313 tic curve, 45 189, 230, 295, 423, 513 Coq, 346 division algorithm, 461 Euclid–Mullin sequence, 513 cosmological theorem, 402 divisor function, 171 , 207, countable, 28 Doctor Who, 67, 85, 145, 499 Cram´er model, 410, 463 267, 393, 487 Euclidean , 272 critical line, 153, 388 dyadic filtration, 25 Euclidean norm, 193 critical strip, 152, 407, 409 Dyck path, 254 Euler characteristic, 347, 509 cryptography, 47, 124, 224, Dyck’s theorem, 509 Euler product formula, 110, 351 dynamical system, 221 140, 151, 188–190, 293, cubic close packing, 476 294, 407, 409 cycle, 386 Earth, 68, 439 Euler totient function, 416 cyclic group, 514 Egorov’s theorem, 555 Euler’s constant, see also e cyclotomic field, 459 eigenvalue trace lemma, 80 Euler’s formula, 8, 35, 40, 118 , 236 election procedure, 199 Electronic Frontier Founda- Euler’s power tower, 72 tion, 464 Euler–Lucas theorem, 422 Data Encryption Standard, elementary function, 302 Euler–Mascheroni constant, 124 ellipse, 146 134, 152 de Morgan’s law, 231 elliptic curve, 513; analytic existential proof, 117 degree, 31 rank, 47; and congruent expected value, 51 Dehn invariant, 369 numbers, 454; definition, extreme point, 194 Delbert Ray Fulkerson Prize, 45; discriminant, 45; Frey, 526 234; group operation, 46; Facebook, 1, 305 density, 339 Hasse–Weil L-function of, factor, 398 derangement, 158 48; largest known rank, fast Fourier transform, 281, DES, 124 47; modular, 234; rank, 453 diagonal argument, 29 47; rational point, 46, 454; feasible, 182 diagonal matrix, 447 torsion subgroup, 47 Fenway Park, 57 diet problem, 182, 184 empirical spectral , Fermat equation, 311, 378, differential equation, 8, 166, 80 461 221, 257, 299 energy-momentum invariant, , 421–423 digit expansion; base B, 113; 11, 12 Fermat prime, 421, 422, 424 binary, 113; decimal, 113 Enigma machine, 122, 157 Fermat’s conjecture, 422 , 439 epsilon conjecture, 234 Fermat’s last theorem, 39, dimension theorem, 196 equation; Black–Scholes, 145, 169, 208, 234, 294, Diophantine approximation, 470; Diophantine, 311, 311, 375, 376, 378, 452, 62, 207, 222, 229, 391 385, 386; Fermat, 311, 457, 460, 476 574 INDEX

Fermat’s little theorem, 351, 539; generating, 448, 497; Gelfond–Schneider constant, 354, 375, 499, 501 inner, 195; iterated expo- 117 Fermat’s nential, 72; Koebe, 393; Gelfond–Schneider theorem, theorem, 448 L-, 3, 48; logarithmic in- 117, 119 fetid dingo’s kidneys, 27, tegral, 107, 153, 409, 411, general comprehension prin- 466, 489 547; matrix exponential, ciple, 85 FFT, 453 69; meromorphic, 233; general theory of relativity, Fibonacci number, 131, 235, modular, 440; multiplica- 11 312, 313, 373, 495, 497, tive, 171; periodic, 281; generating function, 59, 448, 519 prime-counting, 107, 108, 497 129, 151, 516; ramp, 555; Fields Medal, 76, 90, 145, geodesic, 12, 21 170, 228, 269, 287, 288, rational, 233, 377; Rie- , 114 294, 300, 378, 397, 433, mann zeta, 3, 48, 79, 81, geometric progression, 341 441, 446, 491, 505, 507, 110, 139, 151, 169–171, geometric series, 19, 36, 59, 562 187, 189, 190, 363, 364, 78, 151 finite, 28 388, 407, 409, 410, 459; finite extension, 169 sawtooth, 360; - G´eom´etrie alg´ebrique et g´eom´etrie analytique, 233 first category, 481 integrable, 97; square- first incompleteness theorem, wave, 147; sum of divi- GIMPS, 464 86, 88 sors, 172; Takagi, 360; global function field, 169 fixed point, 164, 357, 543 transcendental, 233; von Global Median Matching, Flint Hills series, 229 Mangoldt, 409 265 forcing, 269 function field, 169; global, Global Positioning System, formula; Binet’s, 495, 519; 169; Riemann hypothesis, 14 Simson’s, 496; Stirling’s, 170; zeta function, 169 Goldbach conjecture, 57, 127 539; Wallis’s, 539 functional equation, 152; , 223, 313, 495 forward orbit, 103 Cauchy, 278 Golden State Warriors, 317 fundamental group, 347 four color theorem, 345, 346, Golomb graph, 527 fundamental lemma, 294 476 Goodstein sequence, 86 fundamental polygon, 506 four fours puzzle, 392 Goodstein’s theorem, 87 fundamental theorem of alge- four-square identity, 39, 41 GP-rich, 341 Fourier coefficients, 146 bra, 118, 358, 484 GPS, 14 Fourier matrix, 284 fundamental theorem of Graham’s number, 439, 442 Fourier series, 146, 221, 301 arithmetic, 29, 37 Gram–Schmidt process, 249 , 18, 22, 260, 357 graph, 253, 386; collabora- , 270, 271 GAGA principle, 233 tion, 305; complete, 442; Franklin graph, 347, 349 Gale–Shapley algorithm, 264 Franklin, 347, 349; friend- frequency analysis, 125 Galileo’s paradox, 27 ship, 306; spanning tree in frequency-wave number in- Galois representation, 293 a, 484 variant, 11 Galois theory, 329 gravitational lensing, 14, 362 Fresnel integral, 303 , 163, 543 Frey curve, 234 gamma function, 152, 479, gravitational waves, 14 friendship graph, 306 537, 539 Great Internet Mersenne Fubini–Tonelli theorem, 472 Gauss map, 98 Prime Search, 464 Fulkerson Prize, 526 Gauss measure, 98 greatest common divisor, function; Ackermann’s, 443; , 284 207, 235 analytic, 393; blanc- Gauss’s class number conjec- greedy algorithm, 312 mange, 360; bump, 556; ture, 288 Green–Tao theorem, 2, 58, busy beaver, 122; charac- Gauss’s lemma, 495 340, 511, 512, 521, 522 teristic, 95; Collatz, 101; Gauss–Kuzmin theorem, 97 group; alternating, 439; clas- continuous and nowhere- Gauss–Wantzel theorem, 424 sification of finite simple differentiable, 360; divi- Gaussian, 52, 54, 55 groups, 439; cyclic, 514; sor, 171; elementary, 302; , 511 dihedral, 439; fundamen- Euler totient, 351, 416; Gaussian prime, 511 tal, 347; monster, 439; gamma, 152, 479, 537, GCHQ, 351 of Lie type, 514; pariah, INDEX 575

514; quasithin, 514; Ru- Householder matrix, 248 Kakutani’s problem, 101 bik’s Cube, 333; simple, Householder reflections, 247 KAM theory, 221 439, 513; sporadic, 514 hyperbolic geometry, 272, Kasiski method, 212 Grundgesetze der Arith- 273 Kepler conjecture, 346, 476 metik, 85 hypercube, 531 Khinchin’s constant, 114, 115, 134 Hadamard conjecture, 250 impossibility theorem, 199 Kirby–Paris theorem, 87 Hadamard matrix, 249 incompleteness theorem; Klein bottle, 243, 346, 505, Hadamard’s inequality, 249 first, 86; second, 86 508 Hadwiger–Nelson problem, indicator function, 95 knot; Stevedore, 398; trefoil, 527 inertial frames, 11 395, 505; unknot, 395 Hahn–Banach theorem, 197 infinite, 28 knot polynomial, 396 halting problem, 122 infinitesimal generator, 69 Kolmogorov–Arnold–Moser Hamiltonian, 67, 69, 79, 221 infinity, 269 theorem, 221 Hamiltonian cycle, 386 injective, 27 Korselt’s criterion, 549 Hardy space, 195, 301 inner function, 195 Kronecker product, 296 Hardy–Littlewood k-tuple Institute for Advanced Kronecker’s approximation conjecture, 57, 58, 128 Study, 408 theorem, 134 Hardy–Littlewood conjec- Intel, 33 Kronecker–Weyl theorem, ture (twin primes), 34 interlace, 67 133 Hardy–Weinberg law, 142 intermediate value theorem, Kummer’s congruence, 460, harmonic number, 108, 553 77, 78, 164, 461 461 harmonic series, 35, 40, 110, Internal Revenue Service, 54 301 International Congress of L’Hˆopital’s rule, 555 Hasse–Minkowski local- Mathematicians, 117, 269, L-function, 3, 48, 293; global principle, 20 311, 369, 515, 564 Hasse–Weil, 48; symmet- Hasse–Weil L-function, 48 International Mathematics ric power, 294 Hausdorff maximality princi- Competition for Univer- Lagrange’s four-square theo- ple, 484 sity Students, 76 rem, 445, 448 Hausdorff topology, 231 invariance of domain, 545 Landau’s conjecture, 528 Hawkins prime, 408 invariant, 371, 395 Langlands program, 293, 563 Heawood conjecture, 346 invariant set, 96 Laplace’s method, 539 Heegner number, 343 invariant subspace, 194 large cardinal, 397 , 423 invisible forest, 237 largest known prime, 464 hereditary base-b representa- irrational, 295 Laser Interferometer tion, 86 irrational rotation, 95 Gravitational-Wave Ob- heuristic reasoning, 103, 421, irrationality measure, 227, servatory, 14 422, 424, 463, 553 295 √ LATEX, 363, 413 hexagonal close packing, 476 irrationality of 2, 205 lattice point, 237 hexagonal lattice packing, irrationality type, 133, 222 Laurelin the Golden, 254 475 irreducible representation, Laurent polynomial, 396 Hilbert space, 68, 79, 96, 398, 441 law of complementary prob- 407 IRS, 54 ability, 138 Hilbert’s problems, xi, 117, isologous, 211 , 96 153, 269, 288, 311, 369, iterated exponential func- , 24, 555 385, 451 tion, 72 Legendre symbol, 534 Hilbert–P´olya conjecture, iterated towers, 109 lemma; eigenvalue trace, 80; 407 Gauss’s, 495; Sperner’s, Hodge conjecture, 491 Jacobi symbol, 503 494, 544–546; Zorn’s, 483 homeomorphism, 23, 347, Jacobi’s four-square theo- Leroy P. Steele Prize, 275 544, 545 rem, 445 liar’s paradox, 85, 86, 381 HOMFLY polynomial, 397 Jarn´ık competition, 76 LIGO, 14 Honors Class, 117 Jones polynomial, 395–397, , 181, 403 Horn conjecture, 68 399 Liouville lambda function, 91 Horner’s method, 284, 285 Jones tower, 398 Liouville number, 118 576 INDEX

Liouville’s constant, 119, 62, 69, 248, 249; permu- M¨untz–Sz´asz theorem, 197 227, 329 tation, 439; positive semi- Liouville’s theorem, 118, 119, definite, 67, 250; real or- 329 thogonal, 69; real sym- naive measure theory, 95 Littlewood’s principles, 555 metric, 80, 82; selfadjoint, naive set theory, 85 Local Median Matching, 265 67 , 163, 164, local-global principle, 20 mean, 51 543 mean value theorem for inte- logarithmic derivative, 188 National Institute of Stan- grals, 35 logarithmic integral, 107, dards and Technology, 124 measure zero, 22, 96, 97, 118 153, 409, 411, 522, 547 National Medal of Science, Meissel–Mertens constant, look and say sequence, 402 363 189 Lorentz transformation, 11 National Museum of Mathe- Mercury, 14 Lucas–Lehmer primality matics, 561 Mersenne number, 463, 503 test, 464, 465 National Resident Match Mersenne prime, 463, 473 Lusin’s theorem, 555 Program, 263 Mertens’s theorem, 37, 109, Lyapunov central limit theo- National Science Founda- 110 rem, 411 tion, 451 Mertens’s theorem (prime re- Lychrel number, 104 ciprocals), 189 National Security Agency, method of stationary phase, 209 539 natural density, 339 Maass form, 294 method of undetermined co- natural number, 1 MacArthur Fellow, 76 efficients, 8 Navier–Stokes Equation, 491 Magic Cube, 333 metric space, 22, 481 negative curvature, 21, 23 major arc, 41 Metropolis algorithm, 219 Neptune, 221 Major League Baseball, 311, middle square digits method, Newcomb’s paradox, 383 317, 319 179 Newton fractal, 260, 261 Mandelbrot set, 357 Millennium Prize Problems, Newton’s method, 259, 277 manifold, 241 454 Newton’s second law, 67 Maple, 519 Millennium Problems, 47 Nielsen–Schreier theorem, MapQuest, 57 Miller–Rabin test, 501 484 Markov chain, 219, 220, 318, Mills’s constant, 388 NIST, 124 544 minor arc, 41 Nobel Prize, 76, 79, 163, 199, Markov chain Monte Carlo M¨obius strip, 241, 243, 506 251, 263, 435, 469, 472, algorithms, 219 modular, 440 543, 562 Markov’s theorem, 399 MoMath, 561 non-Euclidean geometry, 273 Mars, 68, 75 moment, 51 nonorientable, 241, 243 Mason–Stothers theorem, Moneyball, 317 nontrivial zeros of the zeta 169, 375–377 monoid, 508 function, 153 Matching; Global Median, monotone sequence property, norm; Euclidean, 193; on a 265; Local Median, 265 37 vector space, 193 Mathematica, 140, 413, 416, monovariant, 371 normal distribution, 51 429, 495, 519 monster group, 439, 442, 515 normal random variable, 470 mathematical induction, 8 monstrous moonshine, 440 normal subgroup, 439 MathOverflow, 399 Monte Carlo method, 175, normal topology, 231 matrix; characteristic poly- 318, 323 nomial, 296; companion, Monty Hall problem, 427, Norwegian Academy of Sci- 296; constraint, 181; con- 428 ence and Letters, 299 sumption, 472; diago- moonshine module, 441 nowhere dense, 22, 481 nal, 447; exponential, 69; Moore–Kline theorem, 25 NP-complete problem, 385, Fourier, 283; Hadamard, Mordell’s conjecture, 378 402 249; Householder, 248; in- Morse theory, 21 NRMP, 263 tegral, 446; left and right Moser spindle, 527 NSA, 209 inverses, 193, 195; multi- Moser’s circle problem, 91 NSA Cryptomathematics In- plication, 282; orthogonal, multiplicative function, 171 stitute, 210 INDEX 577 number; algebraic, 30, 114, orthogonal matrix, 62, 248, Poincar´e disk model, 273 117, 295, 329, 332; alge- 249 point-set topology, 230 braic integer, 343; alge- Ostrowski Prize, 394 Poisson random variable, 539 braic irrational, 133; Ba- Ostrowski’s theorem, 17 pole, 188 con, 306; Bernoulli, 459; Polignac’s conjecture, 33 Carmichael, 501, 504, 549; PversusNPproblem,183, Polish Cipher Bureau, 157 Catalan, 253, 254, 540; 490, 491 Pollard’s p−1 algorithm, 353 class, 446; congruent, 452, p-adic absolute value, 17 P´olya’s conjecture, 91 453; Erd˝os, 305; Erd˝os– p-adic number, 17, 18 polygonal number, 448 Bacon, 306; Fermat, 189, PA, 87 polyhedra, 369 421–423; Fibonacci, 131, packing density, 475 Polymath8 project, 33 235, 312, 313, 373, 495, PageRank algorithm, 465 polynomial; Alexander, 396; 497, 519; Gaussian inte- pair correlation problem, 81 Bernstein, 196; character- ger, 511; Graham’s, 439, pair of pants, 21 istic, 77, 247, 296, 496; 442; harmonic, 553; Heeg- palindrome, 104 cyclotomic, 236; Euler’s, ner, 343; irrational, 95, pan galactic gargle blaster, 91; Fermat’s last theorem, 114, 117, 205, 222, 227, 31 376; fixed points, 358; 258, 295; irrational of type paradox; Banach–Tarski, 61, generating fractal, 261; (K, ν), 222; Liouville, 118; 241, 369, 381; birth- harmonic, 361; HOMFLY, Lychrel, 104; Mersenne, day, 139; Burali–Forti, 88; 397; indecomposable, 300; 189, 463, 503; natural, Galileo’s, 27; liar’s, 85, Jones, 395–397, 399; knot, 1; ordinal, 87, 88; p- 86, 381; Newcomb’s, 383; 396; Laurent, 396; prime- adic, 17, 18; perfect, 473; nonexistence of length, 63; generating, 91, 388; roots, polygonal, 448; prime, 1, Russell’s, 85, 86, 381; 358 57, 289, 382, 409, 414; Smale’s, 241 polynomial-time algorithm, quadratic irrational, 114; paradoxical decomposition, 501 Ramsey, 90; rational, 17, 62 polytope, 185 117, 222, 258; RSA chal- parallel postulate, 273, 369 poset, 483 lenge, 353; Skewes, 107, Pareto condition, 199 positive semidefinite, 250 108, 443, 444; square-free, pariah group, 514 positive semidefinite matrix, 339, 342, 445, 549; taxi- partial order, 483 250 cab, 551; transcendental, partition function, 40, 59 possibility theorem, 199 31, 98, 114, 118, 169, 227– Peano arithmetic, 87 power index, 201 229, 329; triangular, 447; Peano curves, 25 power tower, 72, 109 van der Waerden, 90 Penrose–Banzhaf power in- powerset, 25, 31 number field, 18, 169 dex, 201 matrix, 263 number transcendental, 332 diagram, 295 primality test; Lucas– numbers; algebraic, 169 Pentium processor, 33 Lehmer, 464 perfect, 525 primality testing, 47 perfect graph theorem, 525 prime; Fermat, 421, 424; off-by-one error, 363 perfect number, 473 Gaussian, 511; Hawkins, one-to-one, 27 periodic function, 281 408; largest known, 464; onto, 27 permutation, 157, 158 Mersenne, 463, 473; regu- open mapping theorem, 481 permutation matrix, 439 lar, 459; Sophie Germain, open sector, 472 perturbation theory, 221 460 open set, 230 Picard iteration, 166 prime number, 1, 4, 57, 382, Operation Fortitude, 160 pigeonhole principle, 223, 407, 409, 414; Cram´er operator; selfadjoint, 69; 338, 346 model, 410; Fermat, 422; skew-symmetric, 69; uni- plaintext, 124 of the form x2 + dy2, 289; tary, 69 Planck’s constant, 67 twin, 416 operator theory, 193 Platonic solid, 347, 348 prime number theorem, 34, orbit, 101 Playfair’s axiom, 273 58, 107, 129, 153, 172, order; partial, 483; total, 31, Pluto, 75 181, 187, 189, 303, 339, 483; well, 483 Poincar´e conjecture, 433, 366, 410, 424, 465, 515, ordinal number, 87, 88 491, 505, 507 522, 528, 554 578 INDEX prime-counting function, quadratic form, 20, 287, 446; relatively prime, 3 107, 108, 129, 151, 516 class number, 287; dis- relativity, 11 primitive recursive, 71 criminant, 287; equiva- residue, 188 primitive root, 416 lent, 287; Ramanujan’s Riemann hypothesis, 108– principle; contraction map- ternary, 448; universal, 110, 153, 154, 169, 190, ping, 165; Dirichlet’s box, 446 388, 407, 409, 433, 448, 223; GAGA, 233; general quadratic Gauss sum, 284 490, 501, 503, 521, 552, comprehension, 85; Haus- quadratic irrational, 114 553, 562 dorff maximality, 484; Lit- , 291 Riemann hypothesis for tlewood, 555; local-global, quadrivium, 561 function fields, 170 20; of equivalence, 12, quantum mechanics, 67, 68, Riemann sphere, 188, 377 13; of inclusion-exclusion, 79 Riemann zeta function, 3, 48, 158; pigeonhole, 223, 338, quaternion, 42 79, 81, 110, 139, 151, 169– 346; uniform bounded- 171, 187, 189, 190, 293, ness, 481; well-ordering, R, 487 363, 364, 388, 407, 409, 206, 483 radius of convergence, 152 410, 459 principle of inclusion- Ramanujan conjecture, 294 Riemann–Hurwitz formula, exclusion, 158 Ramanujan sum, 173 377 probability vector, 215 Ramanujan’s constant, 342, Riemann–Roch theorem, 46, problem; 3x + 1, 101; Basel, 534 170 140; birthday, 138; Buf- Ramsey number, 89, 90 Riemannian, 12 fon’s needle, 176; canon- Ramsey theory, 89, 341, 442, Riesz projection, 301 ical linear programming, 444 Riesz representation theo- 183; class number one, random matrix theory, 79, rem, 197 287; congruent number, 408 Risch algorithm, 302, 414 452; cookie, 313; diet, random number generator, Rogers–Ramanujan contin- 182; Dirichlet divisor, 172; 131, 179 ued fraction, 115 Hadwiger–Nelson, 527; random variable; Bernoulli, , 236 halting, 122; Kakutani’s, 55, 179; binomial, 55; Ross Mathematics Program, 101; linear programming, Cauchy, 53; characteris- 235 181; Monty Hall, 427, 428; tic function of, 54; con- Roth’s theorem, 133, 223, NP-complete, 385, 402; of tinuous, 51; convergence 228 small denominators, 221; in probability, 96; density, Roth’s theorem (arithmetic P versus NP, 183, 491; 51; mean of, 51; moments progressions), 228, 340 sleeping beauty, 428; stars of, 51, 80; normal, 470; RSA, 47, 181, 351, 353 and bars, 313; Syracuse, normally distributed, 52; RSA challenge number, 353 101; traveling salesman, Poisson, 539; standard de- Rubik’s Cube, 333; solution, 183, 385, 386; two en- viation of, 51; standard- 334 velopes, 382; Wetzel’s, ized, 52; supported on Rubik’s Cube group, 333 307 primes, 424; uniform, 52; Russell’s paradox, 85, 86, 381 Project Euler, 493 variance of, 51; Weibull, Project Flyspeck, 476 319 sabermetrics, 317 projective plane, 508 random variables; inde- pendent, identically dis- Sage, 452, 453, 519 PROMYS, 421 tributed, 80 SageMath, 519 proof by picture, 206 rank of an elliptic curve, 47 Sato–Tate conjecture, 294 Proofs from The Book, 189 rank-nullity theorem, 299 schlicht, 393 pseudoprime, 500, 549 rational point, 46 Schr¨odinger equation, 67 Putnam Competition, 75 reciprocal sum, 2 Schr¨odinger operator, 407 Putnam Fellow, 75 recreational mathematics, 7 Schwarzschild line element, Pythagorean theorem, 311, reflexive, 28 13 314, 319, 320 refractive index; of a mate- Schweitzer competition, 76 Pythagorean triple, 453 rial, 13; of space-time, 13 Scientific American, 7 Pythagorean won-loss for- , 459 SCIgen, 489 mula, 319 regular topology, 231 second category, 481 INDEX 579 second incompleteness theo- Stevedore knot, 398 The Book, 3 rem, 86, 484 Stigler’s law of eponymy, see and say sequence, 402 495, 552, 556 self-similar, 271 Stirling’s formula, 115, 479, theorem; 15-, 446; 290-, selfadjoint, 67 539 447; Abel–Ruffini, 329; semicircle law, 80 Stone’s theorem, 68 Alexander’s, 399; Ar- sensitive dependence on ini- Strassen algorithm, 283 row’s impossibility, 199; tial conditions, 258 stress-energy tensor, 12 Atiyah–Singer index, 299; sequence; Euclid–Mullin, 513 strong perfect graph theo- Baker–Heegner–Stark, series; finite geometric, 422; rem, 526 288, 534; Beurling’s, Flint Hills, 229; geomet- strong inequality, 18 195; Birkhoff ergodic, ric, 151; Leibniz, 147; Student’s t-distribution, 537 96; Borsuk–Ulam, 165; power series, 324; radius stylo, 488 Brouwer’s fixed-point, of convergence of a power, subfactor, 398 164, 494, 543; Brunn– 152 Sudoku, 401, 403 Minkowski, 23; Cantor Severini–Egorov theorem, sum; of four squares, 445; of surjection, 24; Cantor’s 556 powers of the first n pos- powerset, 31; central Shor’s algorithm, 352 itive integers, 8; of recip- limit, 51, 52, 55, 79, 176, Sierpi´nski triangle, 18, 271 rocals of numbers without 179, 303, 411, 537, 539; sieve of Eratosthenes, 408 a 9 in their decimal rep- Chinese remainder, 238, sigma function, 172 resentation, 35; of recip- 535; Clausen–von Staudt, significand, 54 rocals of perfect squares, 459–461; closed graph, simple, 188 33; of reciprocals of prime 481; cosmological, 402; simple group, 439 powers, 190; of reciprocals dimension, 196; Dirich- simplex algorithm, 182, 185 of primes, 4, 189, 197; of let’s approximation, 223; simplex method, 264 reciprocals of twin primes, Dirichlet’s on primes in 33, 59; of three squares, simply connected, 506 arithmetic progressions, 445, 448; Ramanujan, 173 Simpson’s formula, 496 3, 58, 291, 354, 415, 522, sum of divisors function, 172 Six Degrees of Kevin Bacon, 528, 552, 553; Egorov’s, surface, 505 1 555; Euclid’s, 3, 87, 111, surjective, 27 Skewes’s number, 107, 108, 189, 230, 423, 513; Euler– symmetric, 28 443, 444 Lucas, 422; Fermat’s last, symmetric group, 157 sleeping beauty problem, 428 39, 145, 169, 208, 234, syndetic set, 342 Smale’s paradox, 241 311, 375, 376, 378, 452, Syracuse problem, 101 Society for American Base- 457, 460, 476; Fermat’s ball Research, 317 Szemer´edi’s theorem, 90, little, 351, 354, 375, 499, Space Invaders, 359 228, 340, 341, 511 501, 549; Fermat’s polyg- span, 196 onal number, 448; four special theory of relativity, Takagi function, 360 color, 345, 346, 476; four- 11 Taniyama–Shimura conjec- square (Jacobi), 445; four- spectral theorem, 67 ture, 48 square (Lagrange), 445, speed of light, 11, 13 taxicab Carmichael number, 448; Fubini–Tonelli, 472; Sperner’s lemma, 494, 544– 551 fundamental theorem of 546 Taylor approximation, 139 algebra, 118, 358, 484; sphere eversion, 241 Telperion the Silver, 254 fundamental theorem of sphere packing, 475 Temperly–Lieb(–Jones) rela- arithmetic, 29, 37; Gauss– sporadic group, 514 tions, 399 Kuzmin, 97; Gauss– stable marriages, 264 tensor, 11 Wantzel, 424; Gelfond– stable matching, 263, 264 tensor analysis, 11 Schneider, 117, 119; standard deviation, 51 tensor product, 295 Goodstein’s, 87; Green– standardize, 52 term-by-term multiplication, Tao, 2, 58, 340, 511, 512, Star Trek, 85, 145, 391 109, 171 521, 522; Hahn–Banach, stars and bars problem, 313 ternary Goldbach conjecture, 197; impossibility, 199; Steele Prize, 394 57 intermediate value, 77, steradians, 371 TEX, 363 78, 164, 461; invariance 580 INDEX

of domain, 545; Kirby– Toeplitz operator, 301 van der Waerden’s theorem, Paris, 87; Kolmogorov– topological space, 230, 293, 341 Arnold–Moser, 221; Kro- 481 variance, 51 necker’s approximation, topology, 230, 242; base for Venice, 201 134; Kronecker–Weyl, a, 230; definition, 230; Venus, 68 133; Liouville’s, 118, Hausdorff, 231; noncom- Vigen`ere cipher, 212, 224 119, 329; Lusin’s, 555; mutative, 302; normal, Vitali set, 65, 277 Markov’s, 399; Mason– 231; regular, 231 Volterra integration opera- Stothers, 169, 375–377; torsion subgroup, 47 tor, 195, 197 mean value theorem for torus, 243, 346, 505, 508 von Mangoldt function, 409 integrals, 35; Mertens’s, total order, 31, 483 von Neumann algebra, 163, 37, 109, 110; Mertens’s totient function, 416 396, 397 (prime reciprocals), 189; transcendence degree, 169 Moore–Kline, 25; M¨untz– transcendental, 169, 295, 329 Wallace–Bolyai–Gerwien Sz´asz, 197; Nielsen– transcendental number, 31, theorem, 369 Schreier, 484; open map- 98, 227, 229, 329, 332 Wallis’s formula, 539 ping, 481; Ostrowski’s, 17; transition matrix, 215 Waring’s problem, 39 perfect graph, 525; prime transitive, 28 wave function, 67 number, 34, 58, 107, 129, traveling salesman problem, weak-field approximation, 13 153, 172, 181, 187, 189, 183, 385, 386 Weierstrass M-test, 360 303, 339, 366, 410, 424, tree, 253 Weierstrass approximation 465, 515, 522, 528, 554; trefoil knot, 395, 505 theorem, 193, 196 Pythagorean, 311, 314, triangular number, 447 well-ordered, 278 319, 320; rank-nullity, trivial zeros of the zeta func- well-ordering principle, 206, 299; Riemann–Roch, 170; tion, 152 483 Riesz representation, 197; Tunnell’s theorem, 454 Wetzel’s problem, 307 Roth’s, 133, 227; Roth’s TUNNY, 210 Weyl’s uniform distribution (arithmetic progressions), Turing machine, 121 property, 96 228, 340; second incom- twin prime conjecture, 33, Whitney–Graustein theo- pleteness, 484; Severini– 57, 408, 433, 522, 528 rem, 242 Egorov, 556; Stone’s, 68; twin primes constant, 34, 460 Wiener algebra, 148 strong perfect graph, 526; Twitter, 305 Wiener process, 470 Szemer´edi’s, 90, 228, 340, two envelopes problem, 382 Wiener’s 1/f theorem, 148 341, 511; Thue’s on num- Wigner’s semicircle law, 80, bers with fixed prime fac- Ulam spiral, 522 253 tors, 442; Thue–Siegel– Ulam’s conjecture, 101 Wilf–Zeilberger algorithm, Roth, 228; Toeplitz in- Ultra, 157 416 dex, 301; Tunnell’s, 454; ultrametric, 18 William Lowell Putnam van der Waerden’s, 341; undecidable, 109, 307 Mathematical Competi- Wallace–Bolyai–Gerwien, uniform boundedness princi- tion, 75 369; Weierstrass ap- ple, 481 winding number, 302 proximation, 193, 196; uniformly strict contraction, winning coalition, 201 Whitney–Graustein, 242; 165 Wolfram Alpha, 302, 413, Wiener’s 1/f, 148; Zeck- unique factorization domain, 414 endorf’s, 312, 313, 373 288 Wolfram Mathematica, see Thompson group, 63 universal, 446 also Mathematica Thue’s theorem on numbers universal machine, 121 with fixed prime factors, universal quadratic form, 446 Zaremba’s conjecture, 323, 442 up-arrow, 443 326 Thue–Siegel–Roth theorem, upper density, 340 Zeckendorf decomposition, 228 upper multiplicative density, 312, 373 Thurston’s corrugations, 241 341 Zeckendorf’s theorem, 312, Thwaites conjecture, 101 Uranus, 221 313, 373 time average, 95 Zermelo–Fraenkel axioms, Toeplitz index theorem, 301 van der Waerden number, 90 85, 141, 269 INDEX 581

Zermelo–Fraenkel set theory, ZF, see also Zermelo– zombie infestation, 371 85, 86, 108, 278, 484 Fraenkel set theory Zorn’s lemma, 483 zeta function, see also Rie- ZFC, see also Zermelo– mann zeta function Fraenkel set theory This book is an outgrowth of a collection of 100 problems chosen to celebrate the 100th anniversary of the undergraduate math honor society Pi Mu Epsilon. Each chapter describes a problem or event, the progress made, and connections to entries from other years or other parts of mathematics. In places, some knowledge of analysis or algebra, number theory or probability will be helpful. Put together, these problems will be appealing and accessible to energetic and enthusiastic math majors and aficionados of all stripes.

Stephan Ramon Garcia is WM Keck Distinguished Service Professor and professor of mathematics at Pomona College. He is the author of four books and over eighty research articles in operator theory, complex analysis, matrix analysis, number theory, discrete geometry, and other fields. He has coauthored dozens of articles with students, including one that appeared in The Best Writing on Mathematics: 2015. He is on the editorial boards of Notices of the AMS, Proceedings of the AMS, American Mathematical Monthly, Involve, and Annals of Functional Analysis. He received four NSF research grants as principal investigator and five teaching awards from three different institutions. He is a fellow of the American Mathematical Society and was the inaugural recipient of the Society’s Dolciani Prize for Excellence in Research.

Steven J. Miller is professor of mathematics at Williams College and a visiting assistant professor at Carnegie Mellon University. He has published five books and over one hundred research papers, most with students, in accounting, computer science, economics, geophysics, marketing, mathematics, operations research, physics, sabermetrics, and statistics. He has served on numerous editorial boards, including the Journal of Number Theory, Notices of the AMS, and the Pi Mu Epsilon Journal. He is active in enrichment and supple- mental curricular initiatives for elementary and secondary Photo courtesy of Cesar Silva. mathematics, from the Teachers as Scholars Program and VCTAL (Value of Computational Thinking Across Grade Levels), to numerous math camps (the Eureka Program, HCSSiM, the Mathematics League International Summer Program, PROMYS, and the Ross Program). He is a fellow of the American Mathematical Society, an at-large senator for Phi Beta Kappa, and a member of the Mount Greylock Regional School Committee, where he sees firsthand the challenges of applying mathematics.

For additional information and updates on this book, visit www.ams.org/bookpages/mbk-121

MBK/121