comm-fermat.qxp 9/15/97 2:04 PM Page 1304 Book Review Fermat’s Enigma Reviewed by Allyn Jackson about Fermat’s Last Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem Theorem directed Simon Singh by Singh. (Those in- Walker and Company, New York terested in seeing $22.00 hardcover the program 288 pages should consult their local public Despite the increased interest in Fermat’s Last television stations Theorem since Andrew Wiles announced his proof for broadcast times in 1993, there have been few popular books on the or check the Public subject. In the months immediately following his Broadcasting Sys- announcement, one book capitalized on the mo- tem Web site, ment: The World’s Most Famous Math Problem, by http://www. newspaper columnist Marilyn vos Savant. That pbs.org/. The pro- book suffered from many problems, the worst gram was reviewed being a woefully wrong-headed attempt to dis- in the Notices by credit Wiles’s proof. The only other popular book Andrew Granville, to appear in the U.S. was fortunately much more January 1997, serious. Fermat’s Last Theorem: Unlocking the Secret pages 26–28.) of an Ancient Mathematical Problem by Amir D. Wiles’s proof, which makes use of some of the Aczel, an associate professor of statistics at Bent- deepest and most technically difficult mathemat- ley College, was published in 1996 by Four Walls ics of the twentieth century, presents a formida- Eight Windows. The book received favorable re- ble challenge to any nonexpert who would write views in the popular press (for example, see the about it. Singh, who has a Ph.D. in particle physics New York Times review, available at http://www. from Cambridge University, has done an admirable nytimes.com/books/home/) and was for a short job with an extremely difficult subject. He has also time distributed by the AMS. However, complaints done mathematics a great service by conveying about some mathematical inaccuracies in the book the passion and drama that have carried Fermat’s led the Society to stop selling it. (A review of this Last Theorem aloft as the most celebrated math- book will appear in a future issue of the Notices.) ematics problem of the last four centuries. The Another popular book about Fermat will ap- book landed in the #1 spot on the bestseller list pear in bookstores this month. On October 28 of the The Times of London, proving that “useless” Walker Books will publish Fermat’s Enigma: The mathematics can have a primal fascination for Epic Quest to Solve the World’s Greatest Math- people. ematical Problem by Simon Singh. The publication The book begins with a brief look at that his- is timed to coincide with the broadcast in the toric day, June 23, 1993, when Wiles delivered the United States of The Proof, a BBC documentary last of his three lectures about the proof at the Isaac 1304 NOTICES OF THE AMS VOLUME 44, NUMBER 10 comm-fermat.qxp 9/15/97 2:04 PM Page 1305 Newton Institute in Cambridge, England. He con- equation mod p, for every prime p), though I think cluded by writing Fermat’s famous statement on it is not especially helpful that he decides to call the blackboard and saying, “I think I’ll stop here”; it the E-series instead. (One needs to keep in mind those words provide the title for the first chapter. that the series discussed in the book are power se- Singh then largely leaves Wiles behind and goes into ries, although the book treats them essentially as five chapters’ worth of history about Fermat’s Last numerical sequences consisting of the coefficients Theorem. His discussion of the life and work of of the power series.) He has found a wonderful way Pythagoras makes for absorbing reading and of- to communicate the importance of this series: “In fers the opportunity to fix in the reader’s mind the the same way that biological DNA carries all the idea of proof, using simple examples such as the information required to construct a living organ- Pythagorean theorem. Singh has a knack for fer- ism, the E-series carries the essence of the ellip- reting out interesting historical tidbits and por- tic equation.” traying colorful personalities. However, at times the Not surprisingly, such compelling imagery gets book is unclear about what is fact and what is not. progressively rare as the book wades into the For example, Singh writes that there are no first- deeper waters of Galois theory, modular forms, Iwa- hand accounts of Pythagoras’s life and work, and sawa theory, and the other ingredients of Wiles’s yet a few pages later one finds what is purported proof. Despite a spectacular description of the life to be a direct quotation of Pythagoras. A sentence of Galois and an attempt to describe his work, the about the origin of the quotation would have been book leaves the reader with little understanding of helpful. the power and elegance of Galois theory as it blos- There are other problematic moments in the somed after its creator’s death, and there is little book. In discussing the deep connection that the indication of the role it played in Wiles’s work. Mod- Shimura-Taniyama conjecture proposed between ular forms are a struggle: Singh talks about their the world of elliptic curves and that of modular “inordinate level of symmetry,” but never gets suf- forms, Singh comments on the way that such ficiently specific and vivid to give the reader some- “bridges” in mathematics help mathematicians in thing to carry in mind for the rest of the book. He different areas to share insights. “Mathematics creates more of his own terminology for modular consists of islands of knowledge in a sea of igno- forms: the series that defines a modular form, rance,” as he eloquently puts it. Caught up in the which mathematicians might call the Dirichlet se- imagery, he carries it too far: “The language of ries or the L-series of the form, Singh dubs the M- geometry is quite different from the language of series. The book does not say much about this se- probability, and the slang of calculus is meaning- ries, except to say that it is the list of “ingredients” less to those who speak only statistics.” Not only of the modular form and to liken it also to DNA. does this statement err in implying that statisti- Despite the difficulty of explaining these tech- cians do not know calculus, it also implies that cal- nical points, the discussion of the roots of the culus and geometry are distinct fields. Under some Shimura-Taniyama conjecture is the best part of interpretations, geometry could be said to en- the book. Yutaka Taniyama’s assertion of a con- compass calculus. nection between modular forms and elliptic curves Such small problems occur throughout the book, perplexed mathematicians because it was so far but Singh is such an enthusiastic guide that it is ahead of its time. After Taniyama’s tragic death in easy to forgive them. The best parts of the book 1958, Goro Shimura took Taniyama’s brilliant but intertwine history, personalities, and mathemati- unfinished ideas and built them into what is now cal ideas. Especially effective is the discussion of generally known as the Shimura-Taniyama con- Euler’s work with complex numbers, which brings jecture, which says that the L-series of an elliptic a sense of naturalness and inevitability to an idea curve can be paired with a modular form. This that can seem strange and arbitrary. Singh is story has more poignant drama than many of the equally effective in getting across the difficult no- other historical tales in the book: Singh clearly tion of different “sizes” of infinity, especially in his benefited from having Shimura as a primary source. appeal to the device known as “Hilbert’s Hotel”. Indeed, from that point on many present-day char- Some of the digressions—such as the description acters—Ken Ribet, Barry Mazur, John H. Conway, of public-key cryptography and Gödel’s work on John Coates, and of course Wiles himself—feature undecidable statements—demonstrate the way prominently. that mathematics has influenced nearly every as- As the book explains it, Wiles’s strategy for pect of human endeavor. proving the Shimura-Taniyama conjecture is to The book handles elementary mathematical use induction on both the set of elliptic curves and ideas well, but becomes increasingly vague as the on the set of modular forms—a sort of “double” material becomes more sophisticated. When it induction. There is a clever discussion of proof by comes to elliptic curves, Singh does a good job get- induction, which is likened to toppling dominoes ting at the idea of the L-series (which is defined by knocking over the first one and then proving in terms of the number of solutions to the elliptic that if the nth one goes over, all the rest will fol- NOVEMBER 1997 NOTICES OF THE AMS 1305 comm-fermat.qxp 9/15/97 2:04 PM Page 1306 low. According to the book, Wiles matched the first elements of each of these series and then later was able to show by induction that the rest of the “dominoes” would fall. From what I under- stand of the proof, this is not quite how it goes. Perhaps Singh has presented a reasonable rendi- tion of the proof for a popular work such as this; it would take a reader more expert than I to say for sure.