The Millennium Problems: The Seven Greatest UnsolvedMathematical Puzzles of Our TimeReviewed by Brian E. Blank

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The Millennium Problems: The Seven GreatestUnsolved Mathematical Puzzles of Our TimeKeith J. DevlinBasic Books, 2002256 pp., cloth, $26.00ISBN 0-465-01729-0

On May 24, 2000, the Clay Mathematics Institute(CMI), inspired by the centenary of the Hilbert Chal-lenge and seeking to put its own stamp on the newcentury and perhaps beyond, announced awards ofone million dollars each for solutions to seven“Millennium Prize Problems” [6]. The problems se-lected by the CMI were neither new nor concernedwith pressing practical matters. Ordinarily the pop-ular press would not have found them newswor-thy. But, as the CMI correctly reckoned, seven mil-lion dollars concentrates the mind wonderfully.Journalists worldwide reported the story, stirringup so great an interest that the CMI website wasquickly overwhelmed. It is reasonable to supposethat most of the fuss was concerned not with thezeros of the Riemann zeta function but with thezeros of the prize fund.

The custom of prize problems in mathematicsis an old one. The funds have variously come fromgovernments, societies, interested amateurs, andmathematicians themselves. Among the latter, PaulErdos is no doubt most associated with the prac-tice of placing bounties on obstinate problems. It

is reported that$10,000 was hisgreatest offer,$1,000 the greatestclaim on his funds.Several other math-ematicians have an-nounced prizes of acomparable size.The £1,500 offeredby Thwaites for thesolution of the3x + 1 problem ispossibly the bestknown of these. In1997 Andrew Beal,a Dallas banker withan interest in num-ber theory, upped

the stakes considerably by announcing the BealConjecture and Prize, worth $100,000 at the timeof this writing. During a relatively brief publicitystunt, the Goldbach Conjecture carried a one mil-lion dollar price tag. There have been even morelucrative prize funds. At the time of its endowment,the purchasing power of the most famous of allmathematical prizes, the Wolfskehl Prize for the so-lution of Fermat’s Last Theorem, came to about 1.7million dollars (as measured in 1997 currency) [1].These sums for abstract problems stack up very wellagainst the £20,000 reward (worth more than threemillion dollars in today’s currency) that the Parlia-ment of England established in 1714 for the in-vention of the marine chronometer, an instrument

Book Review

Brian E. Blank is professor of mathematics at WashingtonUniversity, St. Louis. His email address is brian@math.wustl.edu.

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of incalculable commercial and lifesaving impor-tance.

Using the Wolfskehl Prize as an example, Barnerhas convincingly argued that a conspicuous prizecan play an important role in registering mathe-matics on the public’s radar screen [1]. The ques-tion is, now that the initial buzz has died down, willthe Millennium Prize Problems retain an interestoutside the professional community? With his newbook, The Millennium Problems: The Seven Great-est Unsolved Mathematical Puzzles of our Time,Keith Devlin joins a group of authors who are bet-ting that the answer is “yes”. Indeed, there does ap-pear to be a ready market for books of this sort.The enduring popularity of elementary (albeit ar-duous) classics such as Dörrie [3], Khinchin [7],and Klein [8] compellingly illustrates the lure of thedifficult problem. Recently there have appearedpopular accounts of the Kepler Conjecture and theFour-Color Problem, three books on the RiemannHypothesis (of which I have read only [9]), and twobooks on the entire set of Hilbert problems [5], [11].

Several of the cited authors liken unsolved math-ematical problems to challenging mountain peaks.The analogy is a useful one, because it highlightsdifferences as well as similarities. You might be sur-prised to know that climbers make the same com-parison. In his account of the recent expedition thatfound George Mallory’s body on Mount Everest,Doug Bell wrote, “Climbing a truly high mountainis not unlike the academic quest for truth. Both re-quire rigid discipline and yet a flexibility of ap-proach—and to reach either goal a person needsboth the ability to slog along hour after hour andthe finesse to find a way past tricky narrows.” ToDevlin “the seven Millennium Problems are thecurrent Mount Everests of mathematics.” They are,in the order of appearance in his book: the RiemannHypothesis, Yang-Mills existence and the mass gap,the P versus NP problem, the Navier-Stokes exis-tence and smoothness problem, the Poincaré Con-jecture, the Birch and Swinnerton-Dyer Conjecture,and the Hodge Conjecture. The number of pagesdevoted to each ranges downward from 44 to 16in what would have been a monotone decreasingsequence but for the graphics in the Poincaré Con-jecture chapter. Addressing a reader who is as-sumed to have no more than a good high-schoolknowledge of mathematics, Devlin sets himselffour tasks: “to provide the background to eachproblem, to describe how it arose, explain whatmakes it particularly difficult, and give you somesense of why mathematicians regard it as impor-tant.”

Now if Devlin’s audience consisted of well-prepared second-year graduate students with eclec-tic tastes, then his agenda would seem ambitiousbut worthwhile. Devlin, however, is serious aboutreaching a much less sophisticated audience. When

his readers encounter “10!” he feels obliged to tellthem, “You don’t read this aloud as ‘ten’ in anexcited or startled voice.” Surely it is pointless try-ing to convey to such readers just how difficult themillennium problems are. Experience is requiredfor such a judgment. Those of us who have neverclimbed even a moderately difficult mountain can-not appreciate how hard it is to summit Eiger viaits north face. But Reinhold Messner tells us thatit is a scary ascent, and we are prepared to take hisword for it. That is pretty much how Devlin achieveshis goal of explaining the particular difficulties ofthe millennium problems. He shows us no failedroutes to the top, no lines of attack that might getus there if only we could see them through.

Establishing the importance of the millenniumproblems is another task that is not easily accom-plished. Devlin is correct to assume that for his au-dience “We must know! We shall know!” will not suf-fice. I suspect that of the seven millenniumproblems, only P versus NP has the type of sig-nificance that will resonate with the majority of De-vlin’s readers. He explains it to them well. Not sowith the other problems. To affirm the importanceof the Mass Gap Hypothesis, Devlin quotes Witten,who tells us that “the proof would shed light on afundamental aspect of nature.” That is as deepand convincing as it gets. The reader is told that“A proof of the Hodge Conjecture would establisha fundamental link among the three disciplines ofalgebraic geometry, analysis, and topology,” “A so-lution to [the Birch and Swinnerton-Dyer Conjec-ture] will add to our overall understanding of theprime numbers,” “A solution of the Navier-Stokesequations would almost certainly lead to advancesin nautical and aeronautical engineering.” A proofof the Riemann Hypothesis, Devlin writes, “mightwell lead to a major breakthrough in factoringtechniques…with Internet security…hanging in thebalance.” I suppose it is a sign of the times that in-ternet commerce is being used to validate interestin the Riemann Hypothesis.

The last two of Devlin’s goals, explaining thegenesis of each problem and developing its back-ground, can be grouped together. These tasks arethe heart of the book, and Devlin scores some suc-cesses amid the expected failures. The Hodge Con-jecture, for example, completely defies elementarydiscussion. Devlin does not impart a good sense ofthis problem, and he admits it. But it is the conceptof the book that is at fault, not the effort. The Birchand Swinnerton-Dyer Conjecture might also seemto be out of the reach of a popular book, but in thiscase I think that Devlin’s discussion is surprisinglyeffective. The chapter on Yang-Mills Theory andthe Mass Gap Hypothesis is essentially devoid ofmathematics, but it is fun, interesting, and a pain-less summary of a good deal of physics. An intro-duction to topology makes up most of the chapter

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that is devoted to the Poincaré Conjecture. Half thechapter on the Navier-Stokes equation is given overto a crash course on calculus and partial derivatives.Maybe it is the case of a glass-half-full author meet-ing a glass-half-empty reviewer, but I am not con-vinced that it was wise to target this book at an au-dience with such minimal background.

I come now to the two problems that I think arebest suited for Devlin’s concept. The well-knownP versusNP problem was actually a prize problemlong before the CMI put up its one million dollars.In January 1974 Donald Knuth offered one liveturkey to the first person who proves that P = NP .This problem is quite different from the others. Fewundergraduate texts in mathematics focus on theother six millennium problems, but P versus NPhas become a mainstay of the computer science lit-erature. Most books about algorithms, complexitytheory, or theoretical computer science introducethe classes P, NP, and NP-complete (as well asothers such as co-NP). The Traveling SalesmanProblem (TSP) is almost always chosen as an ex-ample. By now most computer scientists antici-pate that many of their readers will already be fa-miliar with TSP. As a rule they supplement it witha few less hackneyed examples. Shasha and Lazere[10] find another way to avoid the commonplaceby interviewing Leonid Levin, overlooked by Devlin,and Stephen Cook. (As an aside, let me note thatwhen scientists write in a popular vein, they oftenconsult colleagues about facts, but they rarely in-terview fellow scientists. The books of Yandell [11]and Sabbagh [9], like that of Shasha and Lazere,demonstrate how effective the technique can be.)

Devlin’s treatment of P versus NP is mundanebut effective. He refers briefly to another NP prob-lem but otherwise sticks closely to TSP. Generallyspeaking, his discussion is a scaled-down versionof the treatment that he gave in The New GoldenAge [2]. The Time-complexity table that appears inboth books is but one illustration of how closelyDevlin sometimes follows his earlier writings. Ajaded reviewer who is apt to regard the new dis-cussion as merely a superfluous rehashing mustconcede that many of Devlin’s readers will be learn-ing these ideas for the first time. Those readers willunderstand the discussion in Devlin’s book andcome away with a good idea of what this millen-nium problem is about. It is therefore a successfulchapter.

We reach the Riemann Hypothesis (RH) at last.Because it is a great problem that can be accessi-ble when approached in a thoughtful way, it shouldbe ideal for Devlin’s type of book. Like Fermat’s LastTheorem, RH originates with a fascinating story thatencourages speculation. Many interesting, easilystated facts about the ζ function are known. Sev-eral curious, seemingly unrelated conjectures areequivalent to RH. Many of the greatest mathemat-

ical characters of the twentieth century were drawnto it. There are anecdotes, false proofs by famousmathematicians, mammoth computer calculations,unimaginably large counterexamples to relatedconjectures—in short, a treasure trove from whichany expositor can profitably plunder if he is so in-clined. On the evidence, Devlin was only lukewarmto the idea.

As mentioned earlier, the chapter on RH, Devlin’slongest, comes to 44 pages. That includes a lot ofstandard padding such as Euclid’s proof of the in-finitude of primes. Dev-lin makes the connectionbetween the zeta func-tion and the sequence ofprimes via the EulerProduct Formula, butafter that everything be-comes needlessly im-precise. The Prime Num-ber Theorem isdiscussed, there is a lotof vague talk about itsrelationship to the equa-tion ζ(s) = 0, but neitherthe fact that ζ(1 + it) �= 0nor its import is men-tioned. Even the criticalstrip does not appear. Aplot of z = sin(xy) findsits way into the chapter,but z =

∣∣ζ(σ + it)

∣∣ is not graphed. Although it will

not bother the typical reader, Devlin’s speculationabout Riemann’s insight (pp. 50–51) is at odds withEdwards’s exposition of the Riemann-Siegel For-mula [4, pp. 136–170]. In the final analysis, muchbetter treatments of RH abound. Indeed, Devlinwrote one of them [2, pp. 193–221].

The job of the popularizer is admittedly diffi-cult. These authors are constantly faced with op-timization problems that are not present in moreadvanced monographs. I confess that I am oftenbaffled by the solutions that they find. For exam-ple, Devlin records the symmetric form of the an-alytic continuation formula for ζ (s) using an un-specified expression, γ (s), for Γ (s/2) or Π (s/2− 1).So far as I can see, little is gained by using non-standard notation—the symmetry is a bit moretransparent—but there is a real loss: the reader whoknows the gamma function but who is learningthe zeta function from Devlin will be deprived ofa beautiful formula. Twice Devlin tells us that RHis the most important unsolved problem in math-ematics. There is something to be gained by re-placing tangled, indecisive discourse with simple,clear-cut assertions. The loss is that such pro-nouncements invite questions that are not an-swered. Why should a topologist consider RH moreimportant than the Poincaré Conjecture? The 3x + 1

To Devlin “theseven

MillenniumProblems arethe current

Mount Everestsof

mathematics.”

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problem, to cite but one example, does not appearto have the gravitas of RH, but until we get to thebottom of it and any number of other unsolvedproblems, is it not premature to bestow highesthonors?

Like journalists, mathematical popularizers arewell served by the precept “the truth, nothing butthe truth, but never the whole truth.” Devlin livesby the third part of that saying but often runsafoul of the second. When anauthor does not bother to checkhis facts, no matter how unim-portant, he puts his credibilityat risk. It is not true that the Rie-mann Hypothesis “is the onlyproblem that remains unsolvedfrom Hilbert’s list” (p. 4). Not allthe seventy-two savants memo-rialized on friezes of the EiffelTower are “nineteenth-centuryFrench scientists” (p. 131). De-vlin writes that “the term ‘imag-inary’ for the square root of anegative quantity seems to havebeen first used by Euler” (p. 37).He is confusing the term “imag-inary” with the notation i .Descartes, Wallis, and Leibnizall used “imaginary” prior toEuler. A footnote on page 54refers to “… the (false) as-sumption that there is nolargest prime.” Finally, it is fool-ish and inappropriate to sug-gest that mathematicians are “the seekers of theonly 100% certain, eternal truth there is.” Surelythere will be curious scientists from other fieldswho will pick up this book. Do we need to pickfights with them?

Devlin is a professional author, and anything hewrites will reflect that. However, there is no wayto disguise a suspect idea. Several editors, he tellsus, approached him with the concept of this book.It shows. Writing that does not burst forth from anauthor is bound to be halfhearted. What I findmost lacking is a real sense of what these problemsare all about. What drives our passion for these andsimilar problems? At the press conference that ac-companied the prize announcements, Landon Clay,founder of the CMI, explained that “It is the desirefor truth and the response to the beauty and ele-gance of mathematics that drives mathematicians.”Put aside the incongruity of such thoughts beingexpressed by a man who has just put up seven mil-lion dollars of incentives. More to the point, his listof driving forces is seriously incomplete. To get thetacky subject of financial gain out of the way, tan-gible benefits are expected to ensue, one way or an-other, from the cracking of a tough nut. I am

reminded that many years ago when Quillen solvedSerre’s Problem, the assistant professor who taughtmy algebra class did not speak of truth, beauty, orelegance; he lamented that a perfectly good prob-lem had been squandered needlessly on a mathe-matician who already had tenure.

There are other powerful driving forces thatClay and Devlin are too discreet to mention. As anundergraduate I attended a lecture that touched

upon the solution of a famousproblem that dominated itsfield for half a century. Atwenty-year-old can be startledto learn that ten years of amathematician’s life might beexpended on the solution ofone problem. How can anyonepersevere so long without ad-mitting defeat? When a fellowstudent phrased that thoughtas a question, my professor,himself a leader in the field,laughed and said “You have toknow Professor ___ . He thinkshe can solve anything.” It wasa funny answer that only mo-mentarily concealed his sincereadmiration for the confidencethat can be so essential for suc-cess. Ego, competition, one-up-manship—not noble forces,perhaps, but mathematics isthe richer for them. In a recentbook devoted entirely to RH,

Karl Sabbagh elicits unusually candid remarks frommany of the top mathematicians in the chase [9].Several mathematicians confront the controversialpractice of hoarding partial results. One admits,“I’m afraid I see all of mathematics as a competi-tion. If someone has a theorem, I always want toprove a better one.” Another expresses relief thata highly publicized proof of RH fell apart, sendingthe unfortunate aspirant back “to oblivion wherehe came from.”

All of this leads one to conclude that Devlin isbeing a tad genteel when he gives high-mindedreasons or even practical reasons that attempt toexplain why mathematicians are drawn to unsolvedproblems. In the preliminary chapter, appropri-ately titled “The Gauntlet Is Thrown”, he comescloser to the truth when he writes “Ultimately,mathematicians pursue these problems for thesame reason the famous British mountaineerGeorge Mallory gave in answer to the newspaperreporter’s question, ‘Why do you want to climbMount Everest?’: ‘Because it is there’.” Although thewords ascribed to Mallory may have been com-posed by an unnamed New York Times reporter asa response to a putative query, alpinist Robert

What I findmost lacking isa real sense of

what theseproblems are

all about. Whatdrives ourpassion forthese and

similarproblems?

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Jasper was asked why he pioneered a new route upthe Matterhorn. He cited “the challenge of the line,which I’ve always regarded as an end in itself.” In-deed! As soon as climbers have conquered a peakthey strive to find new, more taxing routes to thetop. And when those routes have been traversed,they relax the hypotheses and try to be the first tosummit in winter. And so on. It sounds just likemathematics. To paraphrase a credit card com-mercial: CMI Prize Problems—$1,000,000; suc-ceeding where everybody else has failed—price-less.

As I write this review, purported proofs of boththe Riemann Hypothesis and the Poincaré Conjec-ture are circulating. However those claims turnout, we are sure to reap a fine harvest of books andpapers inspired by the Millennium Prize Problems.We may all look forward to the day when a fellowmathematician brings news from the summit andechoes the first words of Edmund Hillary on his de-scent: “We’ve knocked the bastard off!”

References[1] KLAUS BARNER, Paul Wolfskehl and the Wolfskehl Prize,

Notices Amer. Math. Soc. 44 (1997), 1294–1303.[2] KEITH DEVLIN, Mathematics: The New Golden Age, Re-

vised Edition, Columbia University Press, 1999.[3] HEINRICH DÖRRIE, One Hundred Great Problems of Ele-

mentary Mathematics: Their History and Solution,Dover Publications, 1965.

[4] H. M. EDWARDS, Riemann’s Zeta Function, Dover Publi-cations, 2001.

[5] JEREMY J. GRAY, The Hilbert Challenge, Oxford Univer-sity Press, 2000.

[6] ALLYN JACKSON, Million-dollar mathematics prizes an-nounced, Notices Amer. Math Soc. 47 (2000), 877–879.

[7] A. YA. KHINCHIN, Three Pearls of Number Theory, DoverPublications, 1998.

[8] FELIX KLEIN, Famous problems of elementary geometry,in Famous Problems and Other Monographs, SecondEdition, AMS Chelsea Publications, 1962.

[9] KARL SABBAGH, The Riemann Hypothesis: The GreatestUnsolved Problem in Mathematics, Farrar, Straus,Giroux, 2002.

[10] DENNIS SHASHA and CATHY LAZERE, Out of Their Minds:The Lives and Discoveries of Fifteen Great ComputerScientists, Copernicus, 1998.

[11] BENJAMIN H. YANDELL, The Honors Class: Hilbert’s Prob-lems and Their Solvers, A K Peters, 2002.