II'E~-,.m,~-~_'.._[.]m David E. Rowe, Editor]

F ermat's Last Theorem (FLT) was Fermat long known as the most famous of all unsolved mathematical problems. Familiar even to laymen, it Comes to attracted the attention of both profes- sional mathematicians and amateurs, many of whom thought they held the America: key to its solution. Ferdinand Linde- mann, who was suddenly vaulted to Harry Schultz fame when he proved the transcen- dence of ~- in 1882, tried to solve FLT several times over the remainder of his Vandiver career, only to come up with one faulty proof after another. Nor was he alone and FLT in this regard; indeed, FLT stood in a league by itself when it came to the number of incorrect proofs that found (1914-1963) their way into print. As for failed at- tempts by amateurs who sent their "so- LEO CORRY lutions" to mathematicians all over the world but which (thankfully) remained unpublished, the number cannot even be estimated. These circumstances make it easy to understand the general excitement that Figure I. Harry Schultz Vandiver and son, surrounded this story when Andrew Frank, ca. 1930 (HSV). Wiles finally completed his general proof of Fermat's conjecture in 1994. As sionally found even in accounts written news of this impressive achievement for more professional audiences). The rippled through the mathematical com- most striking example of this is surely munity, it also made headlines that at- Simon Singh's best-selling book, Fer- tracted the attention of broad lay audi- mat's Enigma, a book that conveyed to ences who had no chance of grasping very broad audiences the excitement the ideas behind his work. Public in- human beings feel about doing mathe- terest in Wiles's personal sto W and his matics. Its readers learn that Fermat's long quest to solve FLT added an un- conjecture "tormented lives" and "ob- usual sense of drama to the accom- sessed minds" for over three centuries, plishment itself. His curiosity about the and thus constituted "one of the great- problem from childhood, his eight years est stories imaginable." On the front flap of self-imposed seclusion that led to the of some editions one reads that FLT be- breakthrough, the tension that followed came the Holy Grail of and discovery of a non-trivial mistake in his that Euler "had to admit defeat" in his initial proof, and the final resolution attempts to find a proof, while: eight months later in collaboration with Whole and colorful lives were de- Richard Taylor, all these elements only voted, and even sacrified, to finding enhanced interest in this appealing a proof .... took sto W . on the identity of a man to do re- Still, from a broader perspective, in- search in a field forbidden to fe- Send submissions to David E. Rowe, formed individuals might well shake males .... The dashing Evariste Ga- Fachbereich 08--Institut f(Jr Mathematik, their heads when reading the overly lois scribbled down the results of his Johannes Gutenberg University, dramatized popular versions of the research deep into the night before D55099 Mainz, Germany. quest to solve FLT (a tendency occa- venturing out to die in a duel in

30 THE MATHEMATICAL INTELLIGENCER 9 2007 Springer Science+Business Media, Inc. 1832. Yutaka Taniyama... tragically nected with FLT. An interesting, though killed himself in 1958. Paul somewhat elusive historical question Wolfskehl, a famous German indus- that I will attempt to answer at least par- trialist, claimed Fermat had saved tially concerns Vandiver's professional him from suicide. status in the eyes of his contemporaries. On opening that book, one reads that "The Last Theorem is at the heart of an intriguing saga of courage, skulldug- A self-trained mathematician, Harry gery, cunning, and tragedy, involving Schultz Vandiver was born October 21, all the greatest heroes of mathematics." 1882, in Philadelphia. He never com- Not surprisingly, a closer and more pleted high school, and the little col- sober examination of the actual histor- lege- and graduate-level mathematics ical evidence surrounding research on he studied at the University of Penn- Fermat's problem brings to light a far sylvania in 1904-06 was undertaken in less dramatic version of these events. a rather haphazard, non-systematic This is not to say that the history of FLT manner. Thus he never obtained a col- lacks interest, but the sto W hardly war- lege degree, except for an honorary rants the sense of high drama that re- doctorate that the University of Penn- cent writers have brought to it. 1 None sylvania bestowed upon him in 1945 at of the mathematicians who appear in the age of 63. 2 Singh's account (except for Wiles him- In 1900 he started submitting solu- self) ever devoted sustained research ef- tions to problems posed in the Ameri- forts purely focused on an attempt to can Mathematical Monthly, especially solve this famous problem. In fact, on topics in algebra and number the- many of those mentioned in his book ory. 3 This activity seems to have been showed only the slightest interest in his gateway to studying mathematics, Figure 1. Maude, Frank and Harry S. solving it, whereas only one mathe- and it was certainly how he got to know Vandiver (HSV). matician prior to Wiles took a similar George David Birkhoff (1884-1944) passionate lifelong interest in FLT. This well before the latter became the most was Harry Schultz Vandiver (1882- cian's time. Nor has his reputation ben- influential American mathematician of 1973), a figure who does not even ap- efited posthumously from the lavish his generation. The young Birkhoff pear in Singh's book and who is only praise that 's work re- wrote to Vandiver in 1901 commenting mentioned marginally in most other re- ceived. Since Wiles's general proof on the latter's contributions to the cent accounts. came from a completely different di- Monthly while telling him about his Vandiver devoted nearly all of his rection that bore little relationship with own interest in FLT. 4 Thus began a sub- professional life to resolving this famous Vandiver's train of ideas, the latter's stantial correspondence that lasted sev- problem, a quest that set him apart from contributions have either been over- eral years, though they did not meet un- his fellow mathematicians. For although looked or are seen today as devoid of til 1913. Their joint paper, in 1904, was FLT aroused curiosity among number- direct interest for actual research in Birkhoff's first publication. 5 theorists, the problem remained on the . Between 1905 and 1917 Vandiver margins of the field for decades. Re- Nonetheless, from a historical point was working as a customs house bro- markably few" serious efforts were de- of view, Vandiver is a figure of consid- ker and freight agent for his family's voted to it during Vandiver's lifetime. erable interest, not only because of his firm. In his letters to Birkhoff--written Moreover, his research program in- intense involvement with FLT but also on stationery with the letterhead of the volved the kind of massive calculations for the role he played within the Amer- "John L. Vandiver, Custom House Bro- of individual cases that most number- ican mathematical community through- ker" in Philadelphia--he openly ex- theorists consciously avoided. Aided by out his long and, in many ways, exotic pressed his admiration for his friend's electro-mechanical and, later on, elec- career. The present article offers a por- knowledge and "boundless enthusiasm tronic devices for making such calcula- trait of that career, along with a brief for mathematics." Occasionally he sug- tions, Vandiver emerged as a prominent sketch of Vandiver's activities in con- gested ideas intended for possible ad- exponent of a research style that many nection with FLT. In a follow-up to this ditional joint publications, but these of his contemporaries would have con- article I will describe some other aspects never actually materialized. Birkhoff's sidered unworthy of a true mathemati- of Vandiver's work not directly con- juvenile interest in number theory and

1[Corry 2008]. 2There are various sources of information about Vandiver's life, sometimes containing contradictory information. I have drawn here mainly on documents found at the Van- diver Collection, Archives of American Mathematics, Center for American History, The University of Texas at Austin. See also [Greenwood et al., 19731, [Lehmer 1973]. 3See Am. Math. Mo. 7 (May 1900), p. 146. His first number-theoretical problem appears in the Am. Math. Mo. 8 (Aug. 1901), p. 180. A more significant one, dealing with properties of Mersenne numbers appeared in Am. Math. Mo. 9 (Feb. 1902), pp. 34-36. 4[Vandiver 1963, 271]. 5[Birkhoff & Vandiver 1904].

92007 Springer Science+Business Media, Inc., Volume 29, Number3, 2007 31 who was preparing his monumental a network of Moore's academic de- During times of intense History of the Theory ofNumbers. In par- scendants found positions at depart- research effort, he would isolate ticular, Vandiver was actively involved ments throughout the country, and their himselffrom all distractions. in writing the chapter on FLT, and in efforts helped make point-set topology 1928 he was co-author of a supple- a leading field of research in the USA. mentary volume to Dickson's work. 8 Al- Vandiver's network remained far more elementary geometry soon began to re- gebraic Numbers" was produced on circumscribed, reflecting the more lim- cede in favor of his mature pursuits in Dickson's recommendation as the offi- ited interest in FLT and other research analysis and applied mathematics. Van- cial report of the Committee on Alge- topics he pursued throughout the years. diver remained strongly focused on braic Numbers of the National Research Nevertheless, Vandiver traveled ex- number theory and on related algebraic Council, a committee Vandiver chaired tensively and took repeated leaves of disciplines and consistently tried to pull between 1923 and 1928. This collabo- absence to pursue his research. Much Birkhoff back into these fields. In 1915 ration with L. E. Dickson left a deep im- of his correspondence with university Vandiver wrote to him: print on Vandiver. Throughout the authorities revolved around requests re- I am particularly anxious that you years, he continually referred to the lated to these leaves. Thus, it was with become interested in number the- spirit of Dickson's work as an example a touch of irony that in its sympathetic ory. If I can induce you to take up that should be followed in all of math- Memorial Resolution of 1973, Vandiver the subject I am sure you will never ematics, and he attempted to implement was remembered by the Faculty Coun- regret it. Your position in the several initiatives along these lines. cil as a distinguished former UT pro- math[ematical] world is now as- Such undertakings included detailed fessor, whose colleagues "bemoaned sured, and I think you should be bibliographies of various individual the fact that he did not staj' around able to give considerable time to mathematical domains as well as pro- very much. 12 He was constantly apply- these things which virtually consti- posals for significant reforms in mathe- ing for research grants provided by a tuted the life work of such men as matical reviewing and the refereeing number of institutions, including the Gauss, Kummer, Kronecker, Dirich- systems in the USA. National Science Foundation, the let--after your present work is com- Strongly recommended by Dickson, Carnegie Institution, and the Guggen- pleted. 6 Vandiver was appointed in 1924 to a helm Foundation. In 1934, he was the Birkhoff, in turn, consistently encour- professorship at the University of Texas, first mathematician ever to apply for aged his friend to pursue his mathe- Austin. This became his academic home support from the American Philosophi- matical interests beyond what his free until the end of his life, though his re- cal Society. 13 In 1953, at the age of sev- time in business would allow him. Van- lationship with colleagues at this insti- enty-one, he requested (for the sixth diver eventually found himself in a real tution can hardly be described as one time) funding from the Guggenheim dilemma, but it took some time before of peaceful coexistence. Above all, his Foundation for a planned six-month he finally decided to embrace mathe- personal and professional relations with leave of absence. Surprisingly the Foun- matics as a profession. At one point he the almighty Robert Lee Moore dation granted him approval, but then wrote to Birkhoff, half joking-half seri- (1882-1974) were a source of constant Vandiver decided to withdraw his ap- ous, that with business slackening be- strain that reached remarkable peaks of plication. ~4 Even at the age of 76 he re- cause of the war, he now had consid- mutual animosity. 9 Beyond solitary re- ceived a research grant from the NSF. erable time for research. "Perhaps times search, Vandiver's main strengths were Although those within his close circle will become so bad," he added, "that I clearly not in classroom teaching, but of friends would invariably write the will be compelled to look for some rather in direct and personal inter- warmest letters of recommendation in teaching position. ''7 From 1917 to 1919 changes. 1~ While his work involved ac- support of his many applications, this Vandiver served as yeoman in the U.S. tive collaboration with several younger was not always the case with others. Naval Reserve, but soon afterward, mathematicians, and particularly gradu- Birkhoff, for instance, advised the aided by Birkhoff's active endorsement, ate students, he formally directed only Guggenheim Foundation that it would he accepted a teaching position at Cor- five Ph.D. dissertations at Austin. 11 One be better to devote its resources to sup- nell. cannot help but compare him in this re- port younger men, and he saw no rea- With his arrival at Cornell, Vandiver gard with Moore, who devoted a great son why Vandiver could not continue began collaborating with Chicago's deal of his energy to advising many to pursue his research at his home in- (1875-1954), promising Ph.D. candidates. Eventually stitution. ~5 Still, Birkhoff and others con-

6Vandiver to Birkhoff: March 18, 1915 (HUG). 7Vandiver to Birkhoff: May 17, 1915 (HUG). 81Vandiver & Wahlin 1928]. ~1 will deal with this in the follow-up to this article; part of the story is told in [Parker 2005, 226-231]. l~ 1973] describes Vandiver as a "poor lecturer." Some of Vandiver's students expressed similar views elsewhere. 11They are: Ferdinand Biesele (1941), Olin Faircloth (1951), Charles Nicol (1954), Milo Weaver (1956), and Richard Kelisky (1957). 12http://www.utexas.edu/facu~ty/counci~/2~-2~1/mem~ria~s/sCANNED/vandiver.pdf. 13Edwin G. Conklin to Vandiver: July 6, 1934 (APS). ~4Vandiver to Henry Allen Moe: June 17, 1953 (HSV). 15Birkhoff to Foundation: March 7, 1953 (GFA).

32 THE MATHEMATICALINTELLIGENCER 2007 SpringerScience+Business Media, Inc. tinued to stress Vandiver's status as the When it came to baseball, he prided ! am now on a basic research grant acknowledged world-leading expert on himself in taking original views that of the N.S.F., which does not expire FLT and to praise his (not totally unre- went against the flow of mainstream for two years, and as I am now 76 lated) work on cyclotomic fields. In later opinion. Commenting on a recent game years old, I do not feel that I can years, many expressed admiration for played by his "favorite team," the New take on any mathematical work the fact that a man of Vandiver's age York Giants, against their powerhouse aside from what the Foundation ex- could still be an active researcher. ~<' This rivals, the Yankees, Vandiver noted that pects me to do, namely, to do re- admiration was also often conveyed to the Giants' "present line up includes search on number theory. 24 him directly in personal letters. 17 many wonderful ,fielders," adding that Bellman did not give up, however, and Vandiver's frequent travels were part "defense seems to be paid little atten- replied immediately in order to assure of a somewhat nomadic lifestyle that of- tion to these days by the public. They Vandiver that he would "have the re- ten took him, his wife Maude (ride Folms- prefer to look at cheap homeruns. ''21 sponsibility of looking over the papers bee), and their son Frank (1926-2005) Above all, what he enjoyed was the ex- of only one mathematician, namely, around the country and also to Europe. citement of a tight contest, as "a game yourself." And he added: Frank was home-schooled, as experience which ended with a one-sided score As far as your rather young age of had hardened Vandiver's strong distrust was not to his taste. ''22 76 is concerned, I distinctly remem- of public schools, is Frank reported that All kinds of additional oddities were ber that you are the person who in- in his childhood the family never had a associated with this somewhat leg- sisted that he wanted to live to be 95 permanent home nor did they ever own endary figure on (and off) the Texas and to be shot by a jealous husband. a house. They would move from one campus, as reflected in a sympathetic Consequently, if you divided your re- apartment to another, renting the home account written by his Austin colleague maining activities between number of a colleague on leave while Vandiver Robert Greenwood: theory and these pursuits, I feel we was preparing to go on leave himself, H.S. Vandiver was hardly the athletic have the best of the bargain. 25 then going back to another rented apart- type. There is no record of his ever Vandiver's later years were affected by ment and so on. For many years Van- having owned a car, and so he must poor health. Still, he continued to work diver also had a "pemmnent" room at have walked a lot. Between semes- under a "Modified Service" appointment the Alamo Hotel. 19 During times of in- ters in the winters the University until the age of 80; only then did he be- tense research effort (and these were not buildings frequently were heated up come Professor Emeritus. When he died infrequent), he would isolate himself to about 40~ But Professor Vandiver on January 4, 1973, he was 89 years old. from all distractions by checking into this would walk up to campus, go into hotel room. He always kept a suitcase in his office where he had a portable Vandiver and FLT in Context his office, just in case such an eventual- electrical heater with a spherical or Vandiver did not develop new concepts ity might arise. Alternatively, he some- parabolic reflecting surface. He would or overarching theories to deal from times preferred to lock himself in at then work away on a mathematical completely novel perspectives--with home and remain incommunicado for theorem of current interest to him. FLT. Rather, his approach was that of a several days. 2~ He could single-mindedly Usually he would, on these occasions, meticulous technician who fully ex- concentrate on his work, sometimes for- work in his top coat with the electric hausted the unexploited potential of ex- getting even to eat, thereby bringing him- heater warming his feet anti legs. 23 isting theories and refined them further self to the brink of physical collapse. Many friendly exchanges of letters among where necessary. At the center of Van- Two extra-mathematical topics sur- his posthumous papers offer a closer diver's work one finds extensive, highly face fairly often in his letters: classical glimpse into his unique personality. complex calculations of particular cases, music and baseball. Vandiver seems to Richard Bellman (1920--1984) was a bril- along with innovations aimed at im- have owned a remarkable collection of liant and versatile mathematician whose proving existing computational tech- records that he managed to carry along fields of interest were much broader and niques. He was apparently undaunted with him as he moved from place to quite different from Vandiver's. In 1959, by even the most demanding computa- place. He especially loved Mozart, anti Bellman created the Journal e?f Mathe- tions, in part because he was willing to in various letters he mentioned plans to matical AnalysLs" and Applications and in- use a variety of tools (both material and write a mathematician's guide to listen- vited Vandiver to join the editorial board. conceptual) to achieve his task. ing to this genial composer's music. Vandiver initially declined: In order to understand the context of

16Andr6 Weil to Foundation: June 1953 (GFA). 17Alfred Brauer to Vandiver: Dec. 23, 1958 (HSV). Brauer assured Vandiver that he would support his request for a grant with the NSF, and the latter was indeed granted. 18Frank Vandiver later became a distinguished professor of American history and, among other things, Provost of Rice and President of Texas A&M University. 19See [Greenwood et aL, 1973, 10932]. 2~ Vandiver, interview with Ben Fitzpatrick and Albert C. Lewis, June 30, 1999 (MOHP). 2Wandiver to W. L. Ayres: August 31, 1951 (HSV). 22[Greenwood et al., 1973, 10932]. 23Robert Greenwood, "The Benedict and Porter Years, 1903-1937", unpublished oral interview (March 9, 1988), (MOHP), p. 26. 24Vandiver to Bellman: April 20, 1959 (HSV). 25Bellman to Vandiver: April 28, 1959 (HSV).

2007 Springer Science+Business Media, Inc., Volume29, Number3, 2007 Vandiver's work, a few words must be ground is necessary in order to under- ter aspect of his work that most influ- said both about the status of FLT at the stand Vandiver's work. This sketch will enced the development of number the- turn of the twentieth century as well as also underscore the fact that rather few ory in the decades to come, especially to the standing of research in number- significant contributions to solving the through the efforts of Richard Dedekind. theory in the United States. During the problem were made following Ernst Ed- By the turn of the twentieth century, par- first third of the twentieth century, the uard Kummer's work in the 1850s. As ticularly in the wake of Hilbert's influ- American community of number-theo- part of his long-standing efforts to deal ential Zahlbericht, the emphasis on a rists was relatively small and not ex- with questions related to higher reci- "conceptual" perspective (as opposed to tremely prominent. As a rough measure, procity laws, Kummer developed many the more algorithmic approach favored one might note that the index to the first important concepts and techniques that by ) became domi- ten issues of the Transactions oftheAMS turned out also to be relevant for FLT. nant in the discipline. Results based on (1909) lists only two articles under the Thus, it was Kummer who introduced special calculations for particular cases heading of number theory, and in the the notions of regular and irregular were not favored under this view. These following decade, despite a noticeable primes, a distinction based on a prop- general trends in research account to a increase, there were still only thirteen. erty of the "class number" h/, of a cy- large extent for the remarkable fact that In his correspondence, Vandiver often clotomic field k(~'). He then developed Kummer's results relating to FLT were spoke about the lack of interest in num- original methods that enabled him to not essentially improved or extended for ber theory in the US. Looking through prove that FLT is valid for all regular almost sixty years. this correspondence for names of math- primes. In addition, he introduced three Prior to Kummer, Sophie Germain ematicians actively involved in research somewhat complex conditions which, had proved an important result, as a in number theory prior to 1940, one when satisfied by an irregular exponent consequence of which the proof of Fer- finds above all foreign figures, some of p, implied the validity of FLT for that ex- mat's problem can be reduced to deal- whom Vandiver also visited in Europe: ponent. Then, by means of long and te- ing with two separate special cases. Rudolf Fueter, Edmund Landau, Emmy dious computations, he identified all ir- Case I asserts that for p > 2, x p + yP + Noether, Helmut Hasse, Kurt Hensel, regular primes under 164, obtaining ZP = 0 has no integer solutions for x, Philipp Furtw~ingler, Nikolai Grigore- these eight numbers: 37, 59, 67, 101,103, y, z relatively prime to the odd prime vich Chebotarev, Dmitry Mirimanoff, 131, 149, and 157. Applying his criteria p. Case II asserts the same when one Taro Morishima, Trygve Nagell, Felix to the three cases of irregular primes un- and only one of the three numbers x, Pollaczek, and Arnold Walfisz. We also der 100 (37, 49, 67), he achieved his y, z is divisible by p. The little progress find a number of American mathemati- well-known result of 1857 that FLT is that did take place in proving the the- cians, mainly active on the West Coast: valid for all exponents less than 100. Be- orem between Kummer and Vandiver Eric Temple Bell, Hans Frederik Blich- yond this, however, the calculations be- dealt almost exclusively with case I. The feldt, Derrick Norman Lehmer (and came prohibitively complex and, in ad- most important single result was ob- later on Derrick Henry and Emma dition, it became clear to Kummer that tained in 1909 by Arthur Wieferich Lehmer), Robert Daniel Carmichael, Al- some of the criteria he had developed (1884-1954). Wieferich proved that if bert Cooper, Leonard Eugene Dickson, would not apply for certain irregular three integers x, y, z relatively prime to Morgan Ward, and Aubrey Kempner. prime exponents, such as p = 157. p actually did satisfy xP + yP = z p, then Unlike the case with their foreign coun- Kummer also proved that a prime p the congruence 2 p-1 ---= 1 (modp 2) must terparts, this latter group essentially ex- is regular if and only if it does not di- be satisfied. Dimitry Mirimanoff (1861- hausts that of number-theorists active in vide the numerators of any of the 1945) extended this result in 1910 by the American community. After 1940 the Bernoulli numbers B0, Be .... Bp-3, proving that the same p would also sat- field became more active, though it con- which appear as coefficients in the ex- isfy 3 p-1 _- 1 (rood p2). These two re- tinued to remain somewhat on the mar- pansion suits, and some similar ones that were gins of the research community for sporadically added later on, helped es- x = ~ BnX ~ some years. 26 Vandiver's almost exclu- tablish a lower bound for the value of e x- 1 /" n! sive focus on this field, along with the n=O the integers for which the Diophantine unusual circumstances surrounding his At that time values for the Bernoulli equation xP + yP = z p could be satis- early mathematical career, helps to ac- numbers had been calculated up to fied under the conditions of case I (and count for his rather unique situation B62. 2v The ability to identify higher val- this, moreover, only by considering p, within the American mathematical com- ues of regular or irregular primes would and irrespective of the values of x,y,z munity. His decision to devote so much later come to depend on the possibil- that may satisfy the equation). of his professional life to FLT made him ity of calculating higher values of these Based on these results, and directly a truly singular figure. numbers, an effort in which Vandiver motivated by the additional encourage- Although even a mini-history of Fer- was directly involved. ment provided by the creation in 1908 mat's problem is well beyond the scope Kummer was both an avid calculator of the Wolfskehl Prize, several mathe- of this article, a bit of historical back- and a gifted theorist, but it was the lat- maticians decided to attack the problem

26A detailed analysis of the internal structure of this community and its development (along the lines of [Goldstein 1994] for the case of the French community of number-theorists in the second half of the nineteenth century) seems to be an interesting open task for historical research. 2r[Ohm 1840].

34 THE MATHEMATICALINTELLIGENCER 2007 SpringerScience+Business Media, Inc. anew, producing several additional re- Vandiver devised further methods to The most significant progress in cal- sults along the same line of ideas. Thus simplify and speed up the procedures, culations related to FLT resulted from one finds contributions by such leading and also to allow for double-checking. Vandiver's work with the couple Der- figures as Philipp Furtw~ingler and At the same time, however, he was rick Henry Lehmer (1905-1991) and Georg Ferdinand Frobenius, but also by skeptical about the general validity of Emma Lehmer (1906-2007). Their col- the then unknown Vandiver. In 1914, Case II of FLT. Eric Temple Bell wrote laboration started in 1932, though the in his first article on FLT, 28 he proved to him in 1929, first joint publication did not appear un- a Wieferich-like congruence for 5p-1. If I remember rightly, you once said til 1939, when they proved the validity Vandiver's first truly substantial result that you would not be surprised if of FLT for 2 < p < 619. -~5 Above 619, the came in 1920, when he identified a mis- the second case turned out to be calculations became prohibitively long take in Kummer's article of 1857 and false .... You give the limit five hun- and laborious to be carried out with the went on to correct one of its main ar- dred for exponents to be tried. I have kind of desktop calculators available to guments. 29 He continued to refine and no idea of the actual amount of con> the Lehmers. But in 1953 when elec- develop his ideas on FLT over the next putation required for such an under- tronic computers became available, the few years. A summary of his achieve- taking, but I should think it would three mathematicians took a great leap ments appeared in his authoritative arti- be terrific. There is no doubt in my forward, proving that FLT was true for cle of 1929, 3~ for which he was awarded mind that anyone who knows any- all exponents p< 2000. 36 Throughout the AMS's first Cole Prize in number the- thing about the Theory of Numbers their correspondence, Vandiver stressed ory two years later. This award, honor- would say that this work ought to be that, beyond the specific results ob- ing the AMS's long-time secretary Frank done while there is a man not only tained, this research had an enormous Nelson Cole, was established for out- able to do it, but also willing. If in value for advancing research on cyclo- standing work in this field. one of these exponents the compu- tomic fields. In subsequent papers, he tations should give a negative result, further refined the Kummer criteria for Vandiver's Contributions to FIT you will set a problem to exasperate irregular primes, and this led to an ex- Vandiver's article of 1921 went well be- generations of arithmeticians. I rather tension of the results on FLT to 2000 < yond Kummer's results by proving FLT hope that it does turn out that way. 31 p < 2520 in 1954 and then, in 1955, to for exponents up to p = 211. Even be- In the next few years, however, Van- p in the range 2520 < p < 4002. 37 fore it appeared in print, Vandiver re- diver would surpass the exponent 500 This work of Vandiver and his col- alized that his arguments could be used and would continue to confirm the va- laborators using electronic computers to extend the results to p < 269. Under lidity of FLT for ever higher values of did not, however, alter mainstream re- his direction, specific calculations of p, including those covered by case II. 32 search in number theory, at least not in various ranges were performed sepa- In an article in 1934 he remarked that the short run. Nor did it lead to renewed rately by various M.A. students at much of his "work concerning FLT is research efforts in connection with FLT. Austin, including Samuel Wilks (1906- tending toward the possible conclusion Still, seen in retrospect, these pioneer- 1964) and Elizabeth Stafford (1902- that if the second factor of the class ing efforts opened the door to a new 2002). All calculations of values p, number" hi, of K(~)is prime to p, "then direction of research that remains active 100 < p< 211, were performed by FLT is true." This is the famous "Van- today. Additional results along similar Wilks "using Monroe and Marchant diver conjecture," about which he had lines have continued to confirm FLT for electrical computing machines." Van- begun to speculate much earlier. Its im- exponents exceeding one billion, and diver and his team proved that if p di- portance for algebraic number theory in in Case I for higher values still. In fact, vides only one of the numbers t32, B4, general gradually gained recognition even after Wiles's general proof of FLT, .... B/,-3, and if this single Bernoulli over the years, albeit in somewhat mod- new ranges of exponents are still being number is not divisible by p~, then FLT ified versions. -s3 This was by no means tested with ever improved techniques. 3~ is valid for p. This allowed further cal- the only original conjecture that ap- In 1946, following a request from the culations for exponents p < 307. As the peared in his articles, however, as was editors of the American Mathematical difficulty of the calculations increased, pointed out in later research. 34 Monthly, 39 Vandiver published a de-

28[Vandiver 1914]. 29[Vandiver 1920, 1922]. 3~ 1929]. 31 Bell to Vandiver: Jan 15, 1929 (HSV). 32For a detailed account of Vandiver's works during these years and how they eventually led to the use of electronic computers to solve FLT, see [Corry 2007]. 33See, for instance, [Iwasawa & Sims 1965l. [Lang 1978, 142] pointed out that the conjecture had originally been formulated by Kummer [Coll. Vol. 1, 85]. Lang indi- cated that "Vandiver never came out in print with the statement: "1 conjecture etc .... ", but "the terminology 'Vandiver conjecture' seemed appropriate to me. In any case I believe it". 34[Herstein 1950, Denes 1952]. 35[Vandiver 1937, 1937a]. 36[Vandiver, Lehmer & Lehmer 1954}. 37[Vandiver 1954, Vandiver, Selfridge & Nicol 1955]. 38[Wagstaff, 1978; Buhler et aL 1992; Buhler et al. 2001]. 39Lester R. Ford to Vandiver: February 2, 1945 (HSV).

2007 SpringerScience+Business Media, inc., Volume29, Number3, 2007 35 tailed exposition of the state of the art which is now of fundamental impor- As late as the early 1960s Vandiver in research on Fermat's problem. 4~ This tance in many parts of mathematics. was still publishing new results related article became a classical locus of ref- The remarkable character of Kum- to FLT. He also continued to work on erence for many years to come. In sum- mer's achievement has tended, how- a book about FLT and related topics in marizing his opinion about the general ever, to minimize the great number number theory, a project he pursued for validity of the conjecture, about which of connections which the theorem many years. His archive contains hun- he was frequently asked, Vandiver drew has with other subjects. Efforts on my dreds of typewritten pages with whole a clear distinction between the two clas- part to clear up the question have chapters nearly ready for publication, sic cases. He was convinced of the va- led me into the following topics: but for some reason this book was lidity of case I, but not merely because Bernoulli numbers and polynomials never published. it had been proved for very high val- and generalizations; Euler and Gen- ues. Rather, his confidence in this case nochi numbers; Euler and Mirimanoff Correspondence on FLT stemmed from some important theo- polynomials; partitions modulo m; As Vandiver came to know, having your rems he had proved along the way on finite fields and rings, including a name publicly attached to Fermat's trinomial congruences--a topic to great many types of congruences: problem could impose a considerable which he had devoted many efforts. the Dirichlet Zeta Function and the burden on a mathematician. Yet some Case II involved a much more complex related Dedekind Function; the La- experts found efficient ways to duck the situation; thus, while he believed it grange resolvent and Jacobi 4~ num- unwelcome task of reading the steady would ultimately be proven, he did not ber and various generalizations in- stream of faulty proofs submitted by think he had any compelling evidence cluding generalized Gauss sums; the rank amateurs. In the early twentieth to support it. Furthermore, he stated, he theory of Kummer fields, class num- century, Edmund Landau came up with felt less sure than in 1934 about the va- ber, class fields, power characters a nearly ideal solution to this corollary lidity of the Vandiver conjecture, pre- and laws of reciprocity in the theory to Fermat's problem. As G6ttingen's cisely because of its close relationship of algebraic fields; Fermat's quotients leading authority on number theory, with and possible dependence on the and other arithmetic quotient forms; Landau was officially entrusted with validity of FLT. Commenting on the fre- congruence theories as applied to handling all correspondence related to quency with which apparently promis- power series; abstract algebra includ- the Wolfskehl Prize, which offered ing conjectures in number theory are ing, particularly, group theory and 100,000 Marks to anyone who could eventually abandoned, he added: semi-groups; and many types of Dio- solve Fermat's conjecture. Landau had When I visited Furtwtingler in Vienna phantine equations aside from the little interest in the problem, so he took in 1928 he mentioned that he had Fermat relation itself. on this duty with little enthusiasm. conjectured the same thing before I It seems Vandiver was rather carried When the flow of incoming correspon- had brought up any such topic with away with enthusiasm. Only some of dence from amateurs eventually be- him. As he had probably more ex- these topics have substantial, direct con- came unbearable, he became openly perience with algebraic numbers than nections with FLT. On the other hand, disgusted. So he prepared a form letter any mathematician of his generation, Vandiver himself was led to explore that looked something like this: I felt a little more confident. (p. 576) many of these potential payoffs, partly Dear ...... , While mentioning some additional re- because of his interest in questions that Thank you for your manu- suits presented in his current account, arose from, or were thought to be use- script on the proof of Fermat's Last he echoed the opinion voiced several ful for solving FLT. In fact, in 1952-53 Theorem. years earlier in Bell's letter: he published a two-part article on as- The first mistake is on: However it would probably be best sociative algebras and the algebraic the- Page ...... Line ...... if I were wrong about this. I can think ory of numbers, a paper he regarded as This invalidates the proof. of nothing more interesting from the more important and innovative than any Professor E. M. Landau standpoint of the development of of his work directly connected with FLT. An assistant read through the manu- number theory, than to have it turn He was disappointed that this article scripts and filled in the missing details out the Fermat relation has solutions, was seldom cited: in the form letter. for a finite number > 0, of primes /. As far as I know, only one person Fortunately for Vandiver, no rich oil- Concluding, he wrote, has studied thoroughly this paper, man came along to establish a similar Many mathematicians are often in- and he is Alonzo Church. 41 prize fund at the University of Texas for terested in ascertaining how a par- In private correspondence Church a successful proof of FLT. So this cir- ticular topic connects up with other raised some interesting criticisms re- cumstance surely diminished the num- parts of mathematics. In case of Fer- garding the axiomatic debate devel- ber of would-be problem solvers who mat's Last Theorem it is well known oped by Vandiver, and Vandiver was might have written to him. Nevertheless that Kummer's attempts to prove it quick to include Church's comments in he did receive enormous quantities of gave rise to the theory of ideals a follow-up to this article. mail that only grew from year to year.

4O[Vandiver 1946]. 4Wandiver to Mientka: March 13, 1964 (HSV). See also, Vandiver to Herstein, April 2, 1960.

36 THE MATHEMATICALrNTELLIGENCER 2007 Springer Science+Business Media, Inc. In fact, many American mathematicians Born in 1899 in Haiti, he had received that I had an error in my argument. (perhaps all of them?) saw in Vandiver in 1937 a Ph.D. at the Sorbonne, work- After a time I became more and the default address to which any letter ing with Arnaud Denjoy (1884-1974). In more skeptical of any apparent on the topic should be redirected. Van- Haiti he had been professor of mathe- proof that I found using such ele- diver's attitude toward these corre- matics and physics, and director of the mentary means, as I felt that if an spondents was essentially positive, per- Haitian Statistical Institute, and had en- elementary proof existed, it could haps because he had been something joyed a very successful career in the fi- hardly have escaped the attention of of an amateur mathematician himself. nance sector. He was Ambassadorial such great mathematicians as Euler, In response to attempted proofs by rank Representative of Haiti to the Organi- Legendre, Lam~, Abel, Gauss, amateurs, he sent a pre-written, but zation of American States, and later Cauchy, and Kummer, all of whom rather polite reply. His archives contain served in the Ministry of External Af- worked at the problem! . . . no fewer than 225 such answers, sent fairs. As usual, Vandiver read the man- I have looked over the general between 1934 and 1966. To those he uscript fairly carefully and replied po- character of your argument.., and considered qualified mathematicians he litely and in some detail. In his answer, as far as I can see . . . you have usually answered in some detail, though he summarized his general attitude to- used nothing but elementary alge- even in these cases the task became in- ward this matter: bra therein, hence I cannot help be- creasingly onerous with time. Next October 21 I shall start my 80th ing skeptical as to the accuracy of An interesting letter from 1949 attests year of age. Beginning in the year your work. If you regard this as a to this problem in the case of a math- 1914 I published several articles on disparagement of your work, please ematician, Taro Morishima, whose con- the Fermat problem which received now that I have just disparaged tribution Vandiver truly appreciated. He attention from readers to the extent above all rny own e./'forts of this char- was forewarned that Morishima was that many of them wrote letters to acler. 44 about to submit an article on FLT to an me, generally containing their opin- Vandiver further advised Hibbert to American journal. This prompted him ions.., about the problem, and also write up full proofs for various specific to take preemptive action by contacting what they regarded as proof of Fer- cases, to see if they worked. And he several editors (Aurel Wintner of the mat's statement or contributions to added: "In giving you this advice, I am American Journal of Mathematics, that end. For some years I made a assuming that you would prefer to find Rudolph E. Langer, Saunders Mac Lane, practice of replying to such letters the error yourself, if one exists, than to and ) to request that the and giving my estimates of the value have someone else find it." Then, in a article not be sent to him. He would of their work. However, as I con- letter to Stone he explained what he certainly like to read this paper, he said, tinued to publish from time to time really feared about cases like this one: but at his leisure and not under pres- through the years articles pertaining For many years, in connection with sure to finish within some given period to the Fermat problem, my corre- "proofs" of FLT sent to me and of time, however reasonable. Number spondence along that line became which I examined it turned out in theory, he added, "seems to be getting so heavy that if I had continued to nearly every case that if I called the popular," but Vandiver felt he was do this it would have taken most of author's attention to an error in his drowning under the enormous corre- my time .... As an example of this, work, soon after I would receive an- spondence he now had to handle. 42 A I have received four letters within other ms. which he assumed was a few months later, he again complained the last few weeks and about seven correction of his original paper. bitterly about this to another colleague, since the first of the year pertaining Also, if instead of pointing out an while requesting that no f\trther letters to the Fermat problem . . . one of error I would merely state to him be sent to him. Only if he received a them . . . said the full proof of the that there was a step in his argu- manuscript from Siegel, Hasse, or theorem covered ab 50 pages. In ment which I did not understand, Rademacher would he be willing to ex- his resume of the paper given me in then the author would reply that I amine the work in detail. To which he his letter, he made a number of did not understand his [entire] argu- added: "After nearly forty years of look- statements I could not understand at ment. These things would be the be- ing at such manuscripts, good and bad, all: . . . so I told him I was sorry I ginning of a long correspondence don't I deserve a respite? ''43 had to refuse to help him .... that I would have with him. 45 One revealing interchange took Some years prior to Hibbert, Van- place in 1961 around a proposed proof In my 50 years' experience with the diver was involved in another notewor- of FLT by Lucien Hibbert, then Execu- problem I have often been con- thy exchange with a Pakistani air force tive Director of the Inter-American Bank vinced for a time that I had a proof officer named Quazi Abdul Moktader fk)r Development in Washington, D.C. of the theorem using only the tools Mohd Yahya, who was formerly "Pro- Hibbert had been directed to Vandiver of elementary number theory and al- fessor of Mathematics at Brajali Acad- by Israel Herstein and Marshall Stone. gebra, but I found in eve~ such case emy, East Pakistan." In various letters

42Vandiver to various: December 8, 1949 (HSV). 43Vandiver to J. R. Kline: January 12, 1950 (HSV). 44Vandiver to Hibbert: June 23, 1961 (HSV). Emphasis in the original, 45Vandiver to Stone: June 26, 1961 (HSV).

2007 Springer Science+Business Media, Inc., Volume 29, Number 3, 2007 37 written to colleagues about this man, ing these cases, it is apparent that Vandiver referred to him as X, noting the deviation between the sum of that "I do not wish to be sued for libel, the terms in the left-hand member in case the information in this letter of the equation and that of the right- somehow reaches him." He received a hand member increases steadily manuscript from him on FLT that the au- with higher exponent value. There- thor wished to submit to the Proceed- fore, I feel that it is only necessary ings of the National Academy of Sci- to prove n = 3 because this is the ences. As in other cases, Vandiver point of lowest deviation. Any ex- answered the initial letter politely, but ponent value above this is immedi- this led to a lengthy and futile series of ately ruled out as a result of the fact interchanges. Vandiver tried to put an that the deviation is greater than that end to this by suggesting that Yahya of the third power thus making it send his manuscript to a "regular math- impossible to suit the equation. 48 ematical journal," one "preferably in Vandiver, who had written several po- Switzerland or Germany where they lite and possibly helpful letters to Joel seem to have more interest in number along the way, also reacted politely to theory than in the U.S." He feared that Joel's conclusion: most mathematicians, this "dangerous character" might "write he kindly remarked, would not agree Figure 3. Harry S. Vandiver (Creator: me a threatening letter, as some of these with the closing statement of his paper. Walter Barnes Studio (HSV). birds have done in the past. ''46 Eventu- ally Yahya was able to publish his (ob- Recognition and Oblivion viously flawed) proof in a Portuguese In the currently available literature, Van- ematicians in 1926 for the Rockefeller journal in 1976. 47 diver's name is barely mentioned in foundation, Vandiver, then 44 years old, A third interesting correspondence connection with FLT. For instance, in was not on his list. 49 Even during his over FLT took place in 1960-61 when the popular "MacTutor History of Math- most creative phase as a researcher he Vandiver was contacted by a junior high ematics Archive" website, Vandiver seems to have received less recognition school pupil from Baltimore named Joel barely rates a very short entry of his than he probably deserved. Weiss. After learning the names of the own. His name appears only in passing Yet Vandiver received several high three persons in the US who had re- in the site's article on FLT, and he is honors, including the Cole Prize and an cently done work on the Fermat prob- not mentioned at all in the article on honorary doctorate from the University lem, Joel wrote to Vandiver (and also to Dick Lehmer. From the point of view of Pennsylvania; and, of course, he was the Lehmers) for advice on this topic, of current mathematical research asso- the recipient of many research grants. which he had chosen for a school term ciated with the problem, especially fol- Harry Vandiver was the only American paper. He was willing to work hard, and lowing Wiles's dramatic breakthrough, mathematician whose work received so he explained his choice as follows: this may be understandable. But from mention in Edmund Landau's 1927 clas- This theorem, which originally was the point of view of the history of the sic textbook on number theory. He was a curiosity to me, turned out to be problem, this lack of recognition is com- elected vice-president of the AMS for the a stimulating research project well pletely unjustified, though the reasons term 1933-1935, and in 1935 he was an worth the 45 hours of work neces- for this are not difficult to find. AMS Colloquium Lecturer. He served as sary to complete it. I hope that my Although Vandiver was the undis- assistant editor of the Annals of Math- conclusion will start a new train of puted world's leading expert on FLT ematics from 1926 to 1939, and in 1934 thought leading to an eventual proof during his lifetime, contemporaries of- he was elected to the National Academy of Fermat's Last Theorem. ten took an ambivalent attitude toward of Sciences. Still, he always remained Joel later indicated at the end of his fin- him and his passionate quest. Certainly part of a small and rather marginal ished paper what this desired train of he was well-known and respected both sub-community within the larger Amer- thought might be: within the American mathematical com- ican mathematical research enterprise. I conclude that Fermat's Last Theo- munity and abroad, but his interests Strongly fixated on his own work, he rem has been proven all this time, were also viewed as exotic, and evi- was certainly not a shaker and mover. and that its entire proof is that of dence abounds that he was viewed as He would not manage to attract large n = 3. I have reached this conclu- more bizarre than brilliantly original. numbers of young researchers to his sion from an analyzation of a suc- Thus, it is not surprising that when his chosen field; he did not establish a re- cession of cases of the theorem with friend G. D. Birkhoff prepared a list of search school, nor did he develop an exponents 3 through 9. After study- the 10 most prominent American math- influential network of contacts with like-

46Vandiver to Hayman: April 3, 1958 (HSV). 47Mathematical Reviews lists a "private edition" by the author [Yahya 1958], and three additional articles in Portugaliae Mathematica (1973, 1976 and 1977). 48The entire correspondence appears in HSV: File 16-3. 49See [Siegmund-Schuitze 2001, 51]. Birkhoff's list included only mathematicians from three leading centers: Cambridge (Birkhoff, Morse, Osgood, Wiener, Whitehead); Chicago (Bliss, Dickson, E. H. Moore, Moulton); and Princeton (Alexander, Eisenhart, Lefschetz, Veblen).

38 THE MATHEMATICALINTELUGENCER 9 2007 SpringerScience+Business Media, Inc. minded mathematicians. Nor was he an organizational talent who excelled when it came to promoting journals or organizing professional meetings. The honors conferred on Vandiver occasionally betray ambivalence. For example, only after Vandiver himself applied some direct pressure on uni- versity authorities was he named Dis- tinguished Professor at TU in Austin, in 1947. But his title, "Distinguished Pro- fessor of Applied Mathematics and As- tronomy," was certainly odd given his research expertise. More telling still is the context sur- Figure 4. AMS-MAA meeting in Washington D.C. (HSV). Source: Capi- rounding a Festschrifi published in his tol Photo Services, Inc. honor. In 1966 Bellman's Journal of and Applica- tions brought forth the special issue dedicated to Vandiver on his eighty- third birthday. The editors wished to honor him not only for his contributions to FLT and algebraic number theory but also because "he has profoundly influ- enced the development of American mathematics for a period of over sixty years." And yet the American contribu- tions to this volume were all written by his former students and close collabo- rators. Side by side with these papers one finds a score of others written by leading number-theorists from abroad, figures such as Mordell, Hasse, Erd6s, Szemeredi, Gel'fond and Morishima. It's odd that such a collection appeared in a journal far removed from Vandiver's Figure 5. Joel Weiss with a poster presentation of his work on FLT (HSV). own fields of interest. Evidently the de- cision to publish such a Festscbrifi came from close friends who wanted to pay long-overdue tribute to the man and his ally led to the general proof of FLT, and quote. Pictures are reproduced and work, yet sensed that no one outside while opinions may vary as to the in- sources are quoted with permission, us- Vandiver's inner circle would ever un- trinsic mathematical significance of the ing the following abbreviations: dertake it. The honoree, then in deli- ideas developed in his work, one can- HSV= Vandiver Collection, Archives of cate health after undergoing surgery, not make sense of the history of FLT American Mathematics, Center for was deeply touched by this gesture. 5~ without giving prominence to the story American History, The University of Vandiver's lifetime endeavor was of this man, the only one ever to de- Texas at Austin. characterized by remarkable indepen- w)te his entire professional life to solv- MOHP= Oral History Project, The dence and a willingness to pursue self- ing the problem. Legacy of R.L. Moore, Archives of styled, original research programs. As a American Mathematics, Center for researcher, his style was marked by an ACKNOWLEDGMENTS American History, The University of indefatigable appetite for endless cal- Albert C. Lewis and David Rowe read Texas at Austin. culations, by a peculiar style of collab- earlier versions of this article. I thank HUG: George David Birkhoff Papers, oration with small groups of people them for the critical remarks which led Harvard University Archives: Call Num- who were close to him, and by his pi- to significant improvement. ber HUG 4213.2, Box 3, Folder "T-V". oneering use of electronic computers in I have used archival material found in AP$; American Philosophical Society his fields of expertise. While Vandiver's several institutions. I thank the archivists Archive. contributions played no direct role in for assistance in locating and copying the GFA: The John Simon Guggenheim shaping the train of ideas that eventu- originals, and for granting permission to Memorial Foundation Archive.

5~ W. Baker to Bellman: September 29, 1965 (HSV).

92007 Springer Science+Business Media. Inc.. Volume 29. Number 3. 2007 39 REFERENCES Kummer, Ernst E. (Coil) Collected Papers (ed. -- (1934), "Fermat's last theorem and the Birkhoff, George David, and Harry S. Vandiver by Andre Well), Berlin, Springer-Verlag (1975). second factor in the cyclotomic class num- (1904), "On the integral divisors of ah--bh, '' Lang, Serge (1978), Cyclotomic Fields, New ber," Bull AMS 40, 118-126.

Ann. Math. (2) 5, 173-180. York, Springer-Verlag. -- (1937), "On Bernoulli Numbers and Fer- Buhler, J.P., R. E. Crandall, and R. W. Som- Lehmer, Derrick H. (1973), "Harry Schultz mat's Last Theorem," Duke Math. J. 3, polski (1992), "Irregular primes to one mil- Vandiver. 1882-1973," Bull. AMS 80, 817- 569-584. lion," Math. Comp. 59, 717-722. 818. (1937a), "On Bernoulli numbers and Buhler, J.P., R. Crandall, R. Ernvall, T. Met- Lewis, Albert C. (1989), "The Building of the Fermat's last theorem (second paper)," Duke s~tnkyl& and M. Shokrollahi (2001), "Irregu- University of Texas Mathematics Faculty, Math. J. 3, 418-427. lar primes and cyclotomic invariants to 12 1883-1938," in Peter Duren (ed.)A Century -- (1946), "Fermat's Last Theorem," Am. million," J. Symbolic Comput. 31, 89-96. of Mathematics in America--Part III, Provi- Math. Mo. 53 (1946), pp. 555-578.

Corry, Leo (2007), "FLT Meets SWAC: Van- dence, RI, AMS, pp. 205-239. -- (1954), "Examination of methods of attack diver, the Lehmers, Computers and Number Ohm, Martin (1840), "Etwas 0ber die Bernoul- on the second case of Fermat's last theorem," Theory," Annals of History of Computing li'schen Zahlen," Jour. reine u. angew. Math. Proc. Natl. Acad. Sci. USA 40, 732-735.

(Forthcoming). 20, 11-12. -- (1963), "Some of my recollections of -- (2008), "Number Crunching vs. Number Parker, John (2005), R.L. Moore. Mathemati- George David Birkhoff," Jour. Math. Analysis Theory. Computers and Number Theory. cian & Teacher, Washington DC, Mathemat- and Applications 7, 271-283. Computers and Number Theory from Kum- ical Association of America. Vandiver, Harry S., Derrick H. Lehmer, and mer to SWAC," Archive for History of Exact Siegmund-Schultze, Reinhard (2001), Rocke- Emma Lehmer (1954), "An application of Science (Forthcoming). feller and the Internationalization of Mathe- high-speed computing to Fermat's last Denes, Peter (1952), "Beweis einer Vandi- matics between the Two World Wars, Basel theorem," Proc. Natl. Acad. Sci. USA 40, ver'schen Vermutung bezQglich des zweiten and Boston, Birkhauser. 25-33. Falles des letzten Fermat'schen Satzes," Vandiver, Harry S. (1914), "Extensions of the Vandiver, Harry S., John L. Selfridge, and Acta Sci. Math. Szeged 14, 197-202. criteria of Wieferich and Mirimanoff in Con- Charles A. Nicol (1955), "Proof of Fermat's Goldstein, Catherine (1994), "La th6orie des nom- nection with Fermat's Last Theorem," Jour. last theorem for all prime exponents less than bres dans les notes aux Comptes Rendus de reine u. angew. Math. 114, 314-318. 4002," Proc. Natl. Acad. ScL USA 41,970-

I'Academie des Sciences (1870-1914): un pre- -- (1920), "On Kummer's Memoir of 1857 973. mier examen," Riv. Stor. Sci. 2, 137-160. Concerning Fermat's Last Theorem," Proc. Vandiver, Harry S., and George E. Wahlin Greenwood, Robert E. et al. (1973), "In Natl. Acad. Sci. USA 6, 266-269. (1928), Algebraic Numbers--ft. Report of the

Memoriam. Harry Schultz Vandiver, 1882- -- (1922), "On Kummer's memoir of 1857, Committee on Algebraic Numbers, Wash- 1973," Memorial Resolution, Documents concerning Fermat's last theorem (second ington, DC, National Research Council. and Minutes of the General Faculty, The Uni- paper)," Bull AMS. 28, 400-407. Wagstaff, Samuel S. (1978), "The irregular versity of Texas at Austin, 1974, 10926- -- (1929), "On Fermat's Last Theorem," primes to 125000," Math. Comp. 32 (142), 1094O. Trans. AMS 31,613-642. 583-591. Herstein, Israel. (1950), "A Proof of a Conjec- -- (1930), "Summary of results and proofs Yahya, Q. A. M. M. (1958), Complete proof ture of Vandiver," Proc. AMS 1, 370-371. on Fermat's last theorem (fifth paper)," Proc. of Fermat's last theorem. With a foreword Iwasawa, Kenkichi, and Charles Sims (1965), Natl. Acad. Sci. 16, 298-305. by Dr. Razi-Ud-Din Siddiqui. Available from

"Computation of Invariants in the Theory of -- (1930a), "Summary of results and proofs the author, Pakistan Air Force, Kohat, West Cyclotomic Fields," J. Math. Soc. Japan 18, on Fermat's last theorem (sixth paper)," Proc. Pakistan (14 pp. Mimeographed appendix, 86-96. Natl. Acad. Sci. USA 17, 661-673. 3 pp.).

LEO CORRY is head ofthe Cohn Institute. His latest book, Dovid Hilbert dnd the AxiomotJzotion of- Physics, 1898-1918 (Kluwer), was published in 2004. His current research interests include the history of FLT and of computational approaches to number theory.

Cohn Institute for History and Philosophy of Science and Ideas TeI-Aviv University 69978 TeI-Aviv Israel e-mail: corr~post.tau.acil

40 THE MATHEMATICAL INTELLIGENCER 9 2007 Springer Science+BusinessMedEa, Inc.