Fermat Cornes to America: Harry Schultz Vandiver and Flt (1914&
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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by amateurs who sent their "so- LEO CORRY lutions" to mathematicians all over the world but which (thankfully) remained unpublished, the number cannot even be estimated.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 mathematics 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 .... Sophie Germain 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 Andrew Wiles'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 number theory. 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.