Fermat Meets SWAC: Vandiver, the Lehmers, Computers, and Number Theory Leo Corry Tel Aviv University This article describes the work of Harry Schultz Vandiver, Derrick Henry Lehmer, and Emma Lehmer on calculations related with proofs of Fermat’s last theorem. This story sheds light on ideological and institutional aspects of activity in number theory in the US during the 20th century, and on the incursion of computer-assisted methods into pure fields of mathematical research. The advent of the electronic digital computer than 100. Extending this result beyond 100 opened a new era of unprecedented possibil- involved, above all, straightforward (if tedious) ities for large-scale number crunching. Begin- computations. Yet, little work was devoted to ning in the late 1940s, these gradually increas- such computations before 1920, and even ing possibilities were duly pursued in many then, this remained an essentially marginal branches of science. Some of them, like trend within number theory. meteorology, geophysics, or engineering sci- Thus, when electronic computers started to ence, underwent deep and quick transforma- became available in the late 1940s, few tions. Pure mathematical disciplines such as mathematicians working in the core, ‘‘pure’’ number theory can be counted among the less fields of the discipline incorporated them in receptive audiences for these newly opened their research agendas. Even fewer did so for possibilities. One way to account for this Fermat’s last theorem. Harry Schultz Vandiver somewhat ironic situation is to examine the (1882–1973) was one of the few to do so. He main research trends that shaped progress in joined forces with the couple Derrick Henry the algebraic theory of numbers from the Lehmer (1905–1991), and Emma Lehmer second half of the 19th century on. Central (1906–2007). The Lehmers possessed the nec- to such trends was a conscious attempt to essary expertise that combined deep knowl- develop powerful conceptual tools for solving edge in both number theory and the use of theoretical problems ‘‘purely by ideas’’ and electronic computers. They also had the with ‘‘a minimum of blind calculations.’’ institutional connections that facilitated the Indeed, this became an ethos that gradually use of computing resources with SWAC (Stan- came to dominate most fields of pure mathe- dards Western Automatic Computer) on be- matics after 1930. half of this project. This conceptual approach was developed This article describes the work of these three by leading German mathematicians such as mathematicians in connection with Fermat’s Richard Dedekind (1831–1916) and David last theorem, and how they came to introduce Hilbert (1862–1943) on the basis of ideas that electronic computers to research on that first appeared in the work of the great Berlin problem. number-theorist, Ernst E. Kummer (1810– 1893), in the 1850s. But Kummer’s own work From Sophie Germain to Kummer actually features many massive computations Sometime after 1630, Pierre de Fermat wrote with particular cases that would eventually in the margins of a book that the equation xn + disappear from the algebraic theory of num- yn 5 zn has no nontrivial integer solutions bers by the turn of the 20th century. Among x,y,z,whenn . 2. He also claimed to have other things, Kummer’s research led in 1859 to found a proof for this fact, which the margins a famous result, namely, that Fermat’s last of his book were too narrow to contain. theorem is true for all prime exponents less Eventually, this unproved conjecture became 0 IEEE Annals of the History of Computing Published by the IEEE Computer Society 1058-6180/08/$25.00 G 2008 IEEE IEEE Annals of the History of Computing anhc-30-01-cor.3d 11/1/08 22:31:26 0 known as Fermat’s last theorem. Attempts to Adrian Marie Legendre (1752–1833) was a prove it, and indeed to find Fermat’s putative prominent French mathematician who was proof, were unsuccessful. The problem increas- among the first to acknowledge Germain’s ingly attracted the curiosity of mathematicians talents. He corresponded with her and tried to and amateurs alike, while becoming the object use her methods for proving additional cases. of many legends. The mythical dimension of Case II turned out to be much more difficult its story was greatly heightened after the grand from the beginning. For p 5 5, case II was finale provided by the brilliant, surprising, and proved only in 1825 in separate, complementa- highly complex proof, advanced in 1994 by ry proofs of Legendre and Peter Lejeune Dirich- Princeton mathematician Andrew Wiles that let (1805–1859). Dirichlet also proved in 1832 turned the theorem, and many names associ- case II for n 5 14, and he did so while trying to ated with it, into the focus of unusual public prove it for p 5 7. This latter case turned out to attention.1 But as a matter of fact, an attentive be especially difficult, and it was finally proved reading of the historical record shows that in 1839 by Gabriel Lame´ (1795–1870).3 from its very inception, this was an open The next significant contribution came conjecture to which few mathematicians (and from Kummer. His line of attack originated above all few outstanding number theorists) in his efforts to address what he considered to dedicated sustained research efforts worthy of be the most important question in number that name—the kind of attention they devoted theory, namely, the so-called higher reciproc- was mostly passive. ity laws. Kummer’s interest in Fermat’s last Born on the physical margins of a book, for theorem was only ancillary to this, but he more than 350 years Fermat’s theorem essen- relied on an important insight discovered as tially remained at the margins of the mathe- part of these efforts, namely the identification matics profession, brought occasionally into of a special class of prime numbers, later called the limelight before the work of Wiles.2 Still, ‘‘regular.’’ In 1850, Kummer proved that the the theorem’s history is important in many theorem is valid for all regular primes. senses, and this is also the case concerning the Kummer also provided an algorithm based question of the relationship between concep- on the use of so-called Bernoulli numbers, Bn, tual breakthroughs and intense calculation in in order to tell for a given prime number mathematics. whether or not it is regular. Using the values of In the early 19th century, some of the Bn known at the time, he worked out all the sporadic efforts toward proving the theorem computations necessary to see the only non- yield interesting results that set the scene for regular primes he found below 164 were 37, the vast majority of subsequent contributions 59, 67, 101, 103, 131, 149, and 157. He did not (prior to Wiles’ proof that eventually came go beyond 164, possibly because of the from a completely different direction). In this complexity and length of the calculations section, I cursorily present those results and involved. At any rate, Kummer initially be- the concepts associated with them, inasmuch lieved that there would be infinitely many as they are necessary for understanding the regular primes, and indeed that only a few starting point of Vandiver’s work. primes would be irregular. An early, significant result specifically aris- Kummer naturally asked himself how to go ing from attempts to prove the theorem was about the case of irregular primes. He brilliant- advanced by Sophie Germain (1776–1831). As ly developed three criteria that provided a a woman, Germain initially worked outside sufficient condition for the validity of Fermat’s the mathematical establishment of her time, last theorem for a given irregular prime p. and corresponded with Carl Friedrich Gauss Checking the criteria for any given p involves a using a pseudonym. The theorem that was considerable computational effort, but they do later associated with her name implied that yield clear results. Kummer was by no means Fermat’s last theorem can be fully elucidated intimidated by the need to make the necessary by handling the following two separate cases: calculations. And, indeed, in 1857 he pub- lished a famous article that broke new ground, N Case I—none of the integers x,y,z is divis- both conceptually and in terms of specific ible by p; calculations4 It introduced the three said N Case II—one, and only one, of the integers criteria and proved that each of the three x,y,z is divisible by p. irregular primes smaller than 100 satisfies them. He thus achieved the impressive result Germain proved case I for all primes under that Fermat’s last theorem is valid for all 197. exponents under 100. January–March 2008 0 IEEE Annals of the History of Computing anhc-30-01-cor.3d 11/1/08 22:31:27 1 Fermat Meets SWAC: Vandiver, the Lehmers, Computers, and Number Theory Kummer never published his calculations the second half of the 19th century, although nor explained any specific formula that per- these developments had little to do with haps facilitated these calculations. Clearly, the proving Fermat’s last theorem. Mainly under latter were lengthy and demanding. Indeed, the influence of an approach embodied in Kummer’s work turned out to contain some Dedekind’s work on the theory of ideals, new relatively minor inaccuracies, but this was ideas on the theory of algebraic fields gradually found out for the first time only in 1920 by developed in a direction that explicitly dis- Vandiver. It was clear at this point, at any rate, tanced itself form the kind of calculational that Kummer’s results might be extended with efforts developed by Kummer himself.