Still, there is a question of balance between these two aspects of mathematical activity, and the factors that affect this balance throughout history. It is this question that occupies the main focus of the present article. In the case of the examples mentioned above, one can immediately notice the different mathematical circumstances within which these tasks involving in their own ways intensive calculations might be justified. General perceptions about the need, and the appropriate ways for public scrutiny of science, its tasks and its funding, changed very much in the period of time between and , and this in itself would be enough to elicit different kinds of reactions to both undertakings.
But above all, it was the rise of e-commerce and the need for secure encryption techniques for the Internet that brought about a deep revolution in the self-definition of the discipline of number theory in the eyes of many of its practitioners, and in the ways it could be presented to the public.
Whereas in the past, this was a discipline that prided itself above all for its detachment from any real application to the affairs of the mundane world, over the last three decades it turned into the showpiece of mathematics as applied to the brave new world of cyberspace security. The application of public-key encryption techniques, such as those based on the RSA cryptosystem, turned the entire field of factorization techniques and primality testing from an arcane, highly esoteric, purely mathematical pursuit into a most coveted area of intense investigation with immediate practical applications, and expected to yield enormous economic gains to the experts in the field.
Cole, as far as we know, provided no explicit justification for the many hours spent on his pursuit. He was, after all, a man of few words at least according to Bell. Probably he felt no need to provide such a justification to begin with. He developed a life- long interest in computing devices, especially as applied to number theory. One of the various machines he was involved with was a photoelectric sieve he built in for factorizing integers and identifying prime numbers. Lehmer compared this short and accurate calculation to that of a man entrusted with performing the same task: each separate trial would take a man at least six minutes; assuming that the man would work ten hours a day it would take him a hundred thousand years, i.
Technologically speaking, Lehmer was closer to Wagstaff than to Cole, but his justification discourse was of a completely different kind, presumably adequate for Cole as well.
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There is a cowardly and sinking sort of a scientist, no doubt, who is ashamed or afraid to take a walk in the country with the avowed purpose of enjoying the landscape. He is, no doubt, merely trying to avoid the odium that seems to have attached itself to the poet or to the musician who is hard put to it to produce a healthy, bread-and-butter reason for making a sonnet or a symphony.
To listen to the apologists for the study of pure mathematics one would get the impression that this study is sustained, not by the Wonder or Beauty of the subject, but by its external utilities. But how little of the vast field of mathematics has to do with the study of the outside world! And, in addition, Lehmer ended his text by sounding a prophetic note that enhances the main thrust of his opinions.
He thus wrote:. The subtle and expensive determinations of the bending of a ray of light by a gravitational field, or the careful listing of the binary starts in the heavens, can have little application to the making of two squares where only one grew before. Faraday, playing with wires in his laboratory, wrests from the hands of nature a torch that Edison uses to light the world, and Einstein to light the universe. Who can tell? Lehmer passed away precisely at the time when it was becoming clear that the only miscalculation involved in a statement like his was that it would be enough to wait several decades, rather than centuries.
The deep change in the status of time-consuming computational tasks from Cole to Wagstaff, via Lehmer, provides an extreme, most visible example of the more general topic of this article, namely, the changing attitudes of mathematicians towards intensive computations with particular cases as part of the discipline of number theory from the second half of the nineteenth century on. By focusing on the cases of Mersenne primes and irregular primes I will discuss some of the factors that shaped these attitudes in various historical contexts. The section Mersenne primes contains an account of the early history of Mersenne primes up to Cole.
It provides an overview of the mathematicians involved in calculating such numbers and of their scopes of interests, as well as if their main methodological guidelines. The section Irregular primes focuses on work on irregular primes done by Kummer. In spite of their apparent conceptual proximity, these two fields of research in number theory, Mersenne numbers and irregular primes, developed in completely different ways.
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This is particularly the case when it comes to the question of massive computations performed in relation with each of them. The section Number theory and electronic computers provides a comparative overview of the stories of these two kinds of calculations and in doing so, it directs the focus of attention to the topics to be discussed in the following sections. The section The Lehmers, Vandiver, and FLT describes the early work of the Lehmers and the unique approach they followed in their number theoretical investigations, making them ideal candidates to taking a leading role in the early incursion of digital computers into number theory.
This incursion is described in the section The Lehmers, Robinson, and SWAC , after having discussed in the preceding section Traditions and institutions in number theory some institutional, ideological and technological aspects of the development of the discipline of number theory in the USA in the period considered, and the main changes that affected it.
This discussion provides a broad historical context for understanding the work of the Lehmers and its idiosyncratic character within the discipline. Their unique professional and institutional position facilitated a process that could otherwise have taken much longer to materialize, whereby massive calculations with digital computers were incorporated into number theory, first at the margins and gradually into its mainstream.
Nicomachus advanced many additional claims about the perfect numbers, such as, for example, that they all end alternately in digits 6 and 8, or that the n th perfect number has n digits. Many of these claims turned out to be wrong, and some were later proved to be correct. From the point of view of our account here it is important to stress, above all, that even at this early stage of the history of number theory we see two completely different kinds of emphasis embodied in the respective approaches of Euclid and Nicomachus to the same question: the former formulated the general principle and proved the general theorem, whereas the latter set out to look for specific instances of perfect numbers by calculating with particular cases.
This quest for individual instances in the hands of a Pythagorean like Nichomacus finds a clear explanation in his more mystical, rather than purely mathematical motivations. Although after Nicomachus we find discussions about perfect numbers in various sources, especially in the Islamic world, further instances of perfect numbers were discovered only much later, in fifteenth-century Europe. A main figure in this development was Pietro Cataldi — who by was aware of the primality of 2 13 — 1, 2 17 — 1, and 2 19 — 1 but was not the first to add numbers to the list of four numbers known in antiquity.
More importantly, he was the first to realize that if 2 n — 1 is prime, then n has to be prime. The study of what we call now Mersenne primes started as part of this same thread of ideas. Marin Mersenne — was a French Minim friar who became known in the history of mathematics for his role as a clearing house for correspondence between eminent philosophers and scientists—such as Descartes, Pascal and Fermat—as well as for his own enthusiastic interest in questions related with number theory. Like various others with a similar interest before him, Mersenne approached the question of the perfect numbers and of the primality of the factors 2 n — 1.
Mersenne was perfectly aware of the enormous difficulty involved in testing the primality of large numbers of 15 to 20 digits that appear in this context, and it is obvious that he did not actually check all the factors 2 n — 1 for the cases appearing in his list. It is thus all the more curious that he was so certain about his guess as expressed in that statement. By looking at a different text of Mersenne, historian Stillman Drake [Drake ] was able to determine the rule by which he apparently produced the list:.
Nonetheless the list is an amazing achievement not just because of the many insights it involves and the calculational effort involved in producing it, but also because the very long time that passed before its mistakes were first spotted.
This fact was not discovered before In his letters to Mersenne, as was his habitude, Fermat raised many interesting ideas, mentioned many general results proved along the way but usually without revealing his proof and proposed new problems to be solved. Using some of the general results he had proved, Fermat also addressed questions related with particular cases of Mersenne numbers. In a different letter, to Frans van Schooten — in , he proposed the challenge of proving or disproving this assertion.
He also showed, for example, that 23 was a factor of M 11 and that 47 a factor of M Such numbers F m are called Fermat numbers. Eventually, however, in Leonhard Euler — famously showed that F 5 is not a prime.
Among many other things, Euler proved a series of results that allowed him to identify, in many cases, factors of Mersenne numbers. These same results stood behind his proof that F 5 is not prime.
At the same time, he also proved that M 31 is prime, which remained the largest known prime until This result also implies that all even perfect numbers end in either a 6 or 8, but not alternately as stated by Nichomacus. Legendre further developed some of the new methods introduced by Euler to the discipline and used them to find factors of numbers that were quite large at the time, such as 10,, The factorization techniques he introduced had long lasting influence and they deserve a brief discussion here.
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If this congruence has solutions, then a is called a quadratic residue of p and it is called a quadratic nonresidue otherwise. Now, let us assume that for a integer N we can write. Using some additional, elementary properties of the quadratic residues, this property allows determining forms of primes p that are possible candidates for factors of N.
Both Legendre and Gauss wrote very influential textbooks that summarized the state of the art in the discipline and that shaped much of its subsequent development. Subsequent developments on activity related to Mersenne numbers and their possible factors has to be seen against the background of the processes unleashed in number theory by the publication of Disquisitiones Arithmeticae, and by the ways in which these processes left only little room for intensive computations as a main task in the discipline more on this below. And indeed, the person who appears next in our story, as the main contributor in the last third of the nineteenth century to calculations related with Mersenne numbers was not at the mainstream of academic mathematics of his time.
It was only relatively recently, however, that more focused attention was devoted to his research as an object of historical interest, as we see in the illuminating accounts of Hugh Williams  and, from a somewhat different perspective, of Anne-Marie Decaillot [, ]. Then he was artillery office at the Franco-Prussian war of —71, where he distinguished himself in the battlefield.
He published dozens of articles on these topics, which were mainly short research notes generally appearing in relatively minor journals. Among other things, Lucas is well known for the invention of the Tower of Hanoi puzzle [Williams , 65]. His important contributions to questions of factorization and primality testing were developed during a relatively short time he devoted to investigating this field of arithmetic, to Curiously, among the original motivations that led Lucas to his interest in number theory and particularly on prime numbers, questions related to industrial fabrics are prominent.
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In fact, Lucas even formulated a new proof of the reciprocity theorem using weaving-related concepts [Lucas ]. Lucas used a variant formulation of Gauss, which states that:.
Lucas investigated the primality of large numbers by looking at the sequence of Fibonacci numbers, for which he proved several results. From the table of Fibonacci numbers and its divisors Lucas discovered the following two properties:. He presented various successive proofs of this, each of which turned out to be mistaken in its own way. The result was correctly proved only in by Robert Daniel Carmichael — For reasons of space, the details of Lucas interesting calculations cannot be given here. Still, for the purposes of the present account, it is necessary to mention some of the main ideas related with it.
Lucas proved another, related result as follows:. It follows from here that in order to show that M is prime, it suffices to show that M v 2 Notice again: the primality of M is determined, not by checking whether or not this number is divided by certain factors, but rather by checking whether or not the number itself divides another, specified number which itself is typically very large.
This is the core of Lucas innovation. It reduces enormously the amounts of operations to be performed for testing an individual number, but not the complexity and length of the specific computations involved in that case. Now, M is a digit number.