The Development of Number Theory during the Nineteenth Century

Overview

Number theory—the study of properties of the positive integers—is one of the oldest branches of mathematics. It has fascinated both amateurs and mathematicians throughout the ages. The subject is tangible, and a great many of its problems are simple to state yet very difficult to solve. "It is just this," said the great nineteenth-century mathematician Carl Friedrich Gauss (1777-1855), "which gives number theory that magical charm which has made it the favorite science of the greatest mathematicians." Indeed, Gauss himself made seminal contributions to the subject, as did such other nineteenth-century greats as Lejeune Dirichlet (1805-1859), Ernst Kummer (1810-1893), Richard Dedekind (1831-1916), Bernhard Riemann (1826-1866), and Leopold Kronecker (1823-1891). Moreover, since the number-theoretic problems they tackled were very difficult, they often had to resort to "nonelementary" means—mainly algebraic and analytic—to deal with them. ("Elementary" methods are not necessarily simple; rather, they are merely methods that do not use advanced mathematics.)

Background

A supreme masterpiece about number theory that set the stage for the century's advances was Gauss's Disquisitiones Arithmeticae ("Arithmetical Investigations"), published in 1801 but completed in 1798—when Gauss was only 21! The title of his book refers to the fact that in previous centuries "number theory" was called "arithmetic." Pre-nineteenth-century number theory contained many brilliant results but often lacked thematic unity and general methodology. In the Disquisitiones Gauss supplied both. He systematized the subject, provided it with deep and rigorous methods, solved important new problems, and furnished mathematicians with new ideas to guide their researches for much of the nineteenth century.

The fundamental theorem of arithmetic, a cornerstone of the subject, states that every integer greater than 1 is a unique product of primes. Put another way, the primes are the (multiplicative) "building blocks" of the integers: products of primes will generate (uniquely) all the integers. This result was undoubtedly known to mathematicians of past centuries, but Gauss, in the Disquisitiones, was the first to state it formally and give a rigorous proof. Here also appears the first formal definition of the notion of congruence.

Since the primes are the "atoms" that make up the integers, to understand the latter it is imperative to understand the former. In 300 b.c. Euclid (c. 330-260 b.c.) proved that there are infinitely many primes. But how are they distributed among the integers? Do they follow a pattern? (The first 20 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79.) This question baffled mathematicians for centuries. Numerical evidence showed that the primes are spread out irregularly among the integers, in particular that they become scarcer—but not uniformly—as the integers increase in size. For example, there are 8 primes between 9991 and 10090 and 12 primes between 67471 and 67570. Furthermore, arbitrarily large gaps exist between primes: it is easy to produce a sequence of a billion consecutive nonprime integers. On the other hand, considerable evidence suggests that there are infinitely many pairs of primes as close together as can be, namely primes p and q for which q - p = 2 (they are called "twin primes"). This apparent irregularity in the distribution of primes prompted Leonhard Euler (1707-1783) in the eighteenth century to say: "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery which the human mind will never penetrate."

Euler's pessimism was, in an important sense, unjustified. It is true that there is no regularity in the distribution of primes considered individually; in particular, it is most unlikely that we could find a formula that will produce all the primes and only primes. But there is regularity in the distribution of the primes considered collectively. In fact, such regularity was later conjectured by Euler himself, and subsequently by Adrien-Marie Legendre (1752-1833) and Gauss.

In mathematics one must be able not only to give the right answers but to ask the right questions. Instead of looking for a rule that will generate successive primes, one might ask for a description of the number of primes in a given interval. Put differently, one might try to describe not "how" but "how often" the primes occur among the integers. Gauss and others made a conjecture about this issue, but it took close to a century to give a proof, mainly because basic tools were lacking that were developed during the nineteenth century. A major step toward the proof was taken in mid-century by Riemann, who introduced for this purpose what came to be known as the Riemann zeta function. While working on this problem, Riemann introduced a conjecture, still open 150 years later, that came to be known as the Riemann hypothesis. It is arguably the most celebrated unsolved problem in mathematics.

Building on Riemann's work, a proof of Gauss's conjecture was finally given at the century's end, independently by Jacques Hadamard (1865-1963) and Charles Jean de la Vallée-Poussin. The result is now known as the prime number theorem, a central result in number theory. It says that the number of primes less than or equal to x (x being a real number) is approximately equal (or asymptotic) to x/log x. In order to arrive at their proof Hadamard and de la Vallée-Poussin had to introduce important new ideas in complex analysis (the calculus of complex functions).

Two significant observations derive from these considerations. First, that it is often specific problems that motivate the development of theoretical results (in this case, it was attempts to prove the prime number theorem that motivated the introduction of important ideas in complex analysis). Second, that analysis—the study of the continuous—enters to resolve problems in number theory—the study of the discrete. This is surely a surprising phenomenon. In fact, several number-theoretic problems led in the nineteenth century to the founding of a new field, analytic number theory, which is to this day of great importance. (In a most surprising development, Paul Erdös and Atle Selberg proved the prime number theorem in the 1940s by "elementary" methods, without using complex analysis; the proof, however, was far from simple.)

As we mentioned, Euclid proved that there are infinitely many primes. Since 2 is the only even prime, this result can be rephrased to say that there are infinitely many primes in the arithmetic sequence 2n + 1 (n = 0, 1, 2, 3,...) consisting of the odd integers. In the 1830s Dirichlet proved a grand generalization of this result by showing that any arithmetic sequence an + b (n = 0, 1, 2, 3,...), namely b, b + a, b + 2a, b + 3a,..., contains infinitely many primes, with the obvious exclusion of the situation in which a and b have a common factor greater than 1 (in which case, of course, none of an + b is prime). To prove this result Dirichlet introduced important and farreaching ideas from analysis. Here was another celebrated example of analytic number theory.

Despite these triumphs, open problems abound in the distribution of primes. For example, is every even number greater than 2 a sum of two primes (as the evidence suggests)? Is there a prime between n² and (n + 1)² for every positive integer n? Undoubtedly, Euler's statement we quoted earlier about the mysterious nature of the primes has considerable merit.

Impact

Many other important concepts were introduced during the century as a result of work on number theory, and some of these influenced and/or gave rise to other branches of mathematics. For instance, the notions of integral domain, unique factorization domain, Dedekind domain, and ideal—adumbrated or introduced by Gauss, Kummer, Dedekind, and Kronecker—are important concepts in algebra. Their development led to yet another branch of number theory, algebraic number theory, in which the tools of algebra are brought to bear on the study of the integers. In fact, by the end of the century, the very term "integer" could no longer be used with impunity: there were now many types of integers—"ordinary" integers, Gaussian integers, and cyclotomic integers, to name but a few. From these and other developments, we can see that the scope and methods of number theory were enormously enlarged in the nineteenth century.

ISRAEL KLEINER