Chapter 8. Numbers and Number Theory in Modern Mathematics
Beginning with the work of Fermat in the seventeenth century, number theory has
become ever more esoteric and theoretical, developing connections with algebra and
analysis that lie very deep and require many years of study to master. Obviously,
we cannot explain in any satisfactory detail what has happened in this area in
recent years. For that reason, we shall carry the story forward only as far as the
beginning of the twentieth century.
A second topic that we must discuss before leaving the subject of numbers is
the variety of invented number systems, starting with the natural positive integers.
The number zero, negative numbers, and rational numbers do not require a long
explanation, but we need to focus in more detail on real and complex numbers and
the cardinal and ordinal arithmetic that came along with set theory.
Finally, mere counting turns out to be very difficult in some cases; for example,
given twelve points on a circle, each pair of which is joined by a chord, into how
many regions will these chords divide the circle if no three chords intersect in a
common point? To solve such problems, sophisticated methods of counting have
been developed, leading to the modern subject of combinatorics. A survey of its
history concludes our study of numbers.
1. Modern number theory
We are forced to leave out many important results in our survey of modern number
theory. Dickson's summary of the major results (1919, 1920, 1923) occupies 1600
pages, and an enormous amount of work has been added since it was published.
Obviously, the present discussion is going to be confined to a few of the most
significant authors and results.1.1. Fermat. Pierre de Fermat (1603-1665) was a lawyer in Toulouse whose avid
interest in mathematics led him to create, in his spare time, some analytic geometry,
calculus, and modern number theory. According to one source book (Smith, 1929,
p. 214), he was "the first to assert that the equation x^2 — Dy^2 = 1 has infinitely
many solutions in integers." As we have seen, given that it has one solution in
integers, Brahmagupta knew 900 years before Fermat that it must have infinitely
many, since he knew how to create new solutions from old ones. It was mentioned
above that Fermat wrote in the margin of his copy of Diophantus that the sum of
two positive rational cubes could not be a rational cube, and so on (Fermat's last
theorem). Although Fermat never communicated his claimed proof of this fact,
he was one of the first to make use of a method of proof—the method of infinite
descent—by which many facts in number theory can be proved, including the case
of fourth powers in Fermat's last theorem. A proof of the case of fourth powers187