The History of Mathematics: A Brief Course

192 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS

Street in Gottingen named after Gauss.

The 1801 work Disquisitiones arithmeticae became a classical work on the prop-
erties of integers. One the earliest discoveries that Gauss made, when he was still a
teenager, was a proof of the law of quadratic reciprocity. This proof was published
in the Disquisitiones, and over the next two decades he found seven more proofs of
this fundamental fact. The Disquisitiones also contain a proof of the fundamental
theorem of arithmetic and a construction of the regular 17-sided polygon, which is
possible because 17 is a Fermat prime.
A considerable portion of the Disquisitiones is devoted to quadratic binary
forms, in an elegant and sophisticated treatment that contemporaries found difficult
to understand. As Weil says (1984, p. 354),

No doubt the Gaussian theory... is far more elaborate [than Le-

gendre's treatment of the subject]; so much so, indeed, that it

remained a stumbling-block for all readers of the Disquisitiones

step forward. Legendre's estimate for this number, which is now denoted ð(ç) (ð
for prime, of course), was

ç ç / 1.08366 1.08366 2 ë

~ logn - 1.08366 ~ logn. ^ + logn log 2 ç ")'

Here the logarithm is understood as the natural logarithm, what calculus books
7r(n)
usually denote Inn. In particular, the ratio —— tends to 1 as ç tends to
n/ logn
infinity. Legendre did not have a proof of this result, but merely conjecturing it
was an important advance in the understanding of primes.
Legendre also worked on the classification of real numbers; his contributions to
this area are described below.
Number theory blossomed in the nineteenth century due to the attention of
many brilliant mathematicians. Again, we have space to discuss only a few of the
major figures.

1.5. Gauss. Carl Friedrich Wilhelm Gauss (1777-1855), one of the giants of mod-
ern mathematics, lived his entire life in Germany. He studied at the University of
Gottingen from 1795 to 1798 and received a doctoral degree in 1799 from the Uni-
versity of Helmstedt. Thereafter most of his life was spent in and around Gottingen,
where he did profound work in several areas of both pure and applied mathemat-
ics. In particular, he worked in astronomy, geodesy, and electromagnetic theory,
producing fundamental results on the use of observational data (least squares),
mapping (Gaussian curvature), and applied electromagnetism (the telegraph). But
his results in pure number theory are among the deepest ever produced. Here we
look at just a few of them.