In August 1991, the American Mathematical Society sponsored a Short
Course at the summer meeting in Orono, Maine, entitled "The Unreasonable
Effectiveness of Number Theory". Two years earlier, another Short Course
was held on "Cryptology and Computational Number Theory", which em-
phasized cryptologic applications. Therefore, the Short Course in Orono
concentrated on the great breadth of applications outside cryptology. This
volume is based on the lectures given at that Short Course.
Number theory is one of the oldest and noblest branches of mathematics;
indeed, it was already ancient in the time of Euclid. In fact, for almost all of
its history it has seemed to be among the purest branches of mathematics. It
is only within the last few decades that a large number of applications have
been encountered, at least by the mathematical community. The applications
to cryptology are now famous; but it is not as well known that number the-
ory has found an enormous number and variety of real-world applications
in many different fields. Indeed, the standard Mathematics Subject Classifi-
cation includes several codes devoted entirely or heavily to applications of
number theory.
What are the sources of these applications? The largest impetus has been,
not surprisingly, the computer, with its digital and numerical nature. As
with cryptology, the computer has been a driving force in the development
of algebraic coding theory, random number generation, raster graphics, com-
puter arithmetic, fast transforms, and many other areas. Perhaps surpris-
ingly, physics, in spite of its tradition of being continuous rather than dis-
crete, is another rich source of applications. Here, many of these fall into
two categories: periodic phenomena and special functions. Here are just a
few spatially or temporally periodic phenomena in physics: acoustics, diffrac-
tion, antenna design, dynamical systems, resonances of astronomical bodies,
and crystal (and quasicrystal) structure. Number theory also has an affin-
ity with special functions, for instance those arising in statistical mechanics.
Here, the connection often involves the fact that many such special functions
are the generating functions of sequences arising in additive (and sometimes
multiplicative) number theory. It should be mentioned that there are many
applications of number theory that do not come from computers or physics;
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