'Beauty is the first test: there is no
permanent place in the world for
ugly mathematics.'
- G. H. Hardy
Number theory has been considered since time immemorial to be the very paradigm of pure (some would say useless) mathematics. In fact, the Chinese characters for mathematics are Number Science. 'Mathematics is the queen of sciences - and number theory is the queen of mathematics', according to Carl Friedrich Gauss, the life-long Wunderkind, who himself enjoyed the epithet 'Princeps Mathematicorum'.
What could be more beautiful than a deep, satisfying relation between whole numbers? (One is almost tempted to call them wholesome numbers.) In fact, it is hard to come up with a more appropriate designation than their learned name: the integers - meaning the 'untouched ones'. How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex numbers - whose first names virtually exude unsavory involvement with the complex realities of everyday life!
Yet, as we shall see in this book, the theory of integers can provide totally unexpected answers to real-world problems. In fact, discrete mathematics is taking on an ever more important role. If nothing else, the advent of the digital computer and digital communication has seen to that. But even earlier, in physics the emergence of
quantum mechanics and discrete elementary particles put a premium on the methods
and, indeed, the spirit of discrete mathematics.
And even in mathematics proper, Hermann Minkowski, in the preface to his introductory book on number theory, Diophantische Approximationen, published in 1907 (the year he gave special relativity its proper four-dimensional clothing in preparation for its journey into general covariance and cosmology) expressed his conviction that the 'deepest interrelationships in analysis are of an arithmetical nature'.
Yet much of our schooling concentrates on analysis and other branches of continuum mathematics to the virtual exclusion of number theory, group theory, combinatorics and graph theory. As an illustration, at a recent symposium on information
theory, the author met several young researchers formally trained in mathematics and working in the field of primality testing, who - in all their studies up to the Ph.D. - had not heard a single lecture on number theory!

Or, to give an earlier example, when Werner Heisenberg discovered 'matrix'
mechanics in 1925, he didn't know what a matrix was (Max Born had to tell him),
and neither Heisenberg nor Born knew what to make of the appearance of matrices
in the context of the atom. (David Hilbert is reported to have told them to go look for a differential equation with the same eigenvalues, if that would make them happier. They did not follow Hilbert's well-meant advice and thereby may have missed discovering the Schr ̈ dinger wave equation.)

The present book seeks to fill this gap in basic literacy in number theory - not
in any formal way, for numerous excellent texts are available - but in a manner that
stresses intuition and interrelationships, as well as applications in physics, biology,
computer science, digital communication, cryptography and more playful things,
such as puzzles, teasers and artistic designs.
Among the numerous applications of number theory on which we will focus in the subsequent chapters are the following:

1)The division of the circle into equal parts (a classical Greek preoccupation) and
the implications of this ancient art for modern fast computation and random
number generation.

2)The Chinese remainder theorem (another classic, albeit far Eastern) and how
it allows us to do coin tossing over the telephone (and many things besides).

3)The design of concert hall ceilings to scatter sound into broad lateral patterns
for improved acoustic quality (and wide-scatter diffraction gratings in general).

4)The precision measurement of delays of radar echoes from Venus and Mercury
to confirm the general relativistic slowing of electromagnetic waves in graitational fields (the 'fourth' - and last to be confirmed - effect predicted by
Einstein's theory of general relativity).
5)Error-correcting codes (giving us distortion-free pictures of Jupiter and Saturn
and their satellites).

6)'Public-key' encryption and deciphering of secret messages. These methods
also have important implications for computer security.

7)The creation of artistic graphic designs based on prime numbers.

8)How to win at certain parlor games by putting the Fibonacci number systems
to work.

9)The relations between Fibonacci numbers and the regular pentagon, the Golden
ratio, continued fractions, efficient approximations, electrical networks, the
'squared' square, and so on - almost ad infinitum.