Foreword

The gold in ‘them there hills’ is not always buried deep. Much

of it is within easy reach. Some of it is right on the surface

to be picked up by any searcher with a keen eye for detail

and an eagerness to explore. As in any treasure hunt, the

involvement grows as the hunt proceeds and each success

whether small or great adds the fuel of excitement to the

exploration. – A. E. Ross

Number theory is one of the few areas of mathematics where problems of

substantial interest can be described to someone possessing scant mathemat-

ical background. It sometimes proves to be the case that a problem which is

simple to state requires for its resolution considerable mathematical prepa-

ration; e.g., this appears to be the case for Fermat’s conjecture regarding

integer solutions to the equation

xn

+

yn

=

zn.

But this is by no means

a universal phenomenon; many engaging problems can be successfully at-

tacked with little more than one’s “mathematical bare hands”. In this case

one says that the problem can be solved in an elementary way (even though

the elementary solution may be far from simple). Such elementary methods

and the problems to which they apply are the subject of this book.

Because of the nature of the material, very little is required in terms of

prerequisites: The reader is expected to have prior familiarity with number

theory at the level of an undergraduate course. The nececssary background

can be gleaned from any number of excellent texts, such as Sierpi´nski’s

charmingly discursive Elementary Theory of Numbers or LeVeque’s lucid

and methodical Fundamentals of Number Theory. Apart from this, a rig-

orous course in calculus, some facility with manipulation of estimates (in

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