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
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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