xii Foreword
particular, big-Oh and little-oh notation), and a first course in modern al-
gebra (covering groups, rings, and fields) should suffice for the majority of
the text. A course in complex variables is not required, provided that the
reader is willing to overlook some motivational remarks made in Chapter 7.
Rather than attempt a comprehensive account of elementary methods
in number theory, I have focused on topics that I find particularly attractive
and accessible:
Chapters 1, 3, 4, and 7 collectively provide an overview of prime
number theory, starting from the infinitude of the primes, mov-
ing through the elementary estimates of Chebyshev and Mertens,
then the theorem of Dirichlet on primes in prescribed arithmetic
progressions, and culminating in an elementary proof of the prime
number theorem.
Chapter 2 contains a discussion of Gauss’s arithmetic theory of the
roots of unity (cyclotomy), which was first presented in the final
section of his Disquisitiones Arithmeticae. After developing this
theory to the extent required to prove Gauss’s characterization
of constructible regular polygons, we give a cyclotomic proof of
the quadratic reciprocity law, followed by a detailed account of a
little-known cubic reciprocity law due to Jacobi.
Chapter 5 is a 12-page interlude containing Dress’s proof of the
following result conjectured by Waring in 1770 and established by
Hilbert in 1909: For each fixed integer k 2, every natural number
can be expressed as the sum of a bounded number of nonnegative
kth powers, where the bound depends only on k.
Chapter 6 is an introduction to combinatorial sieve methods, which
were introduced by Brun in the early twentieth century. The best-
known consequence of Brun’s method is that if one sums the recip-
rocals of each prime appearing in a twin prime pair p, p + 2, then
the answer is finite. Our treatment of sieve methods is robust
enough to establish not only this and other comparable ‘upper
bound’ results, but also Brun’s deeper “lower bound” results. For
example, we prove that there are infinitely many n for which both
n and n+2 have at most 7 prime factors, counted with multiplicity.
Chapter 8 summarizes what is known at present about perfect
numbers, numbers which are the sum of their proper divisors.
At the end of each chapter (excepting the interlude) I have included several
nonroutine exercises. Many are based on articles from the mathematical
literature, including both research journals and expository publications like
the American Mathematical Monthly. Here, as throughout the text, I have
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