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{\bf{\centerline {Math 431: Theory of Numbers}}}
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{\bf{\centerline{Spring 2000}}}\smallskip
{\bf{\centerline{Instructor: Matthew Dyer}}}
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The natural numbers $1,2,3,\ldots$ are amongst the most familiar of all mathematical objects, and have been a source of  fascination for millennia. The theory of numbers is the area of mathematics concerned with the many deep and subtle relationships that numbers exhibit.

Number theory abounds in  natural and interesting questions. For instance, how does one determine all the right angled triangles whose sides have integer lengths i.e. solutions in natural numbers of $a^2+b^2=c^2$? Which natural numbers are expressible as $a^2+b^2$? as $a^2+b^2+c^2$? As $a^2+b^2+c^2+d^2$?  
Are there any solutions in natural
numbers  of $a^n+b^n=c^n$ for $n\geq 3$ (Fermat's last theorem) ? How many different ways are
there of expressing a number $n$ as a sum of natural numbers? Is there
an explicit formula for the $n$-th prime number? Roughly how big is
the nth prime?  What is the largest currently known prime? Is every
even number greater than two the sum of two prime numbers? Is every
even number expressible as the difference of two prime numbers in
infinitely many ways?   Is there always a prime number between $n$ and
$2n$ for $n\geq 2$? Are there infinitely prime numbers of the form
$4n+1$? of the form $n^2+1$? Why is $22/7$ such a good approximation
to $\pi$? Which numbers  modulo $n$ are squares?

The answers to the above questions are not all known.
In this course, we will discover for ourselves 
 the answers  to some of
these questions,  learn something about the ideas involved in
answering some of the more difficult ones,
 such as Fermat's last theorem, and learn what is
 conjectured to be true about the others.  
Since the subject matter is so familiar and
concrete, number theory can serve as a useful laboratory for the
development of important mathematical skills such as formulating,
testing and proving conjectures. This  will be one of the main
goals of this course, and it will be explicitly incorporated into 
the regular homework.

The  text for the course will be ``A friendly introduction to
number theory'' by Joseph Silverman, supplemented by additional
references. 
Prerequisites for the course will be minimal; certainly, 
familiarity with basic algebra as covered in Math 222 would
 be adequate.







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