|Math 435: Foundations of Number Theory|
Time and Place: MWF 2-2:50pm,
303 Addams Hall
Control Number: 13712 (undergraduate) and 20443 (graduate and graduate non-degree)
Office Phone: (312) 413-3150
Office Hours: Monday 3-4pm, Wednesday 3-4pm, Friday 11am-12.
Prerequisites: Grade of C or better in Math 215.
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DescriptionMathematics is the queen of the sciences and number theory is the queen of mathematics.
Carl Friedrich Gauss
(quoted in: Sartorius von Walterhausen, Gauss zum Gedächtniss, Leipzig, 1856, p.79.)
Number theory is one of the oldest
branches of pure mathematics, and one of the largest; its aim is the
study of the integers. It is a subject abundant with problems which are
easily stated, but whose solution is often extremely difficult and
sometimes requires sophisticated methods from other branches of
mathematics. Examples include:
- Catalan's conjecture (formulated 1844, and proved by Mihailescu only in 2002): 8 and 9 are the only successive integer powers.
- Fermat's last theorem (stated in 1637, but not proved until 1994, by Andrew Wiles): there are no nonzero integers x, y, z such that x^n + y^n = z^n for some integer n>2.
Many other ones are still open, for example:
- The twin prime conjecture: there are infinitely many prime pairs, such as (3,5), (5,7), (11,13), ...
- The Goldbach conjecture: every even number >2 is the sum of two primes, e.g., 4=2+2, 6=3+3, 8=3+5, 10=5+5, ...
Sometimes the answer turns out not to be what someone guessed (centuries ago): In 1769, while thinking about Fermat's last theorem
, Leonhard Euler
conjectured that the equation a^4 + b^4 + c^4 = d^4 would also have no nonzero integer solutions. In 1998, Noam Elkies
of Harvard University found the first counterexample: the equation is
true when a = 2,682,440, b = 15,365,639, c = 18,796,760, and d =
In this course, we will focus on "elementary number theory'': the
division algorithm, the Euclidean algorithm (existence of greatest
common divisors), elementary properties of primes (unique
factorization, the infinitude of primes), congruences, including
Fermat's little theorem, quadratic residues, and so forth. The
term "elementary'' is used in this context only to suggest that no
advanced tools from other areas are used- not that the results
themselves are simple. Indeed, many of the results which we will
discuss (e,g., the Quadratic Reciprocity law, the Chinese Remainder
Theorem) turned out, in retrospect, to presage more sophisticated tools
and themes introduced later in history.
Elementary Number Theory in Nine Chapters, Second Edition (Paperback) by James J. Tattersall, Cambridge University Press, 2005, ISBN 0521615240.
An electronic version of the first edition of this book is accessible here.
There will be three examinations, in class, on Friday, September 29, Friday, November 3, and on Friday, December 8.
Students with examinations which
conflict with the exams in this course are responsible for discussing
makeup examinations with me no later than two weeks prior to the exam
No books, calculators, scratch paper or notes will be allowed
during exams. Students are expected to be thoroughly familiar with the
University’s policy on academic integrity. The University has
instituted serious penalties for academic dishonesty.
Copying work to be submitted for
grade, or allowing your work to be submitted for grade to be copied, is
considered academic dishonesty.
is University policy that students with disabilities who require accommodations
for access and participation in this course must be registered with the
of Disability Services.
Grading policy: Homework: 25%. Exams: 25% each.
below for biographical information about some of the number theorists which we will encounter in this course:
Diophantus of Alexandria
Gustav Lejeune Dirichlet
Eratosthenes of Cyrene
Euclid of Alexandria
Pierre de Fermat
Carl Friedrich Gauss
Carl Gustav Jacobi
August Ferdinand Möbius
Pythagoras of Samos
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