Math 435: Foundations of Number Theory

# General Information

Time and Place: MWF 2-2:50pm, 303 Addams Hall

Instructor: Matthias Aschenbrenner

Homepage: http://www.math.uic.edu/~maschenb

Office:
417 SEO
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.

# Description

Mathematics 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 = 20,615,673.

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.

# Course Text

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.

# Homeworks

There will be a problem set due every two weeks or so, to be handed in at the beginning of class. No late homework will be accepted. Problem sets and (after the due date) their solutions will be posted on this webpage. For more information on homeworks see the handout.

Problem Set 1, due September 15. Solutions.
Problem Set 2, due September 29. Solutions.
Problem Set 3, due October 20. Solutions.
Problem Set 4
, due November 10. Solutions.
Problem Set 5, due December 1. Solutions.

# Exams

There will be three examinations, in class, on Friday, September 29Friday, 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 in question.

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.

It is University policy that students with disabilities who require accommodations for access and participation in this course must be registered with the Office of Disability Services.

Grading policy: Homework: 25%. Exams: 25% each.

# Historical Information

Click below for biographical information about some of the number theorists which we will encounter in this course:

Brahmagupta
Diophantus of Alexandria
Gustav Lejeune Dirichlet
Eratosthenes of Cyrene
Euclid of Alexandria
Leonhard Euler
Pierre de Fermat
Leonardo Fibonacci
Carl Friedrich Gauss
Carl Gustav Jacobi
Joseph-Louis Lagrange