Math 435: Foundations of Number Theory |

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Prerequisites

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Carl Friedrich Gauss

(quoted in: Sartorius von Walterhausen, Gauss zum Gedächtniss, Leipzig, 1856, p.79.)

(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.

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.

An electronic version of the first edition of this book is accessible here.

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.

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.

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 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.

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

BrahmaguptaDiophantus 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

Adrien-Marie Legendre

Marin Mersenne

August Ferdinand Möbius

Blaise Pascal

John Pell

Pythagoras of Samos

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