__Theory of Numbers__

__Fall 2014__

(640:356)

__Course Info__

Instructor: Swastik Kopparty (swastik.kopparty@gmail.com)

Class Time and Place: Mondays and
Thursdays, 10:20am – 11:40am, in SEC-218.

Office Hours: Monday 12:00-1:00
(Hill 432)

Prerequisites: Calculus 3,
familiarity with proofs at the level of Math 300 highly recommended.

References: online sources, links
will be posted.

·
Elementary Number
Theory (Edwin Clark)

·
More
coming soon ..

__Syllabus__

This course is an introduction to
number theory. We will see some of the amazing results of great number
theorists such as Euclid, Fermat, Euler, Lagrange, Legendre and Gauss.

Topics include:

- Divisibility
- GCDs
- Prime
numbers and unique factorization
- Modular
arithmetic
- Pythogorean
triples
- Quadratic
reciprocity
- The
four squares theorem
- Generating
functions
- Applications
of complex numbers
- Applications
to cryptography

There will be
quizzes every 1-2 weeks. There will be
homework every 1-2 weeks. There will be a
final exam. The lowest
homework grade will be dropped. The lowest two
quiz grades will be dropped. Weightage: HW: 30%, quizzes:
35%, final: 35%, class participation: up to 10% bonus. |
No laptops,
tablets, phones, smartwatches etc. can be used during class. You can discuss
homework problems with classmates, but write-ups must be completely your own. You must list all
the people with whom you worked on the homework problems. You are not
allowed to use internet sources for the homework problems. Late homework
will not be accepted unless arrangements are made beforehand. Requests for
turning homework in late will not be granted without a note from a doctor /
dean. Missed quizzes
cannot be made up. |

FINAL
EXAM STUDY GUIDE (last updated: Monday, December 8)

__Homework__

- Homework 0 (due September 7)
- Homework 1 (due September 18)
- Homework 2 (due September 29)
- Homework 3 (due October 20)
- Homework 4 (Due November 17)
- Homework 5 (Due December 1)

__Lecture Schedule__

- September
4: what are numbers? Peano arithmetic, induction
- September
8: more induction, strong induction
- September
11: number systems, converting between bases
- September
15: base q arithmetic, prime numbers
- September
18: the fundamental theorem of arithmetic
- September
22: GCD (guest lecture by Ross Berkowitz)
- September
25: Euclid’s GCD algorithm (guest lecture by John Kim)
- September
29: some consequences of unique factorization
- October
2: congruences
- October
6: Z
_{m}and the units of Z_{m} - October
9: the Chinese remainder theorem
- October
13: the Chinese remainder theorem, continued
- October
16: the Euler totient function
- October
20: order of an element, Fermat’s little theorem
- October
23: Euler’s theorem
- October
27: the RSA cryptosystem
- October
30: perfect squares mod a prime
- November
3: Pythogorean triples
- November
6: the quadratic residue symbol
- November
10: the prime factorization of n!, binomial coefficients
- November
13: the weak prime number theorem
- November
17: generating functions for linear recurrences
- November
20: the Riemann zeta function, Euler’s proof of the infinitude of primes,
integer partitions
- November 24: Gaussian integers
**November 25 (Tuesday with a Thursday Schedule):**primes expressible as the sum of two squares**November 27: NO CLASS (Thanksgiving)**- December
1: divisibility tests
- December
4: decimal expansions of rational numbers, Egyptian fractions
- December
8: beyond this course