MA410 Spring '20 Syllabus

MA 410 2020 Syllabus

* This is a *projected* list and subject to amendment.
Lecture	Topic(s)	Notes	Book(s)
Course Outline *
1. Jan 7	Introduction; Fibonacci		ENT/CINTA
2. Jan 9	Mathematical induction; the binomial theorem		ENT §1
3. Jan 14	Inductive definition of addition, multiplication, exponentiation; divisibility and division with remainder	Maple Worksheet	Class notes; ENT §2
4. Jan 16	Euclid's algorithm		ENT §2
Mon, Jan 20	M. L. King Holiday
5. Jan 21	Extended Euclidean algorithm; diophantine linear equations		ENT §2; class notes
6. Jan 23	Continued fractions; Euclid's lemma		ENT §2
7. Jan 28	Fundamental theorem of arithmetic		ENT §3
8. Jan 30	Theorems on primes: Euclid, Chebyshev, Dirichlet, Hadamard/de la Vallee Poussin, Green-Tao Conjectures on primes: Goldbach, twin, Mersenne, Fermat	sequences of equidistant primes; Barkley Rosser, Lowell Schoenfeld. Approximate formulas of some functions of prime numbers. Illinois J. Math. vol. 6, pp. 64--94 (1962). list of Mersenne primes, factors of Fermat numbers	ENT §3
9. Feb 4	Catch-up; review for first exam
10. Feb 6	Thursday, First exam	Counts 20%
11. Feb 11	Equivalence relations, congruence relations, congruences		Class notes; ENT §4
12. Feb 13	Return of first exam; congruences continued
13. Feb 18	Congruences continued
14. Feb 20	The Chinese remainder theorem	Maple Worksheet	ENT §4.4
15. Feb 25	The little Fermat theorem; pseudoprimes; Fermat primality test;	Carmichael numbers	ENT §5.3
16. Feb 27	Carmichael numbers; Miller-Rabin test	Maple Worksheet	ENT §5.2
Mon, Mar 2, 11:59pm Last day to drop the course
17. Mar 3	Euler's phi function; sums of divisors		ENT §7
18. Mar 5	Public key cryptography; the RSA		ENT §10
Mar 9-13, 2020	Spring Break, no class
19. Mar 17	Catch-up; review for exam
20. Mar 19	Thursday, Second exam	Counts 20%
21. Mar 24	Index calculus: order of an integer modulo n and existence of primitive roots modulo p		ENT §8
22. Mar 26	Return of second exam; primitive roots continued		ENT §8
23. Mar 31	Diffie-Hellman-Merkle key exchange; el-Gamal public key crypto system; digital signatures	Class notes
24. Apr 2	Quadratic and cubic residuosity	Maple Worksheet	ENT §9.1
25. Apr 7	Legendre symbol, the quadratic reciprocity law		ENT §9.2, §9.3
26. Apr 9	Jacobi symbol		ENT §9.3, Problems 16-19
27. Apr 14	Computing squareroots modulo p	Maple worksheet	Tonelli-Shanks Algorithm
28. Apr 16	Pythagorean triples, Fermat's last theorem for n=4		ENT §12.1, §12.2
29. Apr 21	Final exam review
30. Apr 23	Snow day slack lecture
Tuesday, May 5 🎸, 9am-11am, Final exam (counts 30%)
Thursday, May 7, 11:59pm, Grades due

Instruction Personnel

For instructor, office hours, telephone numbers, email and physical address see the homepages of Erich Kaltofen.

Textbook and Online Notes

We will use the books:

I will cover some topics that are not in the book, and will use

A Computational Introduction to Number Theory and Algebra

Shoup's book can be downloaded in pdf format for free. I had considered only using Shoup's book and am interested what you think about that idea. In any case, the syllabus above refers to chapters in these books. For topics in neither book, handouts will be provided.

On-line information: All information on courses that I teach (except individual grades) is now accessible via html browsers, which includes this syllabus. My web page listing all my courses' is at

http://users.cs.duke.edu/~elk27/courses/courses/courses.html

You can also find information on courses that I have taught in the past, and examinations that I have given.

Grading and General Information

Grading will be done with plus/minus refinement.

There will be four homework assignments of approximately equal weight, two mid-semester examinations during the semester, and final examination. Depending on time constraints, I may only grade a selection of homework problems.

I will check who attends class. You will forfeit 5% of your grade if you miss 3 or more classes without a valid justification. I you miss a class because you are sick, etc., please let me know. I may require you to document your reason.

Grade split up
Accumulated homework grade	25%
Final examination	30%
First mid-semester exam	20%
Second mid-semester exam	20%
Class attendance	5%

Course grade	100%

Grade distribution of Spring 2019.

If you need assistance in any way, please let me know (see also the University's policy).

Academic Standards

Examinations:The three examinations will be closed book and closed class notes. However, you will be able to bring note sheets of paper with pertinent information to the examinations (1 for first exam and 2 for second exam and 3 for the final exam).

Collaboration on homeworks: I expect every student to be his/her own writer. Therefore the only thing you can discuss with anyone is how you might go about solving a particular problem. You may use freely information that you retrieve from public (electronic) libraries or texts, but you must properly reference your source.

Late submissions: All programs must be submitted on time. The following penalties are given for (unexcused) late submissions:

up to 1 day late: 20% reduction
up to 2 days late: 50% reduction
after 2 days: no credit (= 100% reduction)

Alleged cheating incidents: I will not decide any penalty myself, but refer all such cases to the proper judiciary procedures.

MA 410 2020 Syllabus

Course Outline*

Instruction Personnel

Textbook and Online Notes

Grading and General Information

Academic Standards

Course Outline *