MA410 Spring '25 Syllabus

MA 410 2025 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, Yitang Zhang 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. Thurs Feb 6	First Exam (8am-10am online on Moodle)	Counts 20%
Tue, Feb 11	Wellness Day
11. Feb 13	Solution of first exam
12. Feb 18	Equivalence relations, congruence relations, congruences		Class notes; ENT §4
13. Feb 20	Congruences continued
14. Feb 25	Sun Zi's Chinese remainder theorem [Suanjing (Math. Manual) ≈ 300-500 CE]	Maple Worksheet	ENT §4.4
15. Feb 27	The little Fermat theorem; pseudoprimes; Fermat primality test;	Carmichael numbers, OEIS-A002997	ENT §5.3
Mon, Mar 3, 11:59pm Last day to drop the course
16. Mar 4	Carmichael numbers; Miller-Rabin test	Maple Worksheet	ENT §5.2
17. Mar 6	Euler's phi function; sums of divisors		ENT §7
Mar 10 - Mar 14	Spring Break
18. Mar 18	Public key cryptography; the RSA		ENT §10
19. Mar 20	Catch-up; review for exam
20. Tues Mar 25	Second exam (8am-10am online on Moodle)	Counts 20%
21. Mar 27	Index calculus: order of an integer modulo n and existence of primitive roots modulo p		ENT §8
22. Apr 1 🤪	Solution of exam 2
23. Apr 3	Using the index for solving polynomial equivalence; Lagrange Theorem on modular roots; proof of existence of primitive roots modulo p	Maple Worksheet	ENT §9.1
24. Apr 8	Quadratic and cubic residuosity; Diffie-Hellman-Merkle key exchange; el-Gamal public key crypto system; digital signatures	Class notes	ENT §9.2, §9.3
25. Apr 10	Legendre symbol; the quadratic reciprocity law; Jacobi symbol		ENT §9.3, Problems 16-19
26. Apr 15	Computing squareroots modulo p	Maple worksheet	Tonelli-Shanks Algorithm
27. Apr 17	Pythagorean triples, Fermat's last theorem for n=4		ENT §12.1, §12.2
28. Apr 22	Catch-up; final exam review
Thursday, Apr 24, 8am-11am online on Moodle, Final Exam (counts 30%)
Friday, May 2, 5pm, Grades due

Instruction Personnel

For instructor, office hours, email and physical address see the homepage 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.

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 get credit by either A. signing the attendance sheet in class or B. go to Moodle and mark your attendance ``P'' (present). You have until 11:59pm on each day of each lecture to do so for that lecture. If you sign the attendance sheet, I will mark you "P". 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 2024.

If you need assistance in any way, please let me know; please see also NC State CARES; there is also the University's policy.

Academic Standards

Examinations:The three examinations will be online with uploading solutions to Moodle, open book/open notes/open Internet query such as wolframalpha.com. But you are not to consult with another human person.

Collaboration on homeworks: I expect every student to be his/her own writer. Therefore please only discuss with one another 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 2025 Syllabus

Course Outline*

Instruction Personnel

Textbook and Online Notes

Grading and General Information

Academic Standards

Course Outline *