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MA 410 2015 Syllabus

Course Outline*

Lecture Topic(s) Notes Book(s)
1. Jan 8 Introduction; Fibonacci
2. Jan 13 Mathematical induction; the binomial theorem

ENT §1
3. Jan 15 Inductive definition of addition, multiplication, exponentiation; divisibility and division with remainder
Class notes; ENT §2
Mon, Jan 19 M. L. King Holiday
4. Jan 20 Euclid's algorithm

ENT §2
5. Jan 22 Extended Euclidean algorithm; diophantine linear equations
ENT §2; class notes
6. Jan 27 Continued fractions; Euclid's lemma
ENT §2
7. Jan 29 Fundamental theorem of arithmetic

ENT §3
8. Feb 3
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 8 Catch-up; review for first exam

10. Feb 10 Tuesday, First Exam Counts 20%
11. Feb 12 Equivalence relations, congruence relations, congruences
Class notes; ENT §4
12. Feb 17 Return of first exam; congruences continued

13. Feb 19 Congruences continued

14. Feb 24 The Chinese remainder theorem

ENT §4.4
15. Feb 26 The little Fermat theorem; pseudoprimes; Fermat primality test;
Carmichael numbers
ENT §5.3
16. Mar 3 Carmichael numbers; Miller-Rabin test

ENT §5.2
Wed, Mar 4, 11:59pm Last day to drop the course
17. Mar 5 Euler's phi function; sums of divisors

ENT §7
Mar 9-13, 2015 Spring Break, no class
18. Mar 17 Public key cryptography; the RSA

ENT §7.5
19. Mar 19 Catch-up; review for exam

20. Mar 24 Tuesday, Second exam Counts 20%
21. Mar 26 Index calculus: order of an integer modulo n and existence of primitive roots modulo p

ENT §8
22. Mar 31 Return of second exam; primitive roots continued

ENT §8
Thursday-Friday, Apr 2-3 Spring Holiday, no class
23. Apr 7 Diffie-Hellman key exchange; el-Gamal public key crypto system; digital signatures
Class notes

24. Apr 9 Quadratic residuosity

ENT §9.1
25. Apr 14 Legendre symbol, the quadratic reciprocity law

ENT §9.2, §9.3
26. Apr 16 Jacobi symbol

ENT §9.3, Problems 16-19
27. Apr 21 Computing squareroots modulo p
Maple worksheet

28. Apr 23 Pythagorean triples, Fermat's last theorem for n=4

ENT §11.1, §11.2
Tuesday, May 5, 9am-11am, Final exam (counts 30%)
* This is a projected list and subject to amendment.

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 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

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.

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:

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

©2010, 2014 Erich Kaltofen. Permission to use provided that copyright notice is not removed.