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MA 410 '05 Syllabus

Course Outline*

Lecture Topic(s) Notes Book(s)
1. Jan 10 Introduction
2. Jan 12 Mathematical induction

ENT §1
3. Jan 14 Inductive definition of addition, multiplication, exponentiation
Class notes
Mon, Jan 17 M. L. King holiday
4. Jan 19 The binomial theorem

ENT §1
5. Jan 21 Divisibility and division with remainder
ENT §2
6. Jan 24 Euclid's algorithm

ENT §2
7. Jan 26 Extended Euclidean algorithm; diophantine linear equations
ENT §2; class notes
8. Jan 28 No class (due to extended class time)

9. Jan 31 Continued fractions; Euclid's lemma
ENT §2
10. Feb 2 Fundamental theorem of arithmetic

ENT §3
11. Feb 4 Theorems on primes: Euclid, Chebyshev, Dirichlet, Hadamard/de la Vallee Poussin, Green-Tao 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).
ENT §3
12. Feb 7 Conjectures on primes: Goldbach, twin, Mersenne, Fermat
list of Mersenne primes, factors of Fermat numbers
ENT §3
13. Feb 9 Catch-up; review for first exam

14. Feb 11 No class (due to extended class time)

15. Feb 14 First Exam Counts 17.5%
16. Feb 16 Equivalence relations, congruence relations

Class notes
17. Feb 18 Return of first exam

18. Feb 21 Congruences
ENT §4
Mon, Feb 21, 5pm Last day to drop the course
19. Feb 23 Congruences continued

ENT §4
20. Feb 25 No class (due to extended class time)

21. Feb 28 The Chinese remainder theorem

ENT §4.4
22. Mar 2 The little Fermat theorem; pseudoprimes
Carmichael numbers
ENT §5.3
23. Mar 4 Fermat primality test; Carmichael numbers

ENT §5.3
Mar 7-11, 2005 Spring Break, no class
24. Mar 14 Miller-Rabin test; Fermat factorization

ENT §5.2
25. Mar 16 Euler's phi function

ENT §7
26. Mar 18 Euler's generalization of the little Fermat theorem Public key cryptography; the RSA

ENT §7.3 and 7.5
27. Mar 21 Properties of Euler's phi function; the tau, sigma and mu functions

ENT §6
28. Mar 23 No class (due to extended class time)
Thursday-Friday, Mar 24-25 Spring Holiday, no class
29. Mar 28 Review for exam

30. Mar 30 Second exam Counts 17.5%
31. Apr 1 The Möbius function and inversion formula

ENT §6.2
32. Apr 4 Return of exam ;-)

33. Apr 6 Index calculus: order of an integer modulo n

ENT §8
34. Apr 8 Index calculus: existence of primitive roots modulo p

35. Apr 11 Catch-up

36. Apr 13 Diffie-Hellman key exchange; el-Gamal public key crypto system; digital signatures
Class notes

37. Apr 15 No class (due to extended class time)

38. Apr 18 Quadratic residuosity

ENT §9.1
39. Apr 20 The quadratic reciprocity law

ENT §9.3
40. Apr 22 Pythagorean triples

ENT §11.1
41. Apr 25 Fermat's last theorem for n=4

ENT §11.2
42. Apr 27 Teaching evaluation

43. Apr 29 No class (due to extended class time)

Wed, May 4, 9am-11am, Harrelson 263: Final exam (counts 25%)
* 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.

I will check who attends class. You will forfeit 10% 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.

©2004 Erich Kaltofen. Permission to use provided that copyright notice is not removed.