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Course Outline*  
Lecture  Topic(s)  Notes  Book(s)  

1. Jan 10  Introduction 

ENT/CINTA  
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, GreenTao 
sequences
of equidistant primes;
Barkley Rosser, Lowell Schoenfeld.
Approximate formulas of some functions of prime numbers.
Illinois J. Math. vol. 6, pp. 6494 (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  Catchup; 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 711, 2005  Spring Break, no class  
24. Mar 14 
MillerRabin 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) 


ThursdayFriday, Mar 2425  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 
Catchup



36. Apr 13 
DiffieHellman key exchange;
elGamal 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, 9am11am, Harrelson 263: Final exam (counts 25%) 
Online 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
There will be four homework assignments of approximately equal weight, two midsemester 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.
Grade split up  
Accumulated homework grade  30% 
Final examination  25% 
First midsemester exam  17.5% 
Second midsemester exam  17.5% 
Class attendance  10% 
Course grade  100% 
If you need assistance in any way, please let me know (see also the University's policy).
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:
©2004 Erich Kaltofen. Permission to use provided that copyright notice is not removed.