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

1. Jan 9  Introduction; Fibonacci 

ENT/CINTA


2. Jan 11  Mathematical induction;
the binomial theorem


ENT §1


Mon, Jan 16  M. L. King Holiday  
3. Jan 18  Inductive definition of addition, multiplication, exponentiation; divisibility and division with remainder 

Class notes;
ENT §2


4. Jan 23  Euclid's algorithm


ENT §2


5. Jan 25  Extended Euclidean algorithm; diophantine linear equations 

ENT §2;
class notes


6. Jan 30  Continued fractions; Euclid's lemma 

ENT §2


7. Feb 1 
Fundamental theorem of arithmetic


ENT §3  
8. Feb 6

Theorems on primes:
Euclid, Chebyshev, Dirichlet,
Hadamard/de la Vallee Poussin,
GreenTao
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. 6494 (1962).
list of Mersenne primes, factors of Fermat numbers 
ENT §3


9. Feb 8  Catchup; review for first exam 



10. Feb 13  Monday, First Exam  Counts 20%  
11. Feb 15 
Return of first exam;
equivalence relations, congruence relations


Class notes


12. Feb 20  Congruences 

ENT §4


13. Feb 22 
Congruences continued




14. Feb 27 
The Chinese remainder theorem


ENT §4.4  
15. Feb 29 
The little Fermat theorem;
pseudoprimes;
Fermat primality test;

Carmichael numbers

ENT §5.3


Mar 59, 2012  Spring Break, no class  
16. Mar 12 
Carmichael numbers;
MillerRabin test


ENT §5.2


Mon, Mar 12, 11:59pm Last day to drop the course  
17. Mar 14 
Euler's phi function;
sums of divisors


ENT §7


18. Mar 19 
Public key cryptography; the RSA


ENT §7.5


19. Mar 21  Catchup; review for exam 

20. Mar 26  Monday, Second exam  Counts 20%  
21. Mar 28 
Return of second exam;
primitive roots


ENT §8


22. Apr 2 
Index calculus: order of an integer modulo n
and
existence of primitive roots modulo p

ENT §8


23. Apr 4 
DiffieHellman key exchange;
elGamal public key crypto system;
digital signatures

Class notes



24. Apr 9 
Quadratic residuosity

ENT §9.1


25. Apr 11 
Legendre symbol,
the quadratic reciprocity law

ENT §9.2, §9.3


26. Apr 16 
Jacobi symbol

ENT §9.3, Problems 1619


27. Apr 18 
Computing squareroots modulo p

Maple worksheet



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

ENT §11.1, §11.2


29. Apr 25 
Final exam review



Monday, May 7, 9am11am, Final exam (counts 30%) 
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. 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 2hour examination  30% 
First 1hour midsemester exam  20% 
Second 1hour midsemester exam  20% 
Class attendance  5% 
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:
©2010 Erich Kaltofen. Permission to use provided that copyright notice is not removed.