Outline  People  Reading  Grading  Academics  Homepage 
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 15  M. L. King Holiday  
3. Jan 16  Inductive definition of addition, multiplication, exponentiation; divisibility and division with remainder 
Maple Worksheet

Class notes;
ENT §2


4. Jan 18  Euclid's algorithm


ENT §2


5. Jan 23  Extended Euclidean algorithm; diophantine linear equations 

ENT §2;
class notes


6. Jan 25  Continued fractions; Euclid's lemma 

ENT §2


7. Jan 30 
Fundamental theorem of arithmetic


ENT §3  
8. Feb 1

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


9. Feb 6  Catchup; review for first exam 



10. Thurs Feb 8  First Exam (8am10am online on Moodle)  Counts 20%  
Tue, Feb 13  Wellness Day  
11. Feb 15  Solution of first exam 



12. Feb 20 
Equivalence relations, congruence relations, congruences


Class notes; ENT §4


13. Feb 22 
Congruences continued




14. Feb 27 
Sun Zi's Chinese remainder theorem
[Suanjing (Math. Manual) ≈ 300500 CE] 
Maple Worksheet

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

Carmichael numbers,
OEISA002997

ENT §5.3


Mon, Mar 4, 11:59pm Last day to drop the course  
16. Mar 5 
Carmichael numbers;
MillerRabin test

Maple Worksheet

ENT §5.2


17. Mar 7 
Euler's phi function;
sums of divisors


ENT §7


Mar 11  Mar 15  Spring Break  
18. Mar 19 
Public key cryptography; the RSA


ENT §10


19. Mar 21  Catchup; review for exam 

20. Tues Mar 26  Second exam (8am10am online on Moodle)  Counts 20%  
21. Mar 28 
Index calculus: order of an integer modulo n
and
existence of primitive roots modulo p

ENT §8


22. Apr 2 
Solution of exam 2




23. Apr 4 
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 9 
Quadratic and cubic residuosity;
DiffieHellmanMerkle key exchange; elGamal public key crypto system; digital signatures

Class notes

ENT §9.2, §9.3


25. Apr 11 
Legendre symbol;
the quadratic reciprocity law;
Jacobi symbol

ENT §9.3, Problems 1619


26. Apr 16 
Computing squareroots modulo p

Maple worksheet

TonelliShanks
Algorithm


27. Apr 18 
Pythagorean triples,
Fermat's last theorem for n=4

ENT §12.1, §12.2


28. Apr 23 
Catchup; final exam review



Thursday, Apr 25, 8am11am online on Moodle, Final Exam (counts 30%)  
Friday, May 3, 5pm, Grades due 
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 examination  30% 
First midsemester exam  20% 
Second midsemester exam  20% 
Class attendance  5% 
Course grade  100% 
If you need assistance in any way, please let me know; please see also NC State CARES ; there is also the University's policy.
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:
©2010, 2014, 2016, 2017, 2018, 2019, 2020, 2021, 2023, 2024 Erich Kaltofen. Permission to use provided that copyright notice is not removed.