Outline | People | Reading | Grading | Academics | Homepage |
Course Outline* | |||||
Lecture | Topic(s) | Notes | Book(s) | ||
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1. Jan 10 | Introduction; Fibonacci |
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ENT/CINTA
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2. Jan 12 | Mathematical induction;
the binomial theorem
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ENT §1
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Mon, Jan 16 | M. L. King Holiday | ||||
3. Jan 17 | Inductive definition of addition, multiplication, exponentiation; divisibility and division with remainder |
Maple Worksheet
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Class notes;
ENT §2
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4. Jan 19 | Euclid's algorithm
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ENT §2
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5. Jan 24 | Extended Euclidean algorithm; diophantine linear equations |
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ENT §2;
class notes
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6. Jan 26 | Continued fractions; Euclid's lemma |
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ENT §2
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7. Jan 31 |
Fundamental theorem of arithmetic
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ENT §3 | ||
8. Feb 2
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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
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9. Feb 7 | Catch-up; review for first exam |
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10. Thurs Feb 9 | First Exam | Counts 20% | |||
11. Feb 14 💘 | Equivalence relations, congruence relations, congruences |
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Class notes;
ENT §4
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Thurs, Feb 16 | Wellness Day | ||||
12. Feb 21 |
Solution of first exam;
congruences continued
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13. Feb 23 |
Congruences continued
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14. Feb 28 |
The Chinese remainder theorem
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Maple Worksheet
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ENT §4.4 | ||
15. Mar 2 |
The little Fermat theorem;
pseudoprimes;
Fermat primality test;
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Carmichael numbers
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ENT §5.3
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Mon, Mar 6, 11:59pm Last day to drop the course | |||||
16. Mar 7 |
Carmichael numbers;
Miller-Rabin test
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Maple Worksheet
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ENT §5.2
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17. Mar 9 |
Euler's phi function;
sums of divisors
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ENT §7
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Mar 13 - Mar 17 | Spring Break | ||||
18. Mar 21 |
Public key cryptography; the RSA
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ENT §10
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19. Mar 23 | Catch-up; review for exam |
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20. Mar 28 | Tuesday, Second exam | Counts 20% | |||
21. Mar 30 |
Index calculus: order of an integer modulo n
and
existence of primitive roots modulo p
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ENT §8
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22. Apr 4 |
Diffie-Hellman-Merkle key exchange;
el-Gamal public key crypto system;
digital signatures
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Class notes
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23. Apr 6 |
Quadratic and cubic residuosity
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Maple Worksheet
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ENT §9.1
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24. Apr 11 |
Legendre symbol,
the quadratic reciprocity law
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ENT §9.2, §9.3
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25. Apr 13 |
Jacobi symbol
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ENT §9.3, Problems 16-19
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26. Apr 18 |
Computing squareroots modulo p
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Maple worksheet
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Tonelli-Shanks
Algorithm
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27. Apr 20 |
Pythagorean triples,
Fermat's last theorem for n=4
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ENT §12.1, §12.2
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Thursday, Apr 27, 8am-10am, Final Exam (counts 30%) | |||||
Thursday, May 4 noon, Grades due |
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
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.
Grade split up | |
Accumulated homework grade | 25% |
Final examination | 30% |
First mid-semester exam | 20% |
Second mid-semester 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 Erich Kaltofen. Permission to use provided that copyright notice is not removed.