| Outline | People | Reading | Grading | Academics | Homepage |
Course Outline* | |||||
| Lecture | Topic(s) | Notes | Book(s) | ||
|---|---|---|---|---|---|
| Mon, Jan 18 | M. L. King Holiday | ||||
| 1. Jan 19 | Introduction; Fibonacci |
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ENT/CINTA
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| 2. Jan 21 | Mathematical induction;
the binomial theorem
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ENT §1
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| 3. Jan 26 | Inductive definition of addition, multiplication, exponentiation; divisibility and division with remainder |
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Class notes;
ENT §2
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| 4. Jan 28 | Euclid's algorithm
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ENT §2
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| 5. Feb 2 | Extended Euclidean algorithm; diophantine linear equations |
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ENT §2;
class notes
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| 6. Feb 4 | Continued fractions; Euclid's lemma |
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ENT §2
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| Tue, Feb 9 | Wellness Day 1 | ||||
| 7. Feb 11 |
Fundamental theorem of arithmetic
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ENT §3 | ||
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8. Feb 16
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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 18 | Catch-up; review for first exam |
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| 10. Feb 23 | Tuesday First Exam | Counts 20% | |||
| 11. Feb 25 | Equivalence relations, congruence relations, congruences |
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Class notes;
ENT §4
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| 12. Mar 2 |
Solution of first exam;
congruences continued
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| 13. Mar 4 |
Congruences continued
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| Fri, March 5 | Wellness Day 2 | ||||
| 14. Mar 9 |
The Chinese remainder theorem
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Maple Worksheet
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ENT §4.4 | ||
| 15. Mar 11 |
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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| 16. Mar 16 |
Carmichael numbers;
Miller-Rabin test
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Maple Worksheet
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ENT §5.2
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| Wed, Mar 17, 11:59pm Last day to drop the course | |||||
| 17. Mar 18 |
Euler's phi function;
sums of divisors
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Maple Worksheet,
Worksheet in pdf
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ENT §7
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| 18. Mar 23 |
Public key cryptography; the RSA
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ENT §10
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| Wed, Mar 24, 2021 | Wellness Day 3 | ||||
| 19. Mar 25 | Catch-up; review for exam |
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| 20. Mar 30 | Tuesday, Second exam | Counts 20% | |||
| 21. Apr 1 🤪 |
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 6 |
Solution of second exam;
primitive roots continued
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ENT §8
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| 23. Apr 8 |
Diffie-Hellman-Merkle key exchange;
el-Gamal public key crypto system;
digital signatures
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Class notes 2011
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| 24. Apr 13 |
Quadratic and cubic residuosity
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ENT §9.1
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| Thurs, Apr 15, 2021 | Wellness Day 4 | ||||
| 25. Apr 20 |
Legendre symbol,
the quadratic reciprocity law,
Jacobi symbol
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ENT §9.2, §9.3;
ENT §9.3, Problems 16-19
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| 26. Apr 22 |
Computing squareroots modulo p
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Pdf of Maple worksheet
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Tonelli-Shanks
Algorithm
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| 27. Apr 27 |
Pythagorean triples
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ENT §12.1
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| 28. Apr 29 |
Fermat's last theorem for n=4
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§12.2
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| Tuesday, May 4, 8:30am-noon, Final exam (counts 30%) | |||||
| Wednesday, May 12, 5pm, 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 (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, 2014, 2016, 2017, 2018, 2019, 2020, 2021 Erich Kaltofen. Permission to use provided that copyright notice is not removed.