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TATA54 Kursinformation/Course information

Examination

Examination is in the form of a written exam.

Miniföreläsning

VT2021 så tänker jag prova att låta studenter hålla ett litet föredrag på ca 20 min om något avsnitt från kursen, eller näraliggande. Det är frivilligt men kommer att ge bonus på tentan.

Förslag på ämnen:

1. Periodlängd för decimal (och annan radix) utveckling av rationella tal
2. Fermattal, Fermatprimtal, Mersenneprimtal
3. Faktoriseringsmetoder
4. Veckodagsberäkning
5. Pseudoprimtal
6. Kappsäckskrypto
7. RSA
8. Pseudoslumptal
10. Faktorisering med kedjebråk
11. Kedjebråksutvecklingen av e
12. Oändlig descent
13. Elliptiska kurvor
14. Modulär aritmetik i Z[i]
15. Z[-1/2 + i sqrt(3)/2]
16. Såll
18. Cyklotoma polynom

Textbook

The textbook for the course is Elementary Number Theory (6'th edition) by Rosen. An older edition will work fine (except that the numbering for the recommended excercises below will be off, use your own judgement as to which exercises to do).

I will also use Elementary Number Theory by Stein, which is available in hardcover from Springer, and freely available in electronic form.

I also recommend the book by Peter Hackman , from our department.

Yet another book is An introduction in elementary number theory by Raji.

I will use the text The Gaussian integers by Conrad when teaching about the subject.

Plan

Contents and scope

The course includes the following parts of the textbook (Rosen): Chapt 1.5, 2.1, 3, 4.1-4, 5.1, 6, 7.1-4, 9, 11.1-4, 12, 13.1-4, 14.

Though I will not lecture about everything listed above, and may briefly talk about topics not on said list, the above defines the contents of the course and what may occur on the exam.

Topics, lectures

1. Integers, divisibility
2. Unique factorization
3. Greatest common divisor, Linear Diophantine equations
4. Congruences, Chinese remainder theorem
5. Multiplicative order, Fermat, Euler
6. Arithmetical functions, Mobius inversion
7. Hensel lifting
8. Lagrange, Primitive roots, Discrete logarithms
10. Continued fractions
11. Pell's equation
12. Sum of squares
13. Gaussian integers

Recommended exercises

In Rosen

Ch 3
3.1: 1ab, 2f,6. 3.2:2. 3.3: 1df,2ef,10,24,30,31 . 3.4: 1bc, 19*, 20*. 3.5: 1il, 2, 4c, 29cd, 32*, 34, 58ac. 3.7: 1cd
Ch 4
4.1: 4,5,7ab,9bc. 4.2:1df, 8c, 9d. 4.3: 3,4. 4.4: 2ab.
Ch 5
5.1:4ab.
Ch 6
6.1:4,5,6,9,10,14,18,19,22,24,24,28,36,38*,41,48*,52. 6.2:2,8,12,16ab. 6.3:1d,6,7,22*
Ch 7
7.1: 2bc,5,8. 7.2: 1bd,2cd,8,34*,35,38*. 7.3: 3b,4,7,8,10,18,22**,28*,29cd. 7.4: 1bd,2be,14,22,23,24,30,31.
Ch 9
9.1: 1ad, 2cd, 5b, 6df, 8, 10, 20*, 22*. 9.2: 1d, 2d, 3e, 6. 9.3: 2,3c, 5c,6c, 8b, 10. 9.4: 1, 2b, 3a, 4, 9, 10, 19, 20*. 9.5: 2,3,10. 9.6: 1bcg, 4def, 13*, 14*.
Ch 11
11.1: 1d, 4, 25. 11.2: 1cd, 2, 4, 5. 11.3: 1ef, 5, 6. 11.4: 1, 2, 5, 6.
Ch 12
12.2: 1bf, 2cd, 3c, 4c. 12.3: 1c, 2a, 6*, 7. 12.4: 1c, 2c, 3b, 4a, 5b, 6b, 7b, 8, 10, 17. 12.5: 2.
Ch 13
13.1: 1, 2, 3, 8, 16. 13.2: 3, 4, 6*, 8, 10, 11*, 20. 13.3: 1b, 2gh,3cd, 6, 7, 9c, 10ef. 13.4: 1c, 2a, 3def, 4ab, 6de, 8*.
Ch 14
14.1: 1ac, 12, 13, 17a. 14.2: 16ab, 17bc. 14.3: 1bc, 2c.

Sidansvarig: jan.snellman@liu.se