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# TATA54 Kursinformation/Course information

## Examination

Examination is in the form of a written exam.

## 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