# MA1301 Number Theory - Fall 2019

Schedule Room
Lectures: Tuesdays 18:15–20:00 Realfagbygg R1
Wednesdays 18:15–20:00 S7 Sentralbygg 2
Exam: see here
Lecturer
Glen Wilson
Office: 1204 Sentralbygg 2
Email: glen [dot] m [dot] wilson [at] ntnu [dot] no
Dag Nilsson
Email: dag [dot] nilsson [at] ntnu [dot] no

Midterm Exam

The date and time for the midterm exam is set to Oct. 9th, 18:15. The midterms will be held in Sentralbygg S5, S6, S8, S7. Please show up ahead of time.

You are allowed to use a specific basic calculator according to NTNU's policy.

Because the language of instruction of this course is English, the midterm exam will be written in English. However, you have the right to write your responses in Norwegian or English.

The grading scheme for the course is in the Course Information document above. The midterm is not mandatory, although you are strongly encouraged to take it.

Students are divided amongst the exam rooms by candidate numbers as follows:

S5: 10000 – 10055

S6: 10056 – 10110

S7: 10111 – 10165

S8: 10166 – 10221

Mattehjelpen01 ​

Lørdager 10:15-13:00, Auditorium K5 i Kjemiblokk 5.

Mattehjelpen01 er et ekstra tilbud for studenter i "01-emnene", dvs. MA0001, MA1101, MA1201 og MA1301. Veiledere er instituttets emeriti Per og Kari Hag. Her kan du få hjelp med oppgaver og andre faglige spørsmål. Første gang lørdag 31.august. ("Drop in"- opplegg. Mulighet for litt tavleundervisning og diskusjonsgrupper.)

## What this course is about

This course gives an introduction to elementary number theory. Topics included are: greatest common divisor, Euclidean algorithm, linear diophantine equations, elementary prime number theory, linear congruences, Chinese remainder theorem, Fermat's little theorem, Euler's phi-function, Euler's theorem with application to cryptography. Additional topics that may change from year to year may include number theoretical functions, Fermat's last theorem for n = 4, continued fractions, rational approximations, Pell's equations, and quadratic reciprocity.

## Prerequisites

There are no prerequisites beyond basic high school mathematics.

## Lecture Plan

This is a rough outline of the course structure that will be updated throughout the semester. We will cover chapters 1–9 and some additional topics depending on the interests of the class.

Week Topic Reference Notes
34.1 Natural numbers, Induction 1.1, 1.2 Lecture 1
34.2 Division algorithm, GCD 2.2, 2.3 Lecture 2
35.1 GCD 2.3 Lecture 3
35.2 Euclidean Algorithm 2.4 Lecture 4
36.1 Diophantine Equations 2.5 Lecture 5
36.2 FTA, primes 3.1, 3.2 Lecture 6
37.1 Goldbach conjecture 3.3 Lecture 7
37.2 Modular arithmetic 4.2 Lecture 8
38.1 Representations of Integers 4.3 Lecture 9
38.2 Chinese Remainder Theorem 4.4 Lecture 10
39.1 Chinese Remainder Theorem 4.4 Lecture 11
39.2 Fermat's Little Theorem 5.2 Lecture 12
40.1 Wilson's Theorem 5.3 Lecture 13
40.2 Number theoretic functions 6.1 Lecture 14
41.1 Review, exam prep Lecture 15
41.2 Midterm exam
42.1 Euler Phi function 7.2 Lecture 16
42.2 Class canceled!
43.1 Euler's theorem 7.3 Lecture 17
43.2 Order mod n 8.1 Lecture 18
44.1 Primitive roots 8.2 Lecture 19
44.2 Quadratic residues 9.1 Lecture 20
45.1 Legendre symbol 9.2 Lecture 21
45.2 Quadratic reciprocity 9.3 Lecture 22
46.1 Pythagorean triples 12.1 Lecture 23
46.2 Fermat's last theorem 12.2 Lecture 24
47.1 Sums of two squares 13.2 Lecture 25
47.2 Summary

## Groups

Your group can now be found in blackboard (under Snarveier/Quick-links → Grupper, eller på venstre side i menyvalget).

Notify Dag Nilsson with your preferred timeslot, if you have not been assigned to a group.

The exercise sessions are not compulsory.

Nr Tid Rom
Gruppe 1 Tirsdag 16:15-18:00 R91 Realfagbygget
Gruppe 2 Tirsdag 14:15-16:00 R91 Realfagbygget
Gruppe 3 Fredag 10:15-12:00 R92 Realfagbygget
Gruppe 4 Tirsdag 14:15-16:00 S23 Sentralbygg 2
Gruppe 5 Onsdag 14:15-16:00 S23 Sentralbygg 2
Gruppe 6 Torsdag 16:15-18:00 K26 Kjemi 4

You can also attend another exercise session than the one you've been assigned, but the assigned groups will be prioritized.

## Exercises

The exercise sessions will start in week 35. The exercises can be handed in electronically in blackboard or physically in boxes marked MA1301 Talteori H19, which can be found 3. etg Nordre lavblokk SBII. The exercise should be delivered to the correct group. Make sure to write your name on the exercise when you hand it in.

35 §1.1: 1d, 3. §1.2: 3e §2.2: 2, 3b 01-09-2019 23:59 Solutions
36 §2.3: 14a, 15. §2.4: 2b, 2c, 3, 4a, 6 ,8 ,9 08-09-2019 23:59 Solutions
37 §2.5: 2 b), 3 b), 5 b), 6. §3.1: 3 a), 4, 6 a), 10. §3.2: 2, 4 a) 15-09-2019 23:59 Solutions
38 §3.3: 2, 3, 9 a). §4.2: 1, 2, 4 a), 8, 13 22-09-2019 23:59 Solutions
39 §4.3: 1, 2 a), 4. §4.4: 1 a) b) c), 3, 4 a) b), 5 29-09-2019 23:59 Solutions
40 §4.4: 17, 20. §5.2: 1, 2, 5, 7, 10, 13 06-10-2019 23:59 Solutions
41 §5.3: 1, 4, 5, 9. § 6.1: 7, 8, 10 a), 11. 20-10-2019 23:59 Solutions
42 No exercises this week due to midterm exam
43 §7.2: 4, 5, 6, 7, 10, 11, 13 27-10-2019 23:59 Solutions
44 §7.3: 1 a), 2, 3, 8. §8.1: 1 a), 5, 9 a), 10. 03-11-2019 23:59 Solutions
45 §8.2: 1, 4 a), 6, 8 a). §9.1: 1 a) b), 3 a), 5, 7 10-11-2019 23:59 Solutions
46 §9.2: 1 a) b), 4, 8 a), 9. §9.3: 1 a) b), 3 a) b), 4, 10 a) 17-11-2019 23:59 Solutions
47 Problems 2, 3 and 7 from 2011v and Problems 2,3 and 5 from 2016s 17-11-2019 23:59 (Note the deadline!) Solutions

## Course material

We will cover the first 9 chapters, plus some additional topics, of the following book.

• David M. Burton: Elementary Number Theory, 7. edition, ISBN 0071289194

## Reference Group

Didrik Nohre Lønvik

Viveka Priya Simhan

Håvard Skjetne Lilleheie

First meeting: September 17, 2019 at 18:00.