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Syllabus:

# Number Theory, 7.5 Credits

Swedish name: Talteori

This syllabus is valid: 2024-09-02 and until further notice

Course code: 5MA123

Credit points: 7.5

Education level: First cycle

Main Field of Study and progress level: Mathematics: First cycle, has at least 60 credits in first-cycle course/s as entry requirements

Revised by: Faculty Board of Science and Technology, 2024-02-20

## Contents

This course treats divisibility properties of the whole numbers, the distribution of primes and modular arithmetic. Further, number theoretic functions are studied, as well as primitive roots of primes, continued fractions, quadratic congruences, reciprocity and quadratic reciprocity. In addition to this, the course includes some modern applications of number theory in cryptography, such as the RSA method and cryptography with elliptic curves.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• account for basic definitions
• use terminology in number theory

Skills and abilities

• analyse and solve problems involving prime factorisations and/or modular congruences
• analyse and solve problems involving the distribution of primes
• analyse and solve problems involving primitive roots
• analyse and solve problems involving continued fractions and/or reciprocity
• analyse and solve problems in some cryptograhpical applications

## Required Knowledge

The course requires courses in Mathematics, minimum 60 ECTS or least two years of university studies including 30 ECTS in mathematics or equivalent. Proficiency in English equivalent to the level required for basic eligibility for higher studies. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

## Form of instruction

The teaching on this course mainly takes the form of lectures and lesson exercises.

## Examination modes

The course is assessed through written examination and hand-in exercises. On written tests, one of the following grades is awarded: Fail (U), Pass (G), Pass with distinction (VG). On hand-in exercises, one of the following grades is awarded: Fail (U), or Pass (G).
The grade for the whole course is determined through a joint assessment of all grades given on the different parts of the examination, and the grade is only set once all compulsory elements have been assessed. To pass the whole course, all elements must have been passed. For the course as whole, one of the following grades is awarded: Fail (U), Pass (G), Pass with distinction (VG).

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for two years following the student's first participation in the course.

Examiners may decide to deviate from the modes of assessment in the course syllabus. Individual adaption of modes of assessment must give due consideration to the student's needs. The adaption of modes of assessment must remain within the framework of the intended learning outcomes in the course syllabus. Students who require an adapted examination must submit a request to the department holding the course no later than 10 days before the examination. The examiner decides on the adaption of the examination, after which the student will be notified.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications.

## Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics

In the event that the syllabus ceases to apply or undergoes major changes, students are guaranteed at least three examinations (including the regular examination opportunity) according to the regulations in the syllabus that the student was originally registered on for a period of a maximum of two years from the time that the previous syllabus ceased to apply or that the course ended.

## Literature

### Valid from: 2024 week 36

Introduction to number theory
Erickson Martin J., Vazzana Anthony
Second edition. : Boca Raton : CRC Press : [2016] : xii, 414 pages :
ISBN: 9781498717496
Mandatory
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