# What are number theoretic concepts

## Module description

### Module: Elementary Number Theory

### Lectures:

title | Type | SWS | Period |
---|---|---|---|

Elementary number theory | lecture | 4 | Summer and winter semester |

Elementary number theory | Group exercise | 2 | Summer and winter semester |

### Responsible for the module:

Prof. Ulf Kühn

### Admission requirements:

No

### Recommended previous knowledge:

Linear Algebra

### Module objectives / desired learning outcomes:

#### Professional competence

##### Knowledge

- Students can describe basic concepts of elementary number theory such as congruences, square remainders, the ring of whole numbers and Diophantine problems and explain them using examples.
- Students are able to discuss logical relationships between these concepts and to explain them using examples.
- They know strategies of evidence and can reproduce them.

##### Skills

- Students can model tasks from elementary number theory with the help of the concepts they have learned and solve them with the methods they have learned.
- Students are able to independently develop further logical connections between the concepts they have learned and can verify them.
- Students can develop a suitable approach to a given problem, pursue it and critically evaluate the results.

#### Personal competencies

##### Social skills

- Students are able to work together in teams and master mathematics as a common language.
- In particular, you can communicate new concepts in a way that is appropriate to the target group and use examples to check and deepen the understanding of fellow students.

##### independence

- Students can independently check their understanding of complex concepts, bring open questions to the point and, if necessary, get specific help.
- Students have developed enough stamina to work on difficult problems in a targeted manner over longer periods of time.

### Credit points module:

9 LP

### Academic achievement:

Oral exam

### Workload in hours:

Self-study: 186, face-to-face study: 84

### Course: Elementary Number Theory

### Language:

German English

### Period:

Summer and winter semester

### Content:

- Calculating with congruences (Chinese remainder theorem, small Fermatian theorem, application to asymmetric encryption)
- Quadratic remainders (Legendre symbol, quadratic reciprocity law)
- Properties of the ring of whole numbers (unit theorem, arithmetic with ideals, ideal classes)
- Application to Diophantine Problems

### Literature:

- A. Beutelspacher, M.-A. Zschiegner: Discrete Mathematics for Beginners. Vieweg
- F. Ischebeck: Invitation to number theory. BI
- J. Kramer: Numbers for beginners. Vieweg
- K. Reiss, G. Schmieder: Basic knowledge of number theory. Jumper

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