# What are number theoretic concepts

## Module description

### Lectures:

titleTypeSWSPeriod
Elementary number theorylecture4Summer and winter semester
Elementary number theoryGroup exercise2Summer and winter semester

Prof. Ulf Kühn

No

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.

9 LP

Oral exam

### Workload in hours:

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

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