Publications

Book Projects

• Fortgeschrittene Geometrie für Mathematikolympioniken
  February 2007
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  This book is a collection of my geometric experience, it is focused on IMO and IMO Shortlist problems and synthetic techniques to solve them.

Appeared in Journals

• Elementary Properties of Cyclotomic Polynomials
  Mathematical Reflections, 2008, Issue 2
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  Elementary number theoretic properties of cyclotomic polynomials are a topic that has become very popular among olympiad mathematics and the discussions about and around several interesting olympiad problems. The purpose of this note is to give an introductory lesson about this issue and the structure behind it and to present some examples of how this knowledge can be used at olympiad problems.

• The Method of Vieta-Jumping
  Mathematical Reflections, 2007, Issue 5
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  The Vieta-Jumping method is a standard procedure for solving easy to recognize types of divisibility problems and was first seen at the IMO in 1988. Since then, this method has become very popular and there have been many problems at famous mathematical competitions which could have been solved straight-forwardly using Vieta-Jumping, most recently at the IMO in 2007.

• A Note on the Carmichael Function
  Mathematical Reflections, 2007, Issue 2
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  For a positive integer m, λ(m) denotes the smallest positive integer k so that x^k≡1 (mod m) for all integers x with gcd(x, m)=1. λ is the so-called Carmichael Function. Some properties of this function are stated and proved in this note.

• Einige Sätze zu Primitivwurzeln
  Die Wurzel, 01/2007 pages 16-22

  An integer g is called a primitive root modulo m if ord_m(g)=φ(m). It is well known that primitive roots modulo m exist if and only if m has the form 2, 4, p^k or 2p^k where p is an odd prime number and k is a positive integer. This and some other properties of primitive roots are presented and proved in this note.

Submitted

• About the Dirichlet Convolution
  October 2008

  The Dirichlet convolution is a binary operation on arithmetic functions. It is of great importance in number theory, especially in connection with multiplicative functions. The purpose of this note is to state and prove some properties of this convolution.

Publications in Conference Proceedings

• Residue Classes with Order 1 or 2 and a Generalisation of Wilson's Theorem
  Junior Mathematical Congress 2008
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  The well-known Wilson's Theorem states that for a positive integer m≥2, (m-1)!≡-1 (mod m) holds if and only if m is a prime number. A generalisation of this theorem states that the product of all residue classes coprime to m is congruent to -1 if and only if m has the form 2, 4, p^k or 2p^k, where p is an odd prime number and k a positive integer (i.e. iff m has primitive roots). In this paper, the generalisation is proved and its connection to primitive roots explored from the group theoretic perspective.

Other

• Formale Sprachen
  June 2008
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  This note on Formal Languages was one of my essays written in the course of my secondary school graduation exam in the subject "Complex Systems". It is focused on regular languages and concludes with some applications in connection with generating functionology. Since this essay should only give an overview of this topic, some proofs are omitted.

• Komplexitätstheorie
  June 2008
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  This note on Complexity Theory was one of my essays written in the course of my secondary school graduation exam in the subject "Complex Systems". Since it should only give an overview of this topic, some proofs are omitted.

• A Point with many Properties
  April 2008
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• Remainders of Aa^x+Bx
  December 2007
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  In recent times, a class of number theoretic problems became quite popular at various competitions, namely analysing the remainders of a^x+bx modulo some positive integer m. In this note, I will try to unite these problems to a more general statement.

• Zyklische Ungleichungen in 3 Variablen und Wege der Symmetrisierung
  August 2006
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  This paper illustrates some techniques of transforming cyclic inequalities in three variables into symmetric inequalities.

• Eigenschaften und Anwendungen des Lotfußpunktdreiecks
  June 2006
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  Let ABC be a triangle and P be a point in its plane. X, Y and Z are the feets of the perpendiculars from P to BC, CA and AB respectively. Then the triangle XYZ is called the Pedal Triangle of P with respect to ABC, it has many particular properties which are often useful at olympiad problems.

• Multiplikative Inverse - Ein Streifzug durch das Bruchrechnen in Restklassen
  January 2006
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  Many people have problems with mixing fractions and residue classes. This short note should give some enlightenment about what is allowed and what is not.

• Die Mathematik von RSA
  August 2005
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  This little paper on the RSA algorithm was my first steps in LaTeX, nothing extraordinary but it was the first mathematical essay I have written.