## Ordinary Generating Functions and Recurrence Equations

1) Ordinary generating functions of a variable Generating functions are an important tool for solving combinatorial problems of various types. A typical problem is the counting of the number of objects as a function of the size $$n$$, which we can denote by $$a_{n}$$. Thus, for each value Read more…

## Lambert Series, the Arithmetic Function $$r(n)$$ and Gauss’s Probability Integral

In this article we will study some properties of Lambert series. Then, using Lambert series relative to the representation of integers as sum of two squares, we will compute the value of Gauss’s classical probability integral. 1) Dirichlet generating functions Let’s briefly recall some properties of Dirichlet generating functions. Definition 1.1Given Read more…

## Iterated Function Systems, Fractals and Sierpinski Triangle

In a previous article we have introduced some examples of fractals, illustrating their main characteristics, both qualitative and quantitative: self-similarity, geometric irregularity, fractional dimension. To continue the study of fractal science, it’s first necessary to give a more rigorous definition of the mathematical context in which fractal objects are defined.Each Read more…

## Dirichlet’s Box Principle and Ramsey Numbers

The box principle is attributed to the German mathematician Dirichlet (1805-1859). It is also called the pigeonhole principle. This article illustrates Dirichlet’s principle and gives a brief introduction to Ramsey’s theory, with some examples of computation of Ramsey numbers. For a deeper understanding of the topics of this article you Read more…

## Euler and Möbius Arithmetic Functions and RSA Cryptography

This article illustrates the properties of the Euler and Möbius functions, which have great importance in Number Theory and in other fields. As an example of application we describe the RSA algorithm for public key cryptography. 1) Arithmetic functions An arithmetic function $$f$$ is a function with real or complex Read more…

## Cardano, Gambling and the dawn of Probability Theory

The birth of probability theory is usually set in the mid-seventeenth century. At that time the two great mathematicians Blaise Pascal (1623–1662) and Pierre de Fermat (1601– 1665) discussed together some gambling problems and defined the theoretical basis of the mathematical theory of classical probability.However, the first studies on the Read more…

## Exercises in Elementary Number Theory (III)

Exercise 1 The Fermat numbers are defined as follows: $F_{n} = 2^{2^{n}} + 1 \quad n =0,1,2,\cdots$ Prove that all Fermat numbers with $$n \gt 1$$ have the last digit equal to $$7$$. HintThe number $$2^{2^{2}} = 16$$ ends with the digit $$6$$. The same Read more…

## Exercises in Elementary Number Theory (II)

Exercise 1 Let $$x_{n}$$ be a sequence of positive integers so defined: $\begin{array}{l} x_{1}= 2 \\ x_{n + 1} = x_{n}^{2} – x_{n} +1 \quad n \gt 1 \\ \end{array}$ Prove that the numbers $$x_{n}$$ are pairwise relatively prime. SolutionThe values of the sequence are obtained Read more…

## The Principle of Inclusion-Exclusion

The principle of inclusion-exclusion is an important result of combinatorial calculus which finds applications in various fields, from Number Theory to Probability, Measurement Theory and others. In this article we consider different formulations of the principle, followed by some applications and exercises. For further information, see for example  and Read more…

## Exercises in Elementary Number Theory (I)

In this article we propose some exercises on Elementary Number Theory; they don’t require advanced mathematical knowledge. Other articles will follow with exercises related to this beautiful branch of mathematics. We recall that the symbol $$\left\lfloor x \right\rfloor$$ denotes the integer part of the real number $$x$$, i.e. the largest Read more…