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In this short article we will present some problems related to the calculation of points with integer coordinates, contained inside or on the boundary of some plane figures. This type of problem has very interesting aspects and finds important applications in various fields of mathematics. 1) The integer lattice in Read more…
The French mathematician Bertrand (1822-1900) formulated the conjecture that for every positive integer \(n\) there is always at least one prime number \(p\) such that \[ n \lt p \le 2n \] This conjecture was proved by the Russian mathematician Chebyshev (1821-1894). In this article we will illustrate the proof found by the Read more…
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…
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…
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…
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…
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…
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…
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…
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…