## Bertrand Hypothesis and Ramanujan Prime Numbers

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…

## 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…

## 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…

## 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…

## Gauss’s Modular Arithmetic and Fermat’s Little Theorem

1) Gauss’s Modular Arithmetic Given a positive integer $$m$$, we say that two integers $$a$$ and $$b$$ are congruent modulo $$m$$ if they give the same remainder when divided by $$m$$. We use the following notation introduced by the German mathematician Gauss: \[ a \equiv b Read more…

## The Divisors of an Integer, Perfect Numbers and Fermat Numbers

The study of integers and their properties is the fundamental object of Number Theory. The properties of prime numbers and divisors of an integer were first studied extensively during the period of ancient Greece (Pythagoras, Euclid, etc.); the study resumed again in the seventeenth century, in particular thanks to the Read more…

## The Mystery of Prime Numbers

A prime number is an integer greater than 1 which is divisible only by 1 and by itself. An integer that is not prime is called composite. Each integer greater than 1 is divisible by at least one prime number. The study of integers and their properties began with the Greeks (Pythagoras, Euclid,…), Read more…