QuaderniCIRD 27 (2023)

Cryptography can be a tool that, enables students to consolidate knowledge and develop mathematical skills if used as a recreational or playful activity. In this paper we introduce Caesar’s cryptosystem, which is one of the simplest encryption schemes. It is very easy to crack the encryption of this system: a brute force attack allows easy recovery of the key and the plaintext. We made a simple modification that allows us to have a greater number of keys, in order to make such a brute force attack infeasible. Finally, we show a statistical method that allows us to force this cryptosystem.

In this contribution, a qualitative analysis is conducted on the responses provided by ChatGPT regarding certain computational exercises in Calculus 1 (such as function studies or limit calculations) and demonstrative exercises. To better understand the context and the reasons behind these responses, the experiments are preceded by a section discussing the architecture of the neural network upon which this type of artificial intelligence is based.

The purpose of this paper is to present the figure of Sophie Germain, setting her in the historical period in which she lived, and to try to explain some of the mathematical tools she used in her attempt to solve one of the most famous theorems in the history of mathematics: Fermat’s Last Theorem. Though the French mathematician gave some important results concerning Fermat’s problem, historically she has not been given credit for what she proved, until recent studies have re-evaluated her works. In the first and second section we present Sophie Germain’s biography and introduce Fermat’s Last Theorem from the algebraic point of view. The following sections are dedicated to the presentation of the most important results by Sophie Germain and the techniques she uses to prove them. Finally, we briefly discuss the failure of her plan, the so called “Grand Plan”, to prove Fermat’s conjecture.

Even if surprising for many mathematicians, quite a large number of the distances described in the Encyclopedia of distances, are not metric distances, i.e., they do not comply with some or other of the metric axioms which appear to be so natural or even unexpendable to those who tackle this multifaceted geometric and topological notion. Using examples taken from linguistics and word strings, we argue that the notion of distance is so rich and fruitful that the metric axioms in some cases risk to be an unreasonably narrow cage.