Congruences

The idea of grouping integers according to the remainder of their division by a given number is as old as arithmetic. From the "Chinese remainder theorem", inherited from ancient China and studied up to the 13th century, to that of Fermat, the field would see many advances, until the arrival of Gauss, who would formalize the notion of "congruence" and sign the birth certificate of modular arithmetic. This new and powerful vision of numbers, which allows encompassing the infinity of integers in a finite model, also applies to concrete domains: the proof by nine, calendar cycles, security codes... It is also found in music and produces some nice curiosities, such as certain magic tricks.

Just as a small drawing is worth more than a long speech, in the realm of mathematics, nothing beats a counterexample to refute a false intuition or an erroneous conjecture. While it is fascinating (or sometimes amusing) for enthusiasts, the search for counterexamples is not a mere distraction. It is fundamental, both in the reasoning process and as a pedagogical tool. It has punctuated the development of mathematics, allowing for example the theory of functions to take shape. More recently, programs using artificial intelligence have highlighted sophisticated counterexamples, refuting numerous conjectures, particularly in graph theory.