Uncharted waters

Mathematics is a very lively science, and the blood that animates it are the problems whose solution we still do not know. This uncharted waters have a very diverse geography: some regions are located very deep inside mathematics themselves, in their purest territories, but they also border on other sciences, or on technology – the Internet, for instance, is today an important source of problems for the world of mathematics. We will dedicate this section to recent mathematical discoveries, and also to exploring the uncharted waters of the problems that are still to be solved.

Sum-free sets

30 September, 2025 Juan Luis Varona0

A set $B$ is said to be sum-free if there are no $x,y,z \in B$ with $x+y=z$. The definition can be given, instead of with integers, in the context of abelian groups, but we will […]

Still hot from the oven

19 March, 2024 Juan Arias de Reyna0

Not all problems are the same. Some are hard to understand, like the Hodge conjecture, for whose solution they offer a $1,000,000\$$ prize, but they might as well give $100,000\$$ for understanding it. Others are […]

Flowers of the hypergeometric garden

31 January, 2024 Juan Arias de Reyna0

Isaac Newton stated: “If I have seen further it is by standing on the shoulders of Giants”. Florian and Sergey our heroes today can say the same. Hypergeometric functions Latin was a nightmare for me. […]

The difficult proof of the corrected Sister Beiter conjecture

27 May, 2023 Juan Arias de Reyna0

Cyclotomic Polynomials The $n$-th roots of unity are the solutions of the equation $x^n-1=0$. Except for $n=1$ the polynomial $x^n-1$ is always reducible, for example $x^4-1=(x^2+1)(x+1)(x-1)$. One can wonder how $x^n-1$ factors into irreducibles over […]

Ramsey numbers

25 April, 2023 Juan Arias de Reyna2

Frank Plumpton Ramsey died at the age of 26 but led a very active life. He excelled in at least three fields Philosophy, Mathematics and Economics. In mathematics he is known for Ramsey’s theorem which […]

Erdös primitive set conjecture

9 February, 2023 Juan Arias de Reyna0

In this entry we will see a young mathematician rejecting his old professor’s advice: stop thinking about this impossible problem, leaving a little aside the problem that his thesis director assigned him, and getting away with […]

Riemann hypothesis and Physics

4 November, 2022 Juan Arias de Reyna0

Physics and Mathematics Pólya was hesitant when it came to choosing his career. He said: I thought I was not good enough for physics and I was too good for philosophy. Mathematics is in between. I […]

Greek geometry, algebraic geometry and transcendence

21 June, 2022 Juan Arias de Reyna1

A few days ago Rafa Nadal won his 14th Roland-Garros title, we all heard about it, but few have heard about a much greater triumph of a Spanish mathematician in Paris, Javier Fresán. A Navarrese […]

Sendov conjecture

29 April, 2022 Juan Luis Varona0

Among the simplest objects in mathematics are polynomials in one variable. Children study them in schools and high schools and, at least many of them, learn to operate with them and understand quite a few […]

Is infinity really necessary?

29 January, 2022 Juan Arias de Reyna0

Two principles on the finite/infinite relation One of the axioms of mathematics is that there exists an infinite set. Without this axiom our mathematics would be much weaker. Many of our theorems would fall like […]

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