A passion for numbers and geometry
Paul Erdős solved many problems in all fields, but his primary passion is certainly number theory. He thus devoted many works to prime numbers. While most of his results, quite advanced, are difficult to popularize, some remain nevertheless accessible, at least in their formulation. This is the case, for example, in geometry, of problems of distances defined by a set of points, of the dissection of a square into squares of different sizes, or of the number of lines passing through given points.
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So many distances!
Here are two problems about distances that interested Paul Erdős. The first studies the distances defined by n points. The second looks for sets of points that define only integer distances.
Some conjectures in algebra
Here are a few conjectures proposed by Erdős that, despite some progress, remain unsolved...
Two geometric jewels
Long after the 19th century, the golden age of geometry, Erdős took an interest in problems whose statements could appear in an elementary textbook. Among these are two jewels of elegance: the Erdős–Mordell theorem, which involves nothing more than a triangle, and the Erdős–de Bruijn theorem, which features only points defining lines.
Trinity College and the dissection of the square | Tangente
In the 1930s, the problem of dissecting a square into smaller squares of different sizes gave rise to two conjectures by Erdős. They would be disproved by four Trinity College students who threw themselves into the research.
Some of Erdős's work in number theory
Analytic and probabilistic number theory was one of Paul Erdős's favorite subjects. Here is a small selection of his contributions in this field, picked here and there from topics that can still be presented accessibly.
A cornucopia
Paul Erdős and Leonidas Alaoglu studied highly abundant and superabundant numbers, rediscovering along the way notions already studied, but not published, by the celebrated Indian mathematician Ramanujan.