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One of the most famous mathematicians of the 20th century, Paul Erdős (1913-1996), was known for setting open problems with elementary and simple formulations, but which were just beyond the reach of known mathematics and required imaginative solutions or the development of new techniques.

It is impossible to collect all the problems that Erdős posed during his lifetime, but the British mathematician Thomas Bloom started a project to collect a large number of Erdős' most interesting problems. Currently, there are about 500 problems on the "Erdős problems" website (http://www.erdosproblems.com/) and only about a fifth of them have been solved to date.

Recently, Vjekoslav Kovač, a full professor at the Mathematical Department of the Faculty of Economics, solved problem #189 by showing that there exists a coloring of the plane in finitely many colors such that there is no rectangle of unit area with all four vertices of the same color.

 Then Adrian Beker, an assistant and doctoral student also at the Mathematical Department of the Faculty of Economics, solved problem #356 by showing that, for every natural number n, there are natural numbers a1<a2<…<ak smaller than n whose consecutive sums take at least cn2 different values.

Their proofs have so far only been published in a preprinted form (arxiv.org/abs/2309.09973 and arxiv.org/abs/2311.10087), but the solutions themselves are very elegant and have already been checked and confirmed on the "Erdős problems" website. This exciting news was recently reported by the Croatian Mathematical Society.

Vjekoslav Kovač and Adrian Beker are so far the only Croatian mathematicians living and working in Croatia who have solved some of Erdős' problems. Among Croatian mathematicians (in terms of origin), the late Branko Grünbaum is also among the solvers.


Author: Božidar Tartaro
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