December 1, 2023 – Professor Vjekoslav Kovač and assistant Adrian Beker, exceptionally talented Croatian mathematicians from Zagreb’s Faculty of Science, achieved incredible success by solving two Erdős problems.

As Poslovni/PIXSELL writes, Paul Erdős was a legendary mathematician of the 20^{th} century, who sacrificed almost all aspects of his private life to constantly and endlessly enthusiastically set and solve mathematical problems. He believed that the progress of mathematics as a science should be guided by open questions – because an open problem is beyond the reach of known methods, so for its successful solution new ideas must be obtained, new techniques developed, or even new theories set. And Erdős asked a lot. He sometimes offered monetary rewards for solutions to problems – starting from $25 and going up to $10,000 that he might not have had, but because of the difficulty of his questions, he rarely had to pay out the rewards anyway.

“He posed the best and most interesting problems in articles and presented them at conferences, and those that have remained unsolved to this day are usually very difficult. In our profession, it is a kind of prestige to solve a problem related to Erdős’ name”, explains Vjekoslav Kovač.

## Anyone Can Try

In the middle of this year, the British mathematician Thomas Bloom started the project of collecting Erdős problems on the website Erdős problems, with the only condition that they were posted in a scientific paper, but there are quite a few of those. Currently, about five hundred of them are listed on the page, with references to the original articles in which they were posted, but only a fifth of them have been solved to date.

“Everyone can try because many of the problems are understandable even with school knowledge of mathematics, but it should be warned that they are notoriously difficult, as they have resisted the greatest mathematical minds for decades,” adds Kovač.

## Problem No. 189

The professor and Adrian Beker are the only Croatian mathematicians who managed to solve some of Erdős’ problems. Vjekoslav Kovač is a full professor at the Faculty of Science and Mathematics in Zagreb, having received his doctorate from the University of California in Los Angeles (UCLA). He deals with mathematical analysis, probability theory, and combinatorics. He solved problem number 189, which Erdős and his colleague Ronald Graham set back in 1979.

“Until now, among other things, I have been involved in one line of research that finds default patterns in large geometric structures. I am exclusively a theoretical mathematician, but some aspects of that theory have applications in everyday life, such as designing and analyzing social networks. Issue number 189 is as old as I am. It caught my attention because it suited my mathematical taste and fit in with earlier research”, says Kovač and adds that he does not know exactly how long it took him to solve, because ideas come over a few months, while the key idea is worked out in a few days.

## One of the Best Students in Croatia

Adrian Beker solved problem number 356 set in 1980. As a high school student, Adrian was one of the most successful Croatian students in national and international mathematics competitions. He won the most Oscars of knowledge, five, as well as gold, silver, and two bronze medals at the International Mathematical Olympiad. After high school, he completed his mathematics studies at the University of Cambridge. He is currently studying for his doctorate at the Zagreb Faculty of Science.

The solutions of two of the Croatia geniuses were published on the open server arXiv, where mathematicians often publish their research before sending it to a scientific journal in order to make it available as soon as possible. Reviews in the best magazines often take as long as two years.

## Mathematics Offers Many Possibilities These Days

Professor Kovač says that today a large number of primary and secondary school students are interested in mathematics and are involved in various competitions. He said that mathematics offers many opportunities in employment with above-average salaries.

“I recommend everyone who is interested to come study mathematics because they will certainly not regret it. Some mathematics graduates choose to stay in science, while others easily find excellent jobs in finance or the IT sector. You can meet mathematicians even in the most unusual workplaces, as all professions are increasingly valuing the exact, abstract, and quantitative way of thinking”, Professor Kovač concluded.