Prime Numbers and Ultra-Complex Problems
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On November 12, 2025, an engaging open lecture on the topic "Prime Numbers and Ultra-Complex Problems" took place at the Faculty of Computer Science, Physics, and Mathematics. The event gathered faculty members and students, as well as alumni of the "Secondary Education (Mathematics)" educational program from previous years.
The lecturer was Professor Oleksandr Hryhorovych Savchenko of the Department of Algebra, Geometry, and Mathematical Analysis.
Professor Savchenko O.H. began the lecture with a profound, motivating introduction. He noted that the Rector's Chess Cup was taking place at the university that day and drew an interesting parallel: although the share of people who play chess professionally is small, most educated individuals are aware of significant events in the chess world. The same applies to mathematics: while there may be few professional mathematicians, educated people follow the events in the world of mathematics.
The lecturer continued this thought by posing rhetorical questions: "Why study prime numbers? Why solve ultra-complex problems related to prime numbers?" Professor Savchenko O.H. pointed out that such questions could also be asked about the existence of Van Gogh's paintings or artworks by other masters. They do not affect people's daily lives. Nevertheless, both art and the resolution of fundamental mathematical problems are the highest achievements of human culture.
A Dive into Number Theory
Following this inspiring start, the professor immersed the audience in the fascinating world of Number Theory. He discussed why problems with an "elementary" formulation often turn out to be ultra-complex (or extremely difficult) to solve, prove, or disprove.
The audience learned about:
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Attempts to find a function for generating a prime number: The famous functions of Leonhard Euler and Adrien-Marie Legendre — $f(n) = n^2 - n + 41$ and $g(n) = n^2 + n + 41$, which yield several prime numbers; as well as the function of Johann Lejeune Dirichlet $f(n) = an+b$, which yields a prime number for infinitely many values of $n$.
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The Twin Prime Problem and recent breakthroughs in its research.
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Fermat numbers and Mersenne numbers and their connection to prime numbers.
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A detailed examination of the Goldbach conjectures (binary and ternary).
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Fermat's Last Theorem, Euler's conjecture and counterexamples to it.
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The Collatz conjecture ($3n + 1$).
Personal Contribution to Modern Mathematics
A highlight of the lecture was Professor Savchenko’s account of his personal contribution to modern mathematics. He shared his experience of inscribing his own problem into the "Lviv Scottish Book" — a collection of mathematical problems or tasks that follows the tradition of the "Scottish Book" (1935-1941) and is a compilation of new mathematical problems formulated by 21st-century mathematicians, visitors to the revived "Scottish."
The lecture sparked keen interest among the audience and demonstrated that countless mysteries and unsolved questions remain in mathematics.
Professor Savchenko concluded the lecture with warm and important words, wishing everyone present peaceful skies and the soonest possible Victory, and expressing hope for a future meeting within the walls of Kherson State University.
