There are mathematical questions that even an elementary school student can understand, yet solving them may require decades or even centuries of work. One such problem, open for 30–40 years, has now been solved by István Pink, a researcher at the University of Debrecen, and his Japanese colleague Takafumi Miyazaki. Their results were published in one of the world’s most prestigious journals, the American Journal of Mathematics, the university’s press center reported.
The research focuses on Diophantine equations—equations with several unknowns whose solutions are sought among integers. A classical example is a² + b² = c², for which (3, 4, 5) is one solution among infinitely many. The problem studied in this case is more general: in the equation aˣ + bʸ = cᶻ, even the exponents are unknown.
The question was the following: if we fix three pairwise relatively prime positive integers greater than one as bases, and raise them to positive integer powers, how many times can it occur that the sum of two powers equals the third? The researchers proved that the equation has at most two positive integer solutions, except in the cases (a, b, c) = (3, 5, 2) and (5, 3, 2), where exactly three solutions exist. With this, a decades-old open problem has been resolved.
The sharpness of the result lies in the fact that there are infinitely many ways to choose base numbers for which exactly two solutions indeed exist. The proof runs to 78 pages and, after an unusually long peer-review process lasting two years, has now been published. The work has received significant international attention and was awarded the Publication Prize of the Count István Tisza Foundation for the University of Debrecen.
Although this is fundamental research, number theory forms the basis of modern digital security: similar theoretical results underpin bank card transactions and encrypted messages. While the present discovery does not directly introduce a new tool, it enriches mathematics with methods that could, in the long term, become foundations of key technologies.
