Diophantine number theory

For more than 30 years, the number theory groups in Austria (led by Robert Tichy) and Hungary (led by Klmn Györy) have been working closely on Diophantine problems, that is on problems whose solutions are sought in integers. Through this project we want to maintain and intensify the existing contact between the members of the groups, focusing on effective methods as well as applications of Siegel`s ineffective approach and Schmidt`s subspace theorem. We are interested in polynomial decompositions and algorithms, mainly related to linear recurrences and arithmetic dynamical systems associated to numeration systems. While individual Diophantine equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations was an achievement of the twentieth century. By Matijasevic (1970) there is no algorithm for solving a general polynomial Diophantine equation. However, in the 1960`s A. Baker established a general effective approach for solving wide classes of polynomial and exponential Diophantine equations. For his achievements in developing a method of linear forms in logarithms and its applications to Diophantine equations, in 1970 A. Baker received the Fields medal. The proposed project offers a possibility to combine the classical approach to Diophantine equations, in which the group in Debrecen is world-known (in particular, Györy), with an analytic and geometric point of view that Austrians have mastered over the years of collaboration with world-class experts (Wüstholz, Zannier, etc.). On the Austrian side there will be 9 researchers involved in the project: Tichy, Fuchs, Ziegler, Kreso, Chim, Planitzer, Ddamulira, and if funding is granted, two new PhD students. On the Hungarian side we will have: Györy, Gal, Pintér, Hajdu, Bérczes, Szalay, Tengely, Pink, Bazs, Györkös-Varga, Szikszai and Bertk. The proposed project covers some of the established common topics of interest, but also features some new topics that we believe we could successfully handle jointly, eg. problems surrouding the generalized Ramanujan-Nagell equation. Finally, we remark that the research on Diophantine problems has applications to various other areas of mathematics, such as combinatorics, finite geometry, cryptography, as well as to combinatorial optimization.

Diophantine number theory deals with problems whose possible solutions are in the set of integers. These include equations, inequalities, and questions with a geometric background. This branch of mathematics is ancient and dates back to the Greek mathematician Diophantus of Alexandria. In modern times, this area of research allows for significant applications in data encryption and coding. Moreover, several long-standing open problems have been solved in the past 50 years, such as Fermat's Last Theorem and Catalan's Conjecture. In the present project, deep results have been achieved regarding Diophantine m-tuples and equations of Catalan-type. These results have attracted significant attention from the professional mathematical community.