Effective Resolution of Exponential Diophantine Equations

Cooperating countries: Uganda and Austria

Coordinating institution: Paris Lodron University of Salzburg, Volker Ziegler, volker.ziegler@plus.ac.at

Partner institutions: Graz University of Technology, Makerere University

Project duration: 1 July 2023 - 30 June 2025

Abstract:

This project between the Paris Lodron University of Salzburg, Graz University of Technology and Makerere University is devoted to number theory. In particular, we want to consider the subject of Diophantine equations which have been considered since ancient times and are still an active field of research. In this project we are especially interested in so-called exponential Diophantine equations and their effective resolution. In particular we are interested in the following three problems:

1. Pillai's problem for linear recurrences: That is we are interested in the number of solutions of the equation U_n-V_m=c for c a large integer and U_n and V_m being members of linear recurrence sequences.

2. Sums of binary recurrence sequences that are perfect powers: That is given a binary recurrence sequence U_n we are interested to find all solutions (n,m,y,k) to the Diophantine equation U_n+U_m=y^k with m,n,y non-zero integers and k an integer >2. We want to obtain finiteness results in the general case. If we restrict the integers y to certain families such as the integers that have at most s non-zero binary digits, then our aim is to obtain effective results.

3. S-unit values in recurrence sequences: Let (U_n) be a linear recurrence sequence and s a fixed number. We are interested to determine a finite set of primes P depending on the sequence U_n such that for any prime numbers  p_1, … ,p_s outside of P the Diophantine equation U_n=p_1^x_1….p_s^x_s has at most s solutions (n,x_1,…,x_s).