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

Budget: EUR 11.600

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). 
 

Summary:

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 has been considered since ancient times and is still an active field of research. In this project we are especially interested in so-called exponential Diophantine equations and their effective resolution.

Solving a Diophantine equation is to solve an equation in two or more unknowns, but only integer solutions are sought. We call a Diophantine equation exponential, if at least one of the unknowns occur in the exponent. The most famous (exponential) Diophantine equation is Fermat’s equation which is the equation x^n+y^n=z^n in positive integers x, y and z and integer exponent n>2.  That this equation has no solution was an open problem for more than 250 years and was finally resolved by Andrew Wiles in 1995.

In this project we are concerned to develop new techniques and algorithms to solve exponential Diophantine equations. In the project we focused on exponential Diophantine equations involving recurrence sequences. Within this project we proved new results on Pillai type equations, representation of S-units as sums of members in linear recurrence sequences and twisted Thue equations. These research effort lead to one publication. Two more publications concerning this topic are in preparation and will be submitted to high quality international, peer reviewed journals.

Beside the scientific progress one aim of the project was also to bring attention to these kind of problems in higher education. That is public talks were given at the involved universities where students could learn recent research progress from the members of the project.