I4406 - Diophantine Number Theory

In this joint project of the FWF and the NKFIH we shall use different techniques from Diophantine number theory (as the subspace theorem, the theory of linear forms in logarithms, Runge's method, hypergeometric methods, etc.) to study various Diophantine problems. E.g. we work on Ritt's decomposition theory and Diophantine applications, we shall study some classical Diophantine equations (as the Erdös-Straus conjecture, Goormaghtigh's equation and the generalized Ramanujan-Nagell equation, Thue and relative Thue equations), and we shall investigate Diphantine problems with recurrence sequences. We emphasize that the Austrian and Hungarian research groups have a long standing and very fruitful cooperation, which is exceptional even by the international standards. Through this project we want to maintain this tight scientific bond and in particular encourage collaboration between younger members of the groups.

In this joint project of the FWF and the NKFIH we shall use different techniques from Diophantine number theory (as the subspace theorem, the theory of linear forms in logarithms, Runge's method, hypergeometric methods, etc.) to study various Diophantine problems. E.g. we work on Ritt's decomposition theory and Diophantine applications, we shall study some classical Diophantine equations (as the Erdös-Straus conjecture, Goormaghtigh's equation and the generalized Ramanujan-Nagell equation, Thue and relative Thue equations), and we shall investigate Diphantine problems with recurrence sequences. We emphasize that the Austrian and Hungarian research groups have a long standing and very fruitful cooperation, which is exceptional even by the international standards. Through this project we want to maintain this tight scientific bond and in particular encourage collaboration between younger members of the groups.

The project has started on 01.04.2020.

Team:

András Bazsó (Debrecen)

Attila Bérczes (Debrecen)

Csanád Bertók (Debrecen)

Kwok Chi Chim (Graz)

Mahadi Ddamulira (Graz)

Clemens Fuchs (Salzburg; co-PI)

István Gaál (Debrecen)

Kalman Gyõry (Debrecen; PI)

Lajos Hajdu (Debrecen; co-PI)

Sebastian Heintze (Salzburg)

Dijana Kreso (Graz)

István Pink (Debrecen)

Ákos Pintér (Debrecen)

László Szalay (Sopron)

Márton Szikszai (Debrecen)

Szabolcs Tengely (Debrecen)

Robert Tichy (Graz; PI)

Nóra Varga (Debrecen)

Ingrid Vukusic (Salzburg; funded by FWF-I4406)

Volker Ziegler (Salzburg)

András Bazsó (Debrecen)

Attila Bérczes (Debrecen)

Csanád Bertók (Debrecen)

Kwok Chi Chim (Graz)

Mahadi Ddamulira (Graz)

Clemens Fuchs (Salzburg; co-PI)

István Gaál (Debrecen)

Kalman Gyõry (Debrecen; PI)

Lajos Hajdu (Debrecen; co-PI)

Sebastian Heintze (Salzburg)

Dijana Kreso (Graz)

István Pink (Debrecen)

Ákos Pintér (Debrecen)

László Szalay (Sopron)

Márton Szikszai (Debrecen)

Szabolcs Tengely (Debrecen)

Robert Tichy (Graz; PI)

Nóra Varga (Debrecen)

Ingrid Vukusic (Salzburg; funded by FWF-I4406)

Volker Ziegler (Salzburg)

In order to foster cooperation in times of Covid-19 we start with a reserach seminar in which results related to the project are presented and discussed. All team members on the Austrian and Hungarian side are cordially invited to participate; link and password of the browser based and self-explanatory Webex meetings will be sent by email. In case you are interested to attend in one of the talks, please send an email to clemens.fuchs AT sbg.ac.at.

Time: Friday, 2.00-3.15pm

Place: Online

Upcoming talks:

- Sebastian Heintze (Salzburg): On the growth of linear recurrences, Friday, 4 December 2020
- Abstract: Let {
*G*}_{n}be a non-degenerate linear recurrence sequence with characteristic roots_{n ≥ 0}*α*. In this talk we will present a function field analogue of the well known result that in the number field case, under some non-restrictive conditions, for_{1},...,α_{t}*n*large the absolute values of*G*grows at least as fast as max(_{n}*|α*)_{1}|,...,|α_{t}|.^{n(1-ε)}

- Jan-Hendrik Evertse (Leiden): Effective methods for Diophantine equations over finitely generated domains, Friday, 11 December 2020
- Abstract: A finitely generated domain over ℤ is a domain containing ℤ generated by finitely many elements, i.e., ℤ[
*z*] where_{1},...,z_{t}*z*may be algebraic or transcendental. In the early 1960s, Lang proved several finiteness results for Diophantine equations with unknowns taken from a finitely generated domain as above, but his proofs are ineffective. In the 1980s, Gyõry developed an effective method to deal with Diophantine equations but this worked only for a restricted class of domains. Gyõry's method combines Baker's theory on linear forms in logarithms with effective methods for Diophantine equations over function fields and an effective specialization method. Some years ago, Gyõry and the speaker extended this to all finitely generated domains. Since then, this has been applied to several classes of Diophantine equations, in works of Gyõry and the speaker, Bérczes and Koymans. In his lecture at the recent workshop Gyõry gave a survey of these applications. In my talk I would like to explain in more detail the method, and also say something on recently developed techniques._{1},...,z_{t} - Sebastian Heintze (Salzburg): On two variants of Diophantine tuples, Friday, 9 October 2020
- Abstract: We will consider two different variants of Diophantine tuples occurring in recent papers. The first one is an S-unit variant in number fields and the second one a polynomial variant using linear recurrences. In both cases we will sketch the proof given in the respective paper.
- István Pink (Debrecen): Special type of unit equations in two variables, Friday, 16 October 2020

- Abstract: For any fixed coprime positive integers
*a,b*and*c*with min{*a*,*b*,*c*}>1, we prove that the equation*a^x+b^y=c^z*has at most two solutions in positive integers*x,y*and*z*, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.A.Bennett, On some exponential equations of S.S. Pillai, Canad. J. Math. 53(2001), no.2, 897-922] which asserts that Pillai's type equation*a^x-b^y=c*has at most two solutions in positive integers*x*and*y*for any fixed positive integers*a,b*and*c*with min{*a*,*b*}>1. In this talk we give a brief summary of corresponding earlier results and present the main improvements leading to this definitive result. This is a joint work with T. Miyazaki. - Ingrid Vukusic (Salzburg): On sums of Fibonacci numbers with few binary digits, Friday, 23 October 2020

- Abstract: We find all natural numbers which are representable as the sum of exactly two Fibonacci numbers and simultaneously as the sum of exactly five powers of two. In addition to complex linear forms in logarithms and the Baker-Davenport reduction method, we use
*p*-adic versions of both tools. This is a joint work with Volker Ziegler. - István Gaál (Debrecen): Power integral bases in algebraic number fields, Friday, 30 October 2020

- Abstract: We recall the basic notions and results of monogenity and power integral bases. After a short survey on the constructive methods to calculate generators of power integral bases in lower degree number fields we shall present some recent results on higher degree number fields, relative extensions, composites of number fields.
- Dijana Kreso (Graz): On Diophantine
*m*-tuples and related*D(n)*-sets, Friday, 6 November 2020

- Abstract: For a nonzero integer
*n*, a set of distinct nonzero integers*{a*such that_{1},a_{2},...,a_{m}}*a*is a perfect square for all_{i}a_{j}+n*1≤i<j≤m*, is called a Diophantine*m*-tuple with the property*D(n)*or*D(n)*-set. The*D(1)*-sets are called simply Diophantine*m*-tuples, and have been studied since the ancient times. In this talk we will discuss the existence and the finiteness of Diophantine*m*-tuples which are simultaneously*D(n)*-sets for some*n≥1*. - Szabolcs Tengely (Debrecen): Integral points via Baker's method, Mordell-Weil sieve and hyperelliptic logarithms, Friday, 13 November 2020

- Abstract: In case of genus 2 curves we know by the celebrated result of Faltings that there are only finitely many rational points. The result cannot be used to actually determine the complete set of points in concrete examples. An older result by Chabauty provides a bound on the number of solutions and sometimes the bound is equal to the number of known points. However, this method does not work if the rank of the Mordell-Weil group is larger (>1). In case of integral points one can get a very large upper bound for the solutions via Baker's method. Using this huge bound one has two different approaches to handle the problem. The first one uses the so-called Mordell-Weil sieve, the second one uses hyperelliptic logarithms. Both procedures have been successfully applied to determine the set of integral points in case of genus 2 curves with Mordell-Weil groups of ranks 3,4,5 and 6. If the rank is larger (>4), then the time of the computation is getting longer and longer (couple of hours). In this talk we show that it is possible to combine the two approaches making possible to reduce the running times.
- Daodao Yang (Graz): Integers representable as differences of linear recurrence sequences and a variant of Pillai's problem with transcendental numbers, Friday, 20 November 2020

- Abstract: Let {
*U*}_{n}and {_{n ≥ 0}*V*}_{n}be two linear recurrence sequences satisfying dominant roots conditions. I will report on asymptotic formulas for the number of integers_{n ≥ 0}*c*in the range [*-x,x*] which can be represented as differences*U*. On the other hand, I will talk about asymptotic formulas for the number of solutions_{n}-V_{m}*(m,n)*∈ ℕto the inequality^{2}*|α*. Here,^{n}-β^{m}|<x*α*or*β*could be transcendental. This is a joint work with Robert Tichy, Ingrid Vukusic and Volker Ziegler.

Virtual conference Diophantine Problems, Determinism and Randomness, 23-27 November 2020 at CIRM (Marseille Luminy, France): Registration and information

C. Fuchs, S. Heintze, Diophantine equations in separated variables and polynomial power sums, arXiv:2008.10342

C. Fuchs, S. Heintze, A function field variant of Pillai's problem, arXiv:2008.10339

C. Fuchs, S. Heintze, Integral zeros of a polynomial with linear recurrences as coefficients, arXiv:2008.10328

R. Tichy, I. Vukusic, D. Yang, V. Ziegler, Integers representable as differences of linear recurrence sequences, arXiv:2008.00844

C. Fuchs, S. Heintze: A polynomial variant of Diophantine triples in linear recurrences, arXiv:2006.12173

C. Fuchs, S. Heintze: Norm form equations with solutions taking values in a multi-recurrence, Acta Arith., to appear, arXiv:2006.11075

C. Fuchs, S. Heintze: On the growth of linear recurrences in function fields, Bull. Austr. Math. Soc., to appear, arXiv:2006.11074

C. Fuchs, S. Heintze: Another S-unit variant of Diophantine tuples, PAMS, to appear, arXiv:1910.09285

TBA

Online research seminar has been launched on 9 October 2020.

Impressum 28.11.2020