FWF 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.

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)
Kálmán Gyõry (Debrecen; PI)
Lajos Hajdu (Debrecen; co-PI)
Sebastian Heintze (Salzburg)
Tobias Hilgart (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)

PhD Defense of Sebastian Heintze: On Friday, 14 May 2021 at 2.15pm Sebastian Heintze has defended his PhD thesis entiteled New results on Diophantine Problems related to linear recurrences.
Research seminar:
In order to foster cooperation in times of Covid-19 we start with a research 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:
The further program will continuously be updated here. Stay safe and stay healthy!
Tobias Hilgart (Salzburg): On a family of cubic Thue Equations involving Fibonacci and Lucas numbers, Friday, 18 June 2021
Abstract: Let Fn denote the n-th Fibonacci number and Ln the n-th Lucas number. We completely solve the family of cubic Thue equations (X-FnY)(X-LnY)X-Y3=±1 and show that there are no non-trivial solutions for n≠1,3.
Previous talks:
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 {a1,a2,...,am} such that aiaj+n is a perfect square for all 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 {Un}n ≥ 0 and {Vn}n ≥ 0 be two linear recurrence sequences satisfying dominant roots conditions. I will report on asymptotic formulas for the number of integers c in the range [-x,x] which can be represented as differences Un-Vm. On the other hand, I will talk about asymptotic formulas for the number of solutions (m,n)∈ ℕ2 to the inequality nm|<x. Here, α or β could be transcendental. This is a joint work with Robert Tichy, Ingrid Vukusic and Volker Ziegler.
Sebastian Heintze (Salzburg): On the growth of linear recurrences, Friday, 4 December 2020
Abstract: Let {Gn}n ≥ 0 be a non-degenerate linear recurrence sequence with characteristic roots α1,...,αt. 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 n large the absolute values of Gn grows at least as fast as max(1|,...,|αt|)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[z1,...,zt] where z1,...,zt 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.
Volker Ziegler (Salzburg): Explicit lower bounds for the Frobenius traces of genus 2 curves, Friday, 18 December 2020
Abstract: Let C be a nonsingular, complete curve of genus 2 over the finite field 𝔽q. Inspired by a paper due to Bombieri and Katz we study the sequence A(n):=|C(𝔽qn)|-(qn+1). In the case that A(n)≠ 0 we prove explicit lower bounds.
Sebastian Heintze (Salzburg): Norm form equations with solutions in a multi-recurrence, Friday, 15 January 2021
Abstract: We are interested in solutions of a norm form equation that takes values in a given multi-recurrence. We show that among the solutions there are only finitely many values in each component which lie in the given multi-recurrence unless the recurrence is of precisely described exceptional shape. This gives a variant of the question on arithmetic progressions in the solution set of norm form equations.
Lajos Hajdu (Debrecen): Powers in arithmetic progressions, Friday, 22 January 2021
Abstract: The question that at most how many squares one can find among N consecutive terms of an arithmetic progression, has attracted a lot of attention. Among many other results and questions, a sharp conjecture of Rudin predicts that for this maximum PN (2), we have PN (2)=P24,1;N (2)=(8N/3)1/2+O(1) for N ≥ 6, where P24,1;N (2) denotes the number of squares in the arithmetic progression 24n+1 for 0≤ n<N. In the talk we take up the problem for arbitrary l-th powers. First we characterize those arithmetic progressions which contain the most l-th powers asymptotically. In fact, we can give a complete description, and it turns out that basically the "best" arithmetic progression is unique for any l. Then we formulate analogues of Rudin's conjecture for general powers l, and we prove these conjectures for l=3 and 4 up to N=19 and 5, respectively. The new results presented are joint with Sz. Tengely.
Kálmán Gyõry (Debrecen): Monogenity, multiple monongenity and power integral bases in number fields, Friday, 5 March 2021
Abstract: First we give a survey on index form equations, power integral bases, monogenity resp. multiple monogenity of number fields and orders. Then we present some recent results on multiply monogenic orders and some generalizations over finitely generated domains.
Nóra Varga (Debrecen): Polynomial Values of Figurate Numbers, Friday, 19 March 2021
Abstract: There are a lot of effective, ineffective and explicit results concerning power values and polynomial values of binomial coefficients. Also, many papers deal with generalizations of these problems, involving polygonal numbers and pyramidal numbers. In this talk we show effective and ineffective theorems concerning polynomial values of figurate numbers. Our results yield common extensions and generalizations of several previous theorems from the literature (joint work with Lajos Hajdu).
László Szalay (Sopron): Fk divides Fn3-1?, Friday, 19 March 2021
Abstract: Let Fn denote the nth Fibonacci number. The well known divisibility properties F(n-ε)/2|(Fn-1) for some ε∈{-2,-1,1,2} and F(n-δ)/2|(Fn2-1) for some δ∈{1,2} induce the following question. Which Fibonacci numbers divide Fn3-1=(Fn-1)(Fn2+Fn+1)? We show that if Fk|Fn2+Fn+1, then k∈{4,7}. Analogously, if Lk|Fn2+Fn+1, where Lk is the nth Lucas number, then k∈{2,4}. This is a joint result with F. Luca and P. Pongsriiam.
Matija Kazalicki (Zagreb): Recent development in the theory of rational Diophantine m-tuples, Friday, 26 March 2021
Abstract: First, we will present two results of the similar flavour - that there are infinitely many rational Diophantine sextuples with square denominators, and that there are infinitely many rational Diophantine quintuples with the property that the product of any of its two elements is a square. This is a joint work with Andrej Dujella and Vinko Petričević. The generalization of the parametrization of Diophantine quadruples (used in the first result) to the D(q)-quadruples will be explained in the rest of the talk. This is joint work with Goran Dražić.
Florian Luca (Wits/Jeddah/Saarbrücken): A variation of a transcendence criterion, Friday, 16 April 2021
Abstract: In this talk, we survey the transcendence criterion for automatic numbers due to Adamczewski, Bugeaud and Luca, and introduce additional conditions in order to make it more useful for particular instances. As a concrete application we prove that if b is algebraic, |b|≥1 and θ is real such that e is algebraic but not a root of unity then ∑cos()/bn is transcendental, where the sum is taken over all n>0 with cos()>0. Further, if ρ is real quadratic and r∈ℚ, then ∑b-⌊nρ+r is transcendental, where the sum is taken over all n≥0. This is joint work with J. Ouaknine and J.B. Worrell.
Ingrid Vukusic (Salzburg): Consecutive tuples of multiplicatively dependent integers, Friday, 23 April 2021
Abstract: A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are (2,8) and (3,9). For triples, we prove the following theorem: If a∉{2,8} is a fixed integer larger than 1, then there are only finitely many triples (a,b,c) of pairwise distinct integers larger than 1 such that (a,b,c), (a+1,b+1,c+1) and (a+2,b+2,c+2) are each multiplicatively dependent. Moreover, these triples can be determined effectively. This is a joint work with Volker Ziegler.
Alan Filipin (Zagreb): The extension of D(-k)-pair {k,k+1}, Friday, 30 April 2021
Abstract: In this talk we will consider the extension of D(-k)-pair {k,k+1} to a quadruple, where k is a positive integer. In the first part of the talk we will prove that if {k,k+1,c,d} is a D(-k)-quadruple with c<d, then c=1. In the second part we will consider the extension of D(-k)-triple {1,k,k+1} to a quadruple. We did not solve the problem completely for general k, but we will present some results when k is of special form, and we will give some conjectures that we believe are true. This is joint work with N. Adzaga and Y. Fujita.
Orsolya Herendi (Debrecen): Finiteness results for polynomial values of surface point counting polynomials, Friday, 7 May 2021
Abstract: In the talk we consider regular polytopes in n-dimensional lattices: the n-dimensional cube, the n-dimensional pyramid and the n-dimensional simplex. It turns out that the number of lattice points on the surfaces of these objects can be described by certain polynomials fn, gn, hn, respectively. We study the Diophantine equations F(x)=G(y), where F is one of fn, gn, hn, and G is a polynomial with rational coefficients, proving various effective and ineffective finiteness theorems. Our results are closely related to those of Bazsó, Bennett, Bilu, Gyõry, Hajdu, Pethõ, Pintér, Tengely, Tichy, Tijdeman, Varga and many others. The presented new results are joint with L. Hajdu.
Péter Sebestyén (Debrecen): Sums of S-units in the solution sets of generalized Pell equations, Friday, 7 May 2021
Abstract: In the talk we give various finiteness results concerning solutions of generalized Pell equations representable as sums of S-units with a fixed number of terms. In case of one term, our result is effective, while in case of more terms we are able to bound the number of solutions only. The presented new results are joint with L. Hajdu.
Yuri Bilu (Bordeaux): Binary polynomial power sums vanishing at roots of unity, Friday, 21 May 2021
Abstract: I will speak on a recent work with Florian Luca, where we show that, apart some obvious exceptions, the polynomials of the form a(x)f(x)n+b(x)g(x)n cannot vanish at roots of unity of order exceeding some effective constant. The qualitative result follows from the Bombieri-Masser-Zannier work on unlikely intersections, but our contribution is making this effective (in fact, totally explicit), which, to our knowledge, cannot be deduced from any available version of the BMZ or similar theorem. Our principal tools are Schinzel's theorem on primitive divisors, and Mann's theorem on vanishing sums of roots of unity.
Rob Tijdeman (Leiden): The Prouhet-Tarry-Escott problem, indecomposability of polynomials and Diophantine equations, Friday, 28 May 2021
Abstract: In this talk we show how the subjects mentioned in the title are related. First we study the structure of partitions of A ⊆ {1,...,n} in k-sets such that the first k-1 symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial fA(x):=ΠaA(x-a). Finally we prove two theorems on the structure of the solutions (x,y) of the Diophantine equation fA(x)=P(y) where P(y)∈ℚ[y]. This is a joint work with L. Hajdu and Á. Papp.
Christopher Frei (Graz): Constructing abelian extensions with prescribed norms, Friday, 4 June 2021
Abstract: Let K be a number field, S a finite set of elements of K, and G a finite abelian group. We explain how to construct explicitly a normal extension L of K with Galois group G, such that all elements of S are norms of elements of L. The construction is based on class field theory and a recent formulation of Tate’s criterion for the validity of the Hasse norm principle. This is joint work with Rodolphe Richard (UCL).
Christian Elsholtz (Graz): Arithmetic progressions in arithmetic sets, Friday, 11 June 2021
Abstract: We study the length of arithmetic progressions in arithmetically interesting sets. For the set of squares this question goes back to Fermat (there are no 4 squares in progression). We discuss new upper bounds in the value set of binary quadratic forms and norm forms (joint work with C. Frei), discuss long gaps between sums of two squares (joint with R. Dietmann, A. B. Kalmynin, S. Konyagin, and J. Maynard), and make a new observation on long progressions in sums of S-units.

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

Publications:
I. Vukusic, On a cubic Family of Thue Equations involving Fibonacci Numbers and Powers of Two, Quaest. Math, to appear, arXiv:2107.03097
T. Hilgart, I. Vukusic, V. Ziegler, On a family of cubic Thue equations involving Fibonacci and Lucas numbers, arXiv:2106.03509
I. Vukusic, V. Ziegler, On a family of unit equations over simplest cubic fields, JNTB, arXiv:2104.12514
I. Vukusic, V. Ziegler, Consecutive tuples of multiplicatively dependent integers, JNT, arXiv:2103.08542
R. Tichy, I. Vukusic, D. Yang, V. Ziegler, On a variant of Pillai's problem with transcendental numbers, arXiv:2011.10387
R. Tichy, I. Vukusic, D. Yang, V. Ziegler, Integers representable as differences of linear recurrence sequences, Res. Number Theory 7, 24 (2021), arXiv:2008.00844
C. Fuchs, S. Heintze: A polynomial variant of Diophantine triples in linear recurrences, Period. Math. Hungar., to appear, arXiv:2006.12173
C. Fuchs, S. Heintze, Diophantine equations in separated variables and polynomial power sums, Monatsh. Math. 196 (2021), 59-65, arXiv:2008.10342
C. Fuchs, S. Heintze: On the growth of linear recurrences in function fields, Bull. Austr. Math. Soc. 104 (2021), 11-20, arXiv:2006.11074
C. Fuchs, S. Heintze, Integral zeros of a polynomial with linear recurrences as coefficients, Indag. Math. 32 (2021), 691-703, arXiv:2008.10328
C. Fuchs, S. Heintze: Norm form equations with solutions taking values in a multi-recurrence, Acta Arith. 198.4 (2021), 427-438, arXiv:2006.11075
C. Fuchs, S. Heintze, A function field variant of Pillai's problem, JNT 222 (2021), 278-292, arXiv:2008.10339
C. Fuchs, S. Heintze: Another S-unit variant of Diophantine tuples, PAMS 149 (2021), 27-35, arXiv:1910.09285

Guests:
TBA

Archive:
Talk entitled "Polynomials, Power Sums and Subspaceology" given by C. Fuchs in the Online Seminar at Würzburg, February 2, 2021
Online research seminar has been launched on 9 October 2020.

Impressum    17.09.2021