I4406 - Diophantine Number Theory

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! 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.

Istvan 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.

Istvan Gaal (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,...,a_m} such that a_ia_j+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 {U_n}_{n ≥ 0} and {V_n}_{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 U_n-V_m. On the other hand, I will talk about asymptotic formulas for the number of solutions (m,n)∈ ℕ² to the inequality |αⁿ-β^m|<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 {G_n}_{n ≥ 0} be a non-degenerate linear recurrence sequence with characteristic roots α₁,...,α_t. 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 G_n grows at least as fast as max(|α₁|,...,|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[z₁,...,z_t] where z₁,...,z_t 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 𝔽_q. Inspired by a paper due to Bombieri and Katz we study the sequence A(n):=|C(𝔽_qⁿ)|-(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 P_N (2), we have P_N (2)=P_24,1;N (2)=(8N/3)^1/2+O(1) for N ≥ 6, where P_24,1;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.

Kalman 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.

Nora 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).

Laszlo Szalay (Sopron): F_k divides F_n³-1?, Friday, 19 March 2021: Abstract: Let F_n denote the nth Fibonacci number. The well known divisibility properties F_(n-ε)/2|(F_n-1) for some ε∈{-2,-1,1,2} and F_(n-δ)/2|(F_n²-1) for some δ∈{1,2} induce the following question. Which Fibonacci numbers divide F_n³-1=(F_n-1)(F_n²+F_n+1)? We show that if F_k|F_n²+F_n+1, then k∈{4,7}. Analogously, if L_k|F_n²+F_n+1, where L_k 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^iθ is algebraic but not a root of unity then ∑cos(nθ)/bⁿ is transcendental, where the sum is taken over all n>0 with cos(nθ)>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 f_n, g_n, h_n, respectively. We study the Diophantine equations F(x)=G(y), where F is one of f_n, g_n, h_n, 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.

Peter Sebestyan (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 f_A(x):=Π_a∈A(x-a). Finally we prove two theorems on the structure of the solutions (x,y) of the Diophantine equation f_A(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.

Tobias Hilgart (Salzburg): On a family of cubic Thue Equations involving Fibonacci and Lucas numbers, Friday, 18 June 2021: Abstract: Let F_n denote the n-th Fibonacci number and L_n the n-th Lucas number. We completely solve the family of cubic Thue equations (X-F_nY)(X-L_nY)X-Y³=±1 and show that there are no non-trivial solutions for n≠1,3.