Computational Aspects of Diophantine Equations
Conference handbook
February 15-19, 2016
University of Salzburg
Contents
Welcome 3
General Information 4
Conference Location . . . . . . . . 4Registration . . . . . . . . . . . . . 4Conference Opening . . . . . . . . 4Technical Equipment of the Lec-
ture Room . . . . . . . . . . . 4Internet Access during Conference . 4Lunch and Dinner . . . . . . . . . 5Co�ee Breaks . . . . . . . . . . . . 5Conference Picture . . . . . . . . . 5Local Transportation . . . . . . . . 5Social Program . . . . . . . . . . . 5Information about the Conference
Venue Salzburg . . . . . . . . 6Information about the University
of Salzburg . . . . . . . . . . 6
Program 7
Summary . . . . . . . . . . . . . . 7Monday . . . . . . . . . . . . . . . 7Tuesday . . . . . . . . . . . . . . . 8Wednesday . . . . . . . . . . . . . 9Thursday . . . . . . . . . . . . . . 9Friday . . . . . . . . . . . . . . . . 10
Abstracts 11
András Bazsó . . . . . . . . . . . . 11Michael A. Bennett . . . . . . . . . 11Attila Bérczes . . . . . . . . . . . . 11Csanád Bertók . . . . . . . . . . . 12Nicolas Billerey . . . . . . . . . . . 13
Yuri F. Bilu . . . . . . . . . . . . . 13Laura Capuano . . . . . . . . . . . 13Andrej Dujella . . . . . . . . . . . 14Bernadette Faye . . . . . . . . . . . 14Christopher Frei . . . . . . . . . . 14Maciej Gawron . . . . . . . . . . . 15Eva Goedhart . . . . . . . . . . . . 17Krisztián Gueth . . . . . . . . . . . 17Kálmán Gy®ry . . . . . . . . . . . 17Lajos Hajdu . . . . . . . . . . . . . 18Christoph Hutle . . . . . . . . . . . 18Angelos Koutsianas . . . . . . . . . 19Dijana Kreso . . . . . . . . . . . . 19Florian Luca . . . . . . . . . . . . 19Takafumi Miyazaki . . . . . . . . . 20Filip Najman . . . . . . . . . . . . 20Roland Paulin . . . . . . . . . . . . 21Attila Peth® . . . . . . . . . . . . . 21László Remete . . . . . . . . . . . . 22Ivan Soldo . . . . . . . . . . . . . . 23Michael Stoll . . . . . . . . . . . . 23Tímea Szabó . . . . . . . . . . . . 23László Szalay . . . . . . . . . . . . 24Márton Szikszai . . . . . . . . . . . 25Petra Tadi¢ . . . . . . . . . . . . . 26Alain Togbé . . . . . . . . . . . . . 26Maciej Ulas . . . . . . . . . . . . . 26Nóra Varga . . . . . . . . . . . . . 27Francesco Veneziano . . . . . . . . 27Martin Widmer . . . . . . . . . . . 27Gisbert Wüstholz . . . . . . . . . . 28
List of participants 30
2
Welcome
The Department of Mathematics and the local organizers welcome you at the conferenceAlgorithmic Aspects of Diophantine Equation, February 15-19, 2016 at the University ofSalzburg. We hope that you will have a very fruitful and enjoyable time in Salzburg.
The aim of this conference is to bring together mathematicians from number theoryespecially working on Diophantine equations, Diophantine analysis or Diophantine geom-etry and to exchange new results and techniques in this area. But, also contributionson applications of Diophantine equations to cryptography, numeration systems, etc. arewelcome. We hope that an active interaction between those areas will take place. Inparticular, the main topics of this conference are gathered around the computational as-pects of Diophantine equations and include: e�ective solutions to Diophantine equations,automatic solutions to Diophantine equations, e�ective results from Diophantine approx-imation, heights in Diophantine analysis, e�ective Diophantine geometry, counting pointson varieties.
The invited plenary speakers are Michael A. Bennett (Vancouver), Yuri Bilu (Bor-deaux), Andrej Dujella (Zagreb), Kálmán Gy®ry (Debrecen), Michael Stoll (Bayreuth),Martin Widmer, and Gisbert Wüstholz (Zurich). The list of participants includes over�fty scientists coming from Austria (Graz, Linz, Salzburg), Canada (Vancouver), Croa-tia (Osijek, Pula, Zagreb), France (Aubiére, Bordeaux), Germany (Bayreuth), Hungary(Debrecen, Sopron, Szombathely), Italy (Pisa), Japan (Maebashi, Wakayama), Poland(Krakow), Senegal (Dakar), Serbia (Belgrade), South Africa (Johannesburg), Switzerland(Basel, Zurich), United Kingdom (London, Warwick), and from the USA (Northamp-ton/MA, Westville/IN).
The scienti�c program starts on Monday, February 15, 2016 at 9:45 in HS402; itcontinues until Friday, February 19 at noon. The conference language is English.
We thank the University of Salzburg, in particular the Department of Mathematics,and the Austrian Science Fund (FWF grants P24801-N26 and P24574-N26) for �nancialsupport.
The OrganizersClemens Fuchs
István PinkVolker Ziegler
3
General Information
Conference Location
The conference takes place in the lecture room HS402 in the Faculty of Natural Scienceof the University of Salzburg. The address is:
Hellbrunner Str. 345020 SalzburgAustria
The Department of Mathematics can be contacted by phone at +43(662)80445300.
Registration
The registration takes place in front of the conference lecture room HS402 on Monday,February 15, from 9:00-10:00 a.m.
Conference Opening
The conference will be opened on Monday, February 15, at 9:45 in HS402. The inauguraladdress will be given by:
Prof. Dr. Fatima Ferreira-BrizaVice-Rector for Research, University of Salzburg
The scienti�c program starts on Monday, February 15 at 10:00 with the plenary talkof Michael A. Bennett.
Technical Equipment of the Lecture Room
The lecture room is equipped with PC and beamer as well as with large blackboards.Technical assistants are present in the breaks to help you transfer your �le to the PC.
Internet Access during Conference
WLAN is available at the campus.
4
Lunch and Dinner
A list of restaurants and supermarkets available in the close vicinity of the conferencelocation is being handed to you with the conference folder.
Co�ee Breaks
Refreshments are served free of charge to participants on:
Monday 11:00-11:30 and 15:30-16:00Tuesday 10:00-10:30 and 15:30-16:00Wednesday, 10:00-10:30Thursday, 10:00-10:30 and 15:30-16:00Friday, 10:00-10:30
In addition, the university cafeteria serves co�ee and snacks.
Conference Picture
The conference picture will be taken on Wednesday at 10:00 immediately after the plenarytalk of Martin Widmer.
Local Transportation
To reach the conference location, take bus 3 or 8 from the city center to stop �Faistauer-gasse�; you can also leave one stop earlier (at �Akademiestraÿe�) or one stop later (at�Jose�au�). For more information, please visit the web page:
https://www.salzburg-ag.at/verkehr/obus/
Social Program
Registration is necessary for the city tour as well as for the conference dinner.
The city tour takes place on Wednesday afternoon. We meet at 14:00 in front of theconference lecture room. The tour takes 2,5-3 hours and will be given in English.
The conference dinner takes place at the restaurant STERNBRÄU (�Jedermannstube�,Griesgasse 23, phone +43(662)842140) at 18:30. The restaurant can be reached on foot,or by bus 8 (leave at stop �F.-Hanusch Platz�).
5
Information about the Conference Venue Salzburg
Salzburg is the capital of Salzburg. For more information about its history and culture,please visit the web page:
http://www.salzburg.info
Information about the University of Salzburg
The Paris-Lodron University of Salzburg was founded in 1622. For more informationabout the university, please visit the web page:
http://www.uni-salzburg.at
For information on the Department of Mathematics of the University of Salzburg,please visit the web page:
http://www.uni-salzburg.at/mathematik
6
Program
Summary
Time Monday Tuesday Wednesday Thursday Friday
09:00-10:00 Registration andOpening (9:45)
Bilu Widmer WüstholzPaulin
09:30-10:00 Faye10:00-10:30
BennettCo�ee Break Co�ee Break Co�ee Break Co�ee Break
10:30-11:00 Luca Szabó Frei Goedhart11:00-11:30 Co�ee Break Varga Remete Kreso Szalay11:30-12:00 Billerey Gawron Miyazaki Tadi¢12:00-12:30 Koutsianas Bertók Veneziano12:30-14:00 Lunch Break14:00-14:30
Stoll Dujella
City Tour
Gy®ry14:30-15:0015:00-15:30 Capuano Soldo Bérczes15:30-16:00 Co�ee Break Co�ee Break Co�ee Break16:00-16:30 Najman Togbe Peth®16:30-17:00 Ulas Hutle Hajdu17:00-17:30 Szikszai Gueth Bazsó
Monday
Time Speaker Title of the talk
09:00-09:45 Registration09:45-10:00 Opening10:00-10:50 Michael A. Bennett Elliptic curves of conductor p2
11:00-11:30 Co�ee break
11:30-11:55 Nicolas Billerey Sum of two S-units via Frey-Hellegouarch curves12:00-12:25 Angelos Koutsianas Elliptic Curves with Good Reduction Outside S
7
12:30-14:00 Lunch break
14:00-14:50 Michael Stoll The Generalized Fermat Equation x2 + y3 = z11
15:00-15:25 Laura Capuano Unlikely Intersections in certain families of abelianvarieties and the polynomial Pell equation
15:30-16:00 Co�ee Break
16:00-16:25 Filip Najman Mordell-Weil groups of elliptic curves over number�elds
16:30-16:55 Maciej Ulas On representing coordinates of points on ellipticcurves by quadratic forms
17:00-17:25 Márton Szikszai Perfect powers in products of terms of an EDS
Tuesday
Time Speaker Title of the talk
9:00-09:50 Yuri Bilu Subgroups of Class Groups and the AbsoluteChevalley-Weil Theorem
10:00-10:30 Co�ee break
10:30-10:55 Florian Luca One the Diophantine equation |τ(n!)| = m!
11:00-11:25 Nóra Varga Equal values of combinatorial numbers11:30-11:55 Maciej Gawron On Decompositions of quadrinomials and related Dio-
phantine equations
12:00-14:00 Lunch break
14:00-14:50 Andrej Dujella There are in�nitely many rational Diophantine sextu-ples
15:00-15:25 Ivan Soldo Diophantine triples in the ring of integers of thequadratic �eld Q(
√t), t > 0
15:30-16:00 Co�ee break
16:00-16:25 Alain Togbé Diophantine triples of Fibonacci Numbers16:30-16:55 Christoph Hutle Diophantine Triples with Values in k-generalized Fi-
bonacci Sequences17:00-17:25 Krisztián Gueth Diophantine triples in Lucas-Lehmer sequences
8
Wednesday
Time Speaker Title of the talk
09:00-09:50 Martin Widmer Weakly admissible lattices, primitive lattice points,and Diophantine approximation
10:00-10:30 Co�ee break
10:30-10:55 Tímea Szabó Power integral bases in quartic �elds and quarticextensions
11:00-11:25 László Remete Power integral bases in pure quartic number �elds:an application of binomial Thue equations
11:30-11:55 Takafumi Miyazaki A polynomial-exponential equation related to theRamanujan-Nagell equation
12:00-12:25 Csanád Bertók On the equation Un = 2a + 3b + 5c
12:30-14:00 Lunch break14:00-16:30 City Tour18:30- Conference dinner
Thursday
Time Speaker Title of the talk
09:00-09:50 Gisbert Wüstholz Billiard on the triaxial ellipsoid
10:00-10:30 Co�ee break
10:30-10:55 Christopher Frei Rational points on smooth cubic surfaces11:00-11:25 Dijana Kreso Diophantine equations and monodromy groups11:30-11:55 Petra Tadi¢ A criterion for injectivity of the specialization ho-
momorphism of elliptic curves and its applications12:00-12:25 Francesco Veneziano Rational points on explicit families of curves
12:30-14:00 Lunch break
14:00-14:50 Kálmán Gy®ry E�ective results for Diophantine equations over�nitely generated domains
15:00-15:25 Attila Bérczes Arithmetic and geometric progressions in the solu-tion set of Diophantine equations
15:30-16:00 Co�ee break
9
16:00-16:25 Attila Peth® On a Binet-type formula for nearly linear recursivesequences and its applications
16:30-16:55 Lajos Hajdu Power values of sums of products of consecutiveintegers
17:00-17:25 András Bazsó On the coe�cients of polynomials related to (al-ternating) power sums of arithmetic progressions
Friday
Time Speaker Title of the talk
09:00-09:25 Roland Paulin Asymptotic Expansion of the Zeros of the Lerch ZetaFunction
09:30-09:55 Bernadette Faye Repdigits as Euler Functions of Lucas Numbers
10:00-10:30 Co�ee break
10:30-10:55 Eva Goedhart The Diophantine equation (a2cxk − 1)(b2czk − 1) =
(abczk − 1)2
11:00-11:25 László Szalay The diophantine equation F |k|x = F|l|y
10
Abstracts
On the coe�cients of polynomials related to (alternating) power sums of
arithmetic progressions
András Bazsó
University of Debrecen, [email protected]
Coauthors: István Mez®
Let a 6= 0, b, k > 0 be integers with gcd(a, b) = 1. In the talk we study the coe�cientsof the polynomials which, at positive integer values n, give, respectively, the sum or thealternating sum of the k-th powers of the n-term arithmetic progression b, a + b, 2a +
b, . . . , (n− 1)a+ b. Joint work with István Mez®.
Elliptic curves of conductor p2
Michael A. Bennett
University of British Columbia, Vancouver, [email protected]
We sketch an approach to �nding all elliptic curves of conductor p2 for prime p
that is computationally reasonably e�cient. There are some nice connections to solv-ing parametrized families of Thue equations and simplest cubic �elds.
Arithmetic and geometric progressions in the solution set of Diophantine
equations
Attila Bérczes
University of Debrecen, [email protected]
In 2004 Bérczes and Peth® started to investigate the arithmetic progressions in thesolution set of norm form equations. Since then the investigation of special progressionsappearing in the solution set of diophantine equations has resulted in a series of interestingresults.
Bérczes and Peth®, Bérczes, Peth® and Ziegler and later Bazsó determined all arith-metic progressions forming solutions of some parametric families of norm form equations.
11
Arithmetic progressions in the solution set of Pell equations were investigated by Peth®and Ziegler, and by Dujella, Peth® and Tadic.
In this talk a survey on these results will be presented, along with new results obtainedby Bérczes and Ziegler on geometric progressions in the solution set of Pell-equations. Fur-ther, Bérczes and Ziegler also determined all geometric progressions consisting of elementsof a given Lucas sequence.
On the equation Un = 2a + 3b + 5c
Csanád Bertók
University of Debrecen, [email protected]
Coauthors: Lajos Hajdu, István Pink, Zsolt Rábai
In the talk, �rst we propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential Diophantine equations. Namely, consider the equation
a1bα1111 · · · b
α1l1l + . . .+ akb
αk1k1 · · · b
αklkl = c (1)
in non-negative integers α11, . . . , α1l, . . . , αk1, . . . , αkl, where ai, bij, are non-zero integersfor every i = 1, . . . , k and j = 1, . . . , l, and c is an integer. Our conjecture is that ifthe equation above has no solutions, then there exists an integer m ≥ 2 such that thecongruence
a1bα1111 · · · b
α1l1l + . . .+ akb
αk1k1 · · · b
αklkl ≡ c (mod m) (2)
has no solutions in non-negative integers αij, i = 1, . . . , k, j = 1, . . . , l.
In the talk we present a result showing that in a sense, the conjecture is valid for"almost all" equations. Further, based upon the conjecture we propose a general methodfor the solution of exponential Diophantine equations, relying on a generalization of aresult of Erd®s, Pomerance and Schmutz concerning Carmichael's λ function.
Finally, we illustrate that our method works not only in Z, but also in the ring ofintegers of Q(α) (where α is a real algebraic number) by generalizing a result of D.Marques and A. Togbé and solving a problem of F. Luca and S. G. Sanchez. Let Un =
A · Un−1 + B · Un−2 (n ≥ 2) with A,B ∈ Z and initial terms U0, U1 ∈ Z be a binarysequence. If a, b, c are non-negative integers, then we give all solutions of the equations
Un = 2a + 3b,
Un = 2a + 3b + 5c,
in the case when (A,B, U0, U1) = (1, 1, 0, 1), (1, 1, 2, 1), (2, 1, 0, 1), (2, 1, 2, 2).
12
Sum of two S-units via Frey-Hellegouarch curves
Nicolas Billerey
Université Clermont - Ferrand 2 - Blaise Pascal, [email protected]
Coauthors: Michael A. Bennett
In this talk we shall describe a method for explicitly �nding all perfect powers thatcan be expressed as the sum of two S-units, where S is a �xed �nite set a primes. Ourapproach, which is based on the modularity of some attached Galois representations,allows us to explicitly solve the problem for some small sets of primes such as S = 2, 3
and S = 3, 5, 7. This is a joint work with Michael A. Bennett.
Subgroups of Class Groups and the Absolute Chevalley-Weil Theorem
Yuri F. Bilu
Université de Bordeaux, [email protected]
The following conjecture is widely believed to be true: given a �nite abelian group G,a number �eld K and an integer d > 1, there exist in�nitely many extensions L/K ofdegree d such that the class group of L contains G as a subgroup. I will speak on someold and recent results on this conjecture, in particular, on my joint work with J. Gillibertin course.
Unlikely Intersections in certain families of abelian varieties and the
polynomial Pell equation
Laura Capuano
Scuola Normale Superiore di Pisa, [email protected]
Coauthors: Fabrizio Barroero
Given n independent points on the Legendre family of elliptic curves of equationY 2 = X(X − 1)(X − c) with coordinates algebraic over Q(c), we will see that there are atmost �nitely many specializations of c such that two independent relations hold betweenthe n points on the specialized curve. This �ts in the framework of the so-called UnlikelyIntersections. We will see a higher-dimensional analogue of this result and explain howit applies to the problem of studying the solvability of the (almost-)Pell equation inpolynomials. This is joint work with Fabrizio Barroero.
13
There are in�nitely many rational Diophantine sextuples
Andrej Dujella
University of Zagreb, [email protected]
Coauthors: Matija Kazalicki, Miljen Miki¢, Márton Szikszai
A rational Diophantine m-tuple is a set of m nonzero rationals such that the productof any two of them increased by 1 is a perfect square. The �rst rational Diophantinequadruple was found by Diophantus, while Euler proved that there are in�nitely manyrational Diophantine quintuples. In 1999, Gibbs found the �rst example of a rationalDiophantine sextuple. In this talk, we describe construction of in�nitely many rationalDiophantine sextuples. This is joint work with Matija Kazalicki, Miljen Miki¢ and MártonSzikszai.
Repdigits as Euler functions of Lucas numbers
Bernadette Faye
UCAD, Dakar, [email protected]
Let ϕ(m) be the Euler function of the positive integer m. Various Diophantine equationsinvolving the Euler function of members of Fibonacci or Lucas numbers have been in-vestigated during the past years. In the current paper, we prove some results about thestructure of all Lucas numbers whose Euler function is a repdigit in base 10. In otherswords, we look at the Diophantine equation
ϕ(Ln) = d
(10m − 1
9
)d ∈ {1, . . . , 9}. (3)
Numbers as the ones appearing in the right�hand side of equation (3) are called rep-digitsin base 10, since their base 10 representation is the string dd · · · d︸ ︷︷ ︸
m times
.
Here, using linear forms in logarithms and the Baker-Davenport reduction method,we show that if Ln is such a Lucas number, then n < 10111 is of the form p or p2 , wherep3|10p−1 − 1.
Rational points on smooth cubic surfaces
Christopher Frei
Graz University of Technology, [email protected]
Coauthors: Efthymios Sofos
Manin's conjecture predicts an asymptotic formula for the number of rational pointsof bounded height on smooth cubic surfaces over number �elds. For a large class of cubic
14
surfaces with a conic bundle structure, we prove that the asymptotic lower bound impliedby Manin's conjecture has the correct order of magnitude. As a consequence, the lowerbound has the correct order of magnitude for all smooth cubic surfaces after a smallextension of the base �eld. The central technical tool is the analysis of divisor sums ofcertain binary forms. This is joint work with Efthymios Sofos (Leiden).
On decompositions of quadrinomials and related Diophantine equations
Maciej Gawron
Jagiellonian University, Krakow, [email protected]
In paper [9] Schinzel, Pintér and Péter give an ine�cient criterion for the Diophantineequation of the form
axm + bxn + c = dyp + eq,
where a, b, c, d, e rationals, ab 6= 0 6= de, m > n > 0, p > q > 0, gcd(m,n) = 1, gcd(p, q) =1, and m, p ≥ 3 to have in�nitely many integer solutions.
In the later paper Schinzel [10] dropped the assumption gcd(m,n) = 1, gcd(p, q) = 1
and gives a necessary and su�cient condition for such equation to have in�nitely manyinteger solutions.
In the recent paper Kreso [6] proved the �nitness of integral solutions for the equation
a1xn1 + a2x
n2 + . . .+ alxnl + al+1 = b1y
m1 + b2ym2 ,
where l ≥ 2 and m1 > m2, n1 > n2 > . . . > nl are �xed positive integers satisfyinggcd(m1,m2) = 1, gcd(n1, n2, . . . , nl) = 1, a1, a2, . . . , al, al+1, b1, b2 are non-zero rationals,except for possibly al+1. With n1 ≥ 3,m1 ≥ 2l(l − 1) and (n1, n2) 6= (m1,m2).
All the mentioned results relies on Bilu-Tichy Theorem [1], and theorems concerningdecompositions of trinomials [2] as main ingredients. No such results for the equationsinvolving at least three non-zero coe�cients at positive powers on both sides are knownmainly because we have no results concerning decompositions of lacunary polynomialswith more than three non-zero terms [6]. Some partial results in this direction are givenin [7].
In this note we describe all possible decompositions of quadrinomials. In the sequelwe use Bilu-Tichy theorem to prove the following generalizations of Schinzel and Kresoresults. More precisely, we prove the following
Theorem. Let f(x) = Axn1 + Bxn2 + Cxn3 + D, g(x) = Exm1 + Fxm2 + Gxm3 + H
with f, g ∈ Q[x], n1 > n2 > n3 > 0, m1 > m2 > m3 > 0, and gcd(n1, n2, n3) = 1,gcd(m1,m2,m3) = 1, (m1,m2,m3) 6= (n1, n2, n3), ABC 6= 0, EFG 6= 0 and n1,m1 ≥ 9.Then the equation
f(x) = g(y)
has only �nitely many integer solutions.
15
Theorem. Let l ≥ 4 and n1 > n2 > . . . > nl > 0, m1 > m2 > m3 > 0 be positiveintegers. Let
f(x) = A1xn1 + A2x
n2 + . . .+ Alxnl + Al+1 and g(x) = Exm1 + Fxm2 +Gxm3
be polynomials with rational coe�cients such that gcd(n1, n2, . . . , nl) = 1, gcd(m1,m2,m3) =
1, A1A2 . . . Al 6= 0, EFG 6= 0 and m1 ≥ 2l(l − 1), n1 ≥ 4. Then the equation
f(x) = g(y)
has only �nitely many integer solutions.
Our results are ine�ective as we use Theorem of Bilu and Tichy which relies on classicaltheorem of Siegel on integral points.
References
[1] Y.F. Bilu, R.F. Tichy, The Diophantine equation f (x) = g(y), Acta Arith. 95 (2000),no. 3, 261-288.
[2] M. Fried, A. Schinzel, Reducibility of quadrinomials, Acta Arith. 21 (1972), 153-171[3] M. Gawron, On decompositions of quadrinomials and related Diophantine equations,
preprint arXiv:1512.02817v1[4] I.M. Gessel, G. Viennot, Binomial determinants, paths, and hook length formulae,
Advances in Math. 58 (1985), 300-321[5] G. Hajós, The solution of Problem 41, (Hungarian). Mat. Lapok, 4, (1953), 40-41[6] D. Kreso, On common values of lacunary polynomials at integer points, New York
J. Math. 21 (2015) 987-1001[7] D. Kreso, R.F. Tichy, Functional composition of polynomials: inde- composability,
Diophantine equations and lacunary polynomials, arXiv:1503.05401[8] R. C. Mason, Diophantine Equations over Function Fields, L. M. S. Lecture Notes
No. 96, Cambridge UP, (1984).[9] G. Péter, Á. Pintér, A. Schinzel, On equal values of trinomials, Monatsh. Math.
162 (2011), no. 3, 313-320.[10] A. Schinzel, Equal values of trinomials revisited. Tr. Mat. Inst. Steklova 276
(2012), Teoriya Chisel, Algebra i Analiz, 255-261; translation in Proc. Steklov Inst.Math. 276 (2012), no. 1, 250-256.
[11] C.L. Siegel, Uber einige Anwendungen Diophantischer Approximationen, Abh. Preuss.Akad. Wiss. Phys.-Math. Kl., 1929, Nr. 1; also: Gesammelte Abhandlungen, Band1, 209-266.
[12] W. W. Stothers, Polynomial identities and hauptmoduln, Quarterly J. Math. Ox-ford, 2, 32, (1981)
[13] U. Zannier, On composite lacunary polynomials and the proof of a conjecture ofSchinzel Invent. Math. 174 (2008), no. 1, 127-138
16
The Diophantine equation (a2cxk − 1)(b2czk − 1) = (abczk − 1)2
Eva Goedhart
Smith College, Northampton, [email protected]
Coauthors: Helen G. Grundman
For a, b, c, k ∈ Z+ with k ≥ 7, the equation
(a2cxk − 1)(b2cyk − 1) = (abczk − 1)2
has no integer solutions x, y, z > 1 with a2xk 6= b2yk. I will dscuss the proof of this resultwhich uses standard results on continued fractions and a Diophantine approximationtheorem due to M.A. Bennett. This is joint work with Helen G. Grundman.
Diophantine triples in Lucas-Lehmer sequences
Krisztián Gueth University of West Hungary, Savaria Campus, [email protected]
In this lecture, I talk about the non-existence of diophantine triples linked to a givenlinear recurrence of order four. A diophantine m-tuple means m distinct positive integerssuch that the product of any two of them is one less then a square of an integer. Weconsider this problem for the terms of a Lucas-Lehmer recurrence sequence instead of thesquare numbers, in the case of m = 3. I proved that there are no integers 0 < a < b < c
such that ab+ 1, ac+ 1 and bc+ 1 all are terms of this sequence.
E�ective results for Diophantine equations over �nitely generated domains
Kalman Gy®ry
University of Debrecen, [email protected]
Recently Evertse and the speaker, partly with Bérczes, have extended the e�ectivetheory of Diophantine equations over number �elds to the case of equations over arbitrary�nitely generated domains over Z (which may contain transcendental elements, too.) Inthe �rst part of the lecture, the most important e�ective results over number �elds willbe formulated in qualitative form. Then we give a survey of their generalizations over�nitely generated domains. Finally, the method of proofs will be brie�y discussed.
17
Power values of sums of products of consecutive integers
Lajos Hajdu
University of Debrecen, [email protected]
For k = 0, 1, 2, . . . put
fk(x) =k∑i=0
i∏j=0
(x+ j).
In the talk we consider the Diophantine equation
fk(x) = yn (4)
in integers x, y, k, n with k ≥ 0 and n ≥ 2. Without loss of generality, we shall assumethat n is a prime. We mention that equation (4) is closely related to several classicalproblems and results.
In the talk we present a general �niteness result concerning (4). Further, we provideall solutions to this equation for k ≤ 10. In our proofs we combine several tools andtechniques, including Baker's method, local arguments, Runge's method, and a methodof Gebel, Peth®, Zimmer and Stroeker, Tzanakis to �nd integer points on elliptic curves.
Diophantine Triples with Values in k-generalized Fibonacci Sequences
Christoph Hutle
Paris Lodron University of Salzburg, [email protected]
One of the oldest problems in number theory is the question of Diophantus, which isabout constructing sets of rationals or integers with the property that the product of anytwo of its distinct elements plus 1 is square. Recently, several variations of this problemhave been investigated. The problem of �nding bounds on the size m for Diophantinem-tuples with values in linear recurrences is one such variation.
It was shown by Fuchs, Hutle, Irmak, Luca and Szalay in 2015 (to appear in Math.Slovaca), that for the Tribonacci sequence {Tn}n≥0 given by T0 = T1 = 0, T2 = 1 andTn+3 = Tn+2 + Tn+1 + Tn for all n ≥ 0, there exist only �nitely many Diophantine tripleswith values in {Tn}n≥0.
In this talk, we will consider the k-generalized Fibonacci sequence given for some k ≥ 2
by F (k)0 = . . . = F
(k)k−2 = 0, F
(k)k−1 = 1 and
F(k)n+k = F
(k)n+k−1 + · · ·+ F (k)
n
for all n ≥ 0. Improving the previous result, we show that there are only �nitely manyDiophantine triples with values in {F (k)
n }n≥0.The proof is not constructive, since it is based on a version of the Subspace Theorem,
one of the most important results in Diophantine approximation.
18
Elliptic Curves with Good Reduction Outside S
Angelos Koutsianas
University of Warwick, United [email protected]
In this talk, we will present a new algorithm that �nds all elliptic curves E over anumber �eld K with good reduction outside a �nite set of primes S in K by solvingS-unit equations.
Diophantine equations and monodromy groups
Dijana Kreso
Graz University of Technology, [email protected]
Coauthors: Robert F. Tichy
Jointly with Robert Tichy, I have studied Diophantine equations of type f(x) = g(y),where f and g have at least two distinct critical points and equal critical values at atmost two distinct critical points. Our results cover and generalize several results in theliterature on the �niteness of integral solutions of such equations. In so doing, we analyzethe properties of the monodromy groups of such polynomials. We show that if f hascoe�cients in a �eld of characteristic 0, at least two distinct critical points and all distinctcritical values, then the monodromy group of f is a doubly transitive permutation group.This in particular means that f can not be represented as a composition of lower degreepolynomials. We further show that if f has at least two distinct critical points and nothree distinct critical points with equal critical values, and f(x) = g(h(x)) with g, h withcoe�cients in K and deg g > 1, then either deg h ≤ 2, or f is of special type. In the lattercase, in particular, f has no three simple critical points, nor �ve distinct critical points.
On the Diophantine equation |τ(n!)| = m!
Florian Luca
University of the Witwatersrand, Johannesburg, South [email protected]
Coauthors: Jhon Jairo Bravo Grijalba
Let τ(n) be the Ramanujan function given by
q( ∞∏n=1
(1− qn))24
=∑n≥1
τ(n)qn (|q| < 1).
In 2006, jointly with I. E. Shparlinski, we proved that the Diophantine equation |τ(n!)| =m! has only �nitely many positive integer solutions (n,m). In my talk, I take this onestep further and report that the only such solutions are (n,m) = (1, 1), (2, 4). The proofuses linear forms in logarithms. This is joint work with Jhon Jairo Bravo Grijalba fromUniversidad del Cauca, Colombia.
19
A polynomial-exponential equation related to the Ramanujan-Nagell
equation
Takafumi Miyazaki
Gunma University, [email protected]
Let us consider the equation
x2 +Dm = pn in positive integers x,m, n (1)
where p is a �xed prime and D > 1 is a �xed integer not divisible by p. This is ageneralization of the Ramanujan-Nagell equation for (p,D) = (2, 7) with m = 1. Manyworks on equation (1) concern to bound its number of solutions (x,m, n). In this direction,the de�nitive results in the case of m = 1 are already obtained by F. Beukers (1990) forp = 2, and by Y. Bugeaud and T.N. Shorey (2001) for odd p. In 2001, Bugeaud completelydescribed the solutions of equation (1) for p = 2, and also showed that equation (1) hasat most two solutions when p is odd and D > 2, except for some speci�c pairs of (p,D).After his work, P. Yuan and Y. Hu (2005) succeeded in dealing with those exceptionalcases. The combination of their works implies the following result:
Proposition. Let p be odd and D > 2. Then equation (1) has at most two solutions,unless (p,D) = (5, 4). The solutions in this exceptional case are given by (x,m, n) =
(1, 1, 1), (3, 2, 2), (11, 1, 3).
In this talk, we consider equation (1) when D takes a concrete form in terms of p. In sucha case, we show that all solutions x,m, n can be bounded by an e�ectively computableconstant depending only on p. Moreover, we solve the equation completely under someconditions on p. The motivation for this study is to apply a classical lemma obtained byV.A. Dem'janenko (1965) which is used in the study of an unsolved problem concerningprimitive Pythagorean triples.
Mordell-Weil groups of elliptic curves over number �elds
Filip Najman
University of Zagreb, [email protected]
The Mordell-Weil group E(K) ofK-rational points of an elliptic curve E over a number�eld K, is a �nitely generated abelian group and hence isomorphic to the direct productof its torsion subgroup and Zr, where r is the rank of E/K.
In this talk we will consider the question of what this group can be over number�elds of certain type, e.g. over all number �elds of degree d or over a �xed number �eld.After surveying known results, both old and new, about torsion groups, we will showthat prescribing the torsion over number �elds (as opposed to over Q!) can force various
20
properties on the elliptic curve. For instance, all elliptic curves with points of order 13
or 18 over quadratic �elds have to have even rank and elliptic curves with points of order16 over quadratic �elds are base changes of elliptic curves de�ned over Q. We show thatthese properties arise from the geometry of the corresponding modular curves.
Asymptotic Expansion of the Zeros of the Lerch Zeta Function
Roland Paulin
Paris Lodron University of Salzburg, [email protected]
We discuss the asymptotic behavior of the zeros s of the Lerch zeta function L(λ, α, s) =∑∞n=0
e2πiλn
(n+α)swhen z = e2πiλ → 0. In particular we de�ne recursively a series expansion,
and we also give an explicit, non-recursive formula for the coe�cients. The obtainedresults considerably strengthen some of the statements of a 1975 paper of Fornberg andKölbig. The methods used are quite general, and can be applied to describe the asymp-totic behavior of the zeros of a wide class of general Dirichlet series.
On a Binet-type formula for nearly linear recursive sequences and its
applications
Attila Peth®
University of Debrecen, [email protected]
Coauthors: Shigeki Akiyama, Jan-Hendrik Evertse
This talk is based on a joint work with Shigeki Akiyama, Tsukuba University, Japan,and Jan-Hendrik Evertse, Universiteit Leiden, The Netherlands.
We de�ne a nearly linear recursive sequence (an) and give a Binet-type formula. Weprove that the �uctuation of a linear recursive sequences can be extremely large, then weanalyze the distance of a nlrs to a naturally chosen lrs.
We investigate the zero multiplicity and common terms of nlrs's. We show for two nlrs,having single dominating roots, which are algebraic and multiplicatively independent,that the indices of consecutive common terms grow exponentially. We prove that theSkolem-Lech-Mahler theorem, does not hold generally for nlrs with at least two dominatingroots with equal absolute values. This implies that unbounded nlrs with multiplicativelyindependent characteristic roots exist with in�nitely many common values.
21
Power integral bases in pure quartic number �elds: an application of
binomial Thue equations
László Remete
University of Debrecen, [email protected]
A number �eld K of degree n is monogene if there exist an integer ϑ ∈ K suchthat ZK = Z[ϑ], that is {1, ϑ, ϑ2, . . . , ϑn−1} is an integral basis (so called power integralbasis) in K.
We consider the problem of monogenity and generators of power integral bases in purequartic �elds K = Q( 4
√m) where m is a square free integer with m ≡ 2, 3 (mod 4). Set
α = 4√m. For 1 < m < 107 we determine all generators
ϑ = a+ xα + yα2 + zα3
of power integral bases of K where a, x, y, z ∈ Z with
max(|x|, |y|, |z|) < 101000.
To obtain this result we calculated solutions (x, y) with max(|x|, |y|) < 10500 of binomialThue equations of type
x4 −my4 = ±1
for 2 ≤ m ≤ 107. This extensive calculation was performed on a supercomputer. Weextended this calculation also to exponents n = 3, 4, 5, 7, 11, 13, 17, 19, 23, 29.
We generalized these results also to the relative case. Let d be one of
d = 3, 7, 11, 19, 43, 67, 163
and let L = Q(i√d). Let m ≡ 2, 3(mod 4), assume (d,m) = 1 and set α = 4
√m.
For 1 < m ≤ 5000 we calculate all generators ϑ = A + Xα + Y α2 + Zα3 of relativepower integral bases of K over L with A,X, Y, Z ∈ ZL with max(|X|, |Y |, |Z|) < 10500.To prove this later result we solved binomial Thue equations in the relative case, overimaginary quadratic �elds.
We also proved that these octic �elds K does not admit any generators of (absolute)power integral bases of the form
ϑ = A+ ε(Xα+ Y α2 + Zα3)
where A,X, Y, Z ∈ ZL, ε a unit in L and
max(|X|, |Y |, |Z|) < 10500.
22
Diophantine triples in the ring of integers of the quadratic �eld Q(√−t), t > 0
Ivan Soldo
University of Osijek, [email protected]
Let R be a commutative ring and z ∈ R. A set {a1, a2, . . . , am} in R such that ai 6= 0,i = 1, . . . ,m, ai 6= aj and aiaj + z is a square in R for all 1 ≤ i < j ≤ m is called aDiophantine m-tuple with the property D(z), or simply a D(z)-m-tuple in the ring R.We study D(−1)-triples of the form {1, b, c} in the ring Z[
√−t], t > 0, for positive integer
b such that b is a prime, twice prime and twice prime squared. We prove that in thosecases c has to be an integer. By using that result we obtained some results about theexistence of D(−1)-quadruples in certain ring of integers Z[
√−t] of the quadratic �eld
Q(√−t), t > 0.
The Generalized Fermat Equation x2 + y3 = z11
Michael Stoll
Bayreuth University, [email protected]
Coauthors: Nuno Freitas, Bartosz Naskrecki
Generalizing Fermat's original problem, equations of the form xp+yq = zr, to be solvedin coprime integers, have been quite intensively studied. It is conjectured that there areonly �nitely many solutions in total for all triples (p, q, r) such that 1/p + 1/q + 1/r <
1 (the `hyperbolic case'). The case (p, q) = (2, 3) is of special interest, since severalsolutions are known. To solve it completely in the hyperbolic case, one can restrictto r = 8, 9, 10, 15, 25 or a prime ≥ 7. The cases r = 7, 8, 9, 10, 15 have been dealtwith by various authors. In joint work with Nuno Freitas and Bartosz Naskrecki, weare now able to solve the case r = 11 and prove that the only solutions (up to signs)are (x, y, z) = (1, 0, 1), (0, 1, 1), (1,−1, 0), (3,−2, 1). We use Frey curves to reduce theproblem to the determination of the sets of rational points satisfying certain conditionson certain twists of the modular curve X(11). A study of local properties of mod-11Galois representations cuts down the number of twists to be considered. The main newingredient is the use of the `Selmer group Chabauty' techniques developed recently by thespeaker to �nish the determination of the relevant rational points.
Power integral bases in quartic �elds and quartic extensions
Tímea Szabó
University of Debrecen, [email protected]
The existence of power integral bases is a classical topic in algebraic number theory. Itis well known that if a number �eld admits a power integral basis of type (1, θ, . . . , θn−1)
23
then up to equivalence it admits only �nitely many of them. There is an extensiveliterature of calculating power integral bases in special algebraic number �elds. Thisproblem is equivalent to solving diophantine equations, so called index form equationsThere are e�cient algorithms for calculating power integral bases in lower degree (≤ 6) andin special higher degree (6, 8, 9) number �elds. The problem of power integral bases wasalso considered in relative extensions. Algorithms for calculating relative power integralbases were given in relative cubic and in relative quartic extensions. It is an especiallydelicate problem if we solve the index form equation not only in a speci�c number �eldbut in an in�nite parametric family of number �elds, where the index form equation isgiven in a parametric form. Such results are known in certain parametric families of cubic,quartic and quintic number �elds. Similar results for calculating relative power integralbases in in�nite parametric families of relative extensions were not known before.In this talk we present the resolution of the index form equations in two families oftotally complex biquadratic �elds depending on two parameters and prove that up toequivalence, they admit only one generator of power integral bases. Note that these arethe �rst families of number �elds with two parameters where all generators of powerintegral bases determined.In the second half of my talk considering in�nite parametric families of octic �elds, thatare quartic extensions of quadratic �elds, we describe all relative power integral bases ofthe octic �elds over the quadratic sub�elds and then we check if there exist correspondinggenerators of absolute power integral bases.
The diophantine equation F[k]x = F
[`]y
László Szalay
University of West Hungary, Sopron, [email protected]
Let {Fn} denote the sequence of Fibonacci numbers de�ned by F0 = 0, F1 = 1, andFn = Fn−1 + Fn−2 for n ≥ 2. The notion of hyper-Fibonacci numbers F [k]
n (n ≥ 0, k ≥ 0)was introduced by Dil and Mez® [1] as follows. Let F [0]
n = Fn, F[k]0 = 0, and
F [k]n = F
[k]n−1 + F [k−1]
n , kn > 0.
Considering the title equation, we have the following theorems.
Theorem. Given the positive integer `, the equation F [k]x = F
[`]y has �nitely many solu-
tions in the nonnegative integers x, y and k < `, which are e�ectively computable.
Theorem. Beside the trivial solutions, the equation F [k]x = F
[`]y possesses only the solu-
tions
(k, `, x, y) = (0, 11, 14, 4), (0, 16, 16, 4), (1, 2, 4, 3), (1, 7, 12, 5), (1, 20, 11, 3), (5)
(2, 8, 6, 3), (2, 11, 7, 3), (2, 33, 11, 3), (4, 6, 5, 4), (4, 12, 5, 3), (6, 12, 4, 3).
if 0 ≤ k < ` ≤ 50.
The trivial solutions are considerd as
24
• F [k]0 = 0 = F
[`]0 ,
• F [k]1 = 1 = F
[`]1 ,
• F [0]2 = 1 = F
[`]1 ,
• F [k]x = F
[F
[k]−1x
]2 .
We conjecture, that (5) and the trivial solutions give all the solutions to F [k]x = F
[`]y .
References
[1] Dil, A. � Mez®, I., A symmetric algorithm for hyperharmonic and Fibonacci num-bers, Appl. Math. Comp., 206 (2008), 942-951.
Perfect powers in products of terms of an EDS
Márton Szikszai
University of Debrecen, [email protected]
Let E be a non-singular elliptic curve and P ∈ E(Q) be a point of in�nite order. Writethe multiples of P as
nP =
(AnB2n
,CnB2n
)with An, Bn, Cn ∈ Z, Bn > 0 and (AnCn, Bn) = 1. The sequence B = (Bn)
∞n=1 is called
an elliptic divisibility sequence, an important class of non-linear recurrences. Let ` ≥ 2.Set
P`(B) = {i : Bi is `−th power}
and
N` = #P`(B), M` = maxP`(B).
Consider the diophantine equation
BmBm+d . . . Bm+(k−1)d = y`
in unknown positive integersm, d, k, y with (m, d) = 1 and k ≥ 2. In this talk we show thatthe above equation admits �nitely many solutions. Further, if P`(B) is given explicitly,then we prove that for every solution (m, d, k, y) we have max(m, d, k, y) ≤ C(N`,M`),where C is an e�ective constant depending on N` and M` only. We combine several tools,including the arithmetic properties of B and bounds concerning the greatest prime divisorand the number of prime divisors of blocks of consecutive terms of arithmetic progressions.
25
A criterion for injectivity of the specialization homomorphism of elliptic
curves and its applications
Petra Tadi¢
Juraj Dobrila University of Pula, [email protected]
Let K be a number �eld. Let E be a nonconstant elliptic curve over K(t) with at leastone nontrivial K(t)-rational 2-torsion point. By the Silverman specialization theorem thespecialization homomorphism t 7→ t0 is injective for all but �nitely many t0 ∈ K. For Kof class number one, Gusi¢ and Tadi¢ obtained a method for �nding t0 ∈ K for which thecorresponding specialization homomorphism is injective. In principal they extended themethod to arbitrary number �elds K.
This method can be used to calculate the rank of elliptic curves of the form as above,by looking at the structure of a chosen specialized curve. This method has been used byGusi¢, Dujella, Peral and Tadi¢ to calculate exactly the rank of some record high rankelliptic curves over Q(t).
Diophantine triples of Fibonacci Numbers
Alain Togbé
Purdue University North Central, Westville, [email protected]
Coauthors: Bo He, Florian Luca
Let Fm be the mth Fibonacci number. In this talk, We will prove that if F2nFk + 1
and F2n+2Fk + 1 are both perfect square, then k = 2n + 4, for n ≥ 1 or k = 2n − 2, forn ≥ 2, except when n = 2 case in which we can additionally have k = 1. This talk isbased on a joint paper with B. He and F. Luca.
On representing coordinates of points on elliptic curves by quadratic forms
Maciej Ulas
Jagiellonian University, Krakow, [email protected]
Coauthors: Andrew Bremner
Given an elliptic quartic of type Y 2 = f(X) representing an elliptic curve of positiverank over Q, we investigate the question of when the Y -coordinate can be representedby a quadratic form of type ap2 + bq2. In particular, we give examples of equations ofsurfaces of type c0 + c1x + c2x
2 + c3x3 + c4x
4 = (ap2 + bq2)2, a, b, c ∈ Q where we candeduce the existence of in�nitely many rational points. We also investigate surfaces oftype Y 2 = f(ap2 + bq2) where the polynomial f is of degree 3. Joint work with AndrewBremner.
26
Equal values of combinatorial numbers
Nóra Varga
University of Debrecen, [email protected]
Coauthors: Lajos Hajdu, Ákos Pintér, Szabolcs Tengely
Let k,m be integers with k ≥ 3 and m ≥ 3 and denote by
fk,m(X) =X(X + 1) · · · (X + k − 2)((m− 2)X + k + 2−m)
k!
the Xth �gurate number with parameters k and m. In this talk some e�ective �nitenessstatements for the general equation
fk,m(x) = f2,n(y) (6)
in integers x and y are presented. Further, it is proved that the unique solution of theequation
fk,k+2(x) = f2,4(y) (7)
in integers k ≥ 5, x ≥ k − 2 and y ≥ 1 is (k, x, y) = (5, 47, 3290). (The talk is based on ajoint work with Lajos Hajdu, Ákos Pintér and Szabolcs Tengely).
Rational points on explicit families of curves
Francesco Veneziano
University of Baselfrancesco.veneziano-at-unibas.ch
Coauthors: Sara Checcoli, Evelina Viada.
I will present a method, of easy application, to compute the rational points on curvesin some products of elliptic curves. We prove some explicit and very sharp estimates forthe height of such rational points. The bounds are so good that we can implement acomputer search. I will present several explicit examples in which this has been done. Allresults are in collaboration with Sara Checcoli and Evelina Viada.
Weakly admissible lattices, primitive lattice points, and Diophantine
approximation
Martin Widmer
Royal Holloway University of London, United [email protected]
After surveying some impressive results of Skriganov on counting lattice points inaligned boxes for weakly admissible lattices we present some new counting results in amore general framework. Our error estimates are inferior to Skriganov's regarding the
27
dependence on the volume of the box but superior regarding the dependence on thelattice. It is this improvement that allows for counting results for primitive lattice points.When time permits we will also discuss applications to Diophantine approximation byprimitive points as studied by Chalk and Erd®s and more recently by Dani, Laurent, andNogueira.
Billiard on the triaxial ellipsoid
Gisbert Wüstholz
ETH Zürich, [email protected]
Coauthors: Ronaldo Garcia
The triaxial ellipsoid seems on the �rst sight an extremely trivial object. Topologicallyit is just a sphere and there is not much to say about it. However at least since Jacobi weknow that this is not the case. Jacobi has shown in his wonderful monograph VorlesungenÜber Dynamik, Gesammelte Werke 8, how rich the mathematics around the ellipsoid is.One of the key object is the pencil of confocal ellipsoids
x2
a− λ+
y2
b− λ+
z2
c− λ= 1
with 0 < a < b < c and λ ∈ R. Jacobi introduced the famous elliptic coordinateswhich are nowadays indispensable for geodesists and earth scientists. It is further of greatimportance for studying the di�erential geometry of the ellipsoid. As an outcome of thesestudies a large variety of elliptic and hyperelliptic curves became visible and with thatthe theory of dynamical and Hamiltonian systems entered. One of the foci were the so-called principal curvature lines which appear as the intersection of two such ellipsoids inthe pencil. Much more complicated were the genus two hyperelliptic curves which areintimately related to the geodesics on the ellipsoid which turn out to have a very strangetopological property. In particular they are lines on the ellipsoid which in general windaround without getting closed in �nite time. In a joint project with Ronaldo Garcia welooked at the problem of characterizing closed geodesics in terms of the geometry of theassociated abelian surface which is the Jacobian of some genus 2 hyperelliptic curve. Toour knowledge so far no curvature line which takes a non-zero angle with respect to oneof the principal curvature lines has been shown to be not closed. It is expected they arenot closed in general so that a billiard ball turning around along these lines never comesback. Both of these problems lead to abelian integrals, in the curvature lines case to anelliptic integral of the third kind and in the geodesic situation to an abelian integral ofthe �rst kind and the questions can be answered by solving two of the problems of Th.Schneider about the transcendence of such integrals. In both cases one has to determinethe structure of the endomorphism algebra of special commutative algebraic groups. Thisis not so di�cult (surprise-surprise!) in the case of geodesics. In the case of the curvature
28
lines the algebraic group is an extension of an elliptic curve by a product of additive andmultiplicative groups and here things become much more complicated.
29
List of participants
1. Cristoph Aistleitner Linz Austria2. András Bazsó Debrecen Hungary3. Michael A. Bennett Vancouver Canada4. Attila Bérczes Debrecen Hungary5. Csanád Bertók Debrecen Hungary6. Nicolas Billerey Aubière France7. Yuri F. Bilu Bordeaux France8. Laura Capuano Pisa Italy9. Kwok Chi Chim Graz Austria10. Andrej Dujella Zagreb Croatia11. Bernadette Faye Dakar Senegal12. Christofer Frei Graz Austria13. Clemens Fuchs Salzburg Austria14. Stevan Gaiovi¢ Belgrade Serbia15. Maciej Gawron Krakow Poland16. Eva Goedhart Northampton USA17. Krisztián Gueth Szombathely Hungary18. Kálmán Gy®ry Debrecen Hungary19. Lajos Hajdu Debrecen Hungary20. Peter Hellekalek Salzburg Austria21. Markus Hittmaier Salzburg Austria22. Christoph Hutle Salzburg Austria23. Christina Karolus Salzburg Austria24. Hidetaka Kitayama Wakayama Japan25. Angelos Koutsianas Warwick United Kingdom26. Dijana Kreso Graz Austria27. Pierre Lezowski Aubière France28. Florian Luca Johannesburg South Africa29. Antoine Marnat Graz Austria30. Takafumi Miyazaki Maebashi Japan31. Filip Najman Zagreb Croatia
30
32. Roland Paulin Salzburg Austria33. Attila Peth® Debrecen Hungary34. István Pink Salzburg/Debrecen Austria/Hungary35. Ákos Pintér Debrecen Hungary36. László Remete Debrecen Hungary37. Adrian Scheerer Graz Austria38. Ivan Soldo Osijek Croatia39. Michael Stoll Bayreuth Germany40. Tímea Szabó Debrecen Hungary41. László Szalay Sopron Hungary42. Márton Szikszai Debrecen Hungary43. Petra Tadi¢ Pula Croatia44. Robert F. Tichy Graz Austria45. Niclas Technau Graz Austria46. Alain Togbé Westville USA47. Maciej Ulas Krakow Poland48. Nóra Varga Debrecen Hungary49. Francesco Veneziano Basel Switzerland50. Martin Widmer London United Kingdom51. Gisbert Wüstholz Zürich Switzerland52. Volker Ziegler Salzburg Austria
31
32
Ka
rte
nd
ate
n ©
20
16
Go
og
le1
00
m
Sa
lzb
urg
33
Salzach
Salz
ach
Mönch
s-ber
g
Ka
pu
zinerbe
rg
Staa
tsgre
nze
Deu
tsch
lan
d
MIR
ABEL
L
WAL
LFAH
RTSK
IRCH
EM
ARIA
PLA
IN
ZEN
TRUM
CENTRE
SCHL
OSS
LEO
POLD
SKRO
N
SCHL
OSS
HEL
LBRU
NN
WAS
SERS
PIEL
E
GAI
SBER
G-
SPIT
ZE
nac
h S
chw
arza
ch /
St. V
eit
nac
h S
traß
wal
chen
nac
h M
ün
chen
nac
h B
erch
tesg
aden
110
nac
h V
og
gen
ber
g
nac
h M
üh
ldo
rf
S2
S3
S3
nac
h B
ad R
eich
enh
all
S4
28 n
ach
Han
gen
den
stei
n84
0 n
ach
Ber
chte
sgad
en35
nac
h R
if25
nac
h U
nte
rsb
erg
bah
n
8
1
1
4
4
S1 ·
S11
6 7
6
S1S1
1
20,2
1,27
,28
21,2
2
21
21
27
27
21
2222
,25
25
2228
,840
21,2
2,15
1
25,8
40
A,20,21
,22,24
,27,28
20,2
8
28
20,2
8
20
2022
21
23
21
23
23
23
23
21
21
20,25
,28,84
0
A,2
2
24
24
28
2
2
22
27,A
20,25
,28,84
0
34
34
34
3535
28
10
10
Nur
im F
rühj
ahrs
-,So
mm
er- u
ndH
erbs
tfah
rpla
n
151
2
20
2522
24
78
65
3
14
10
A
12
12
10
20
A
25
27
27
28,8
407
8
65 14
3
1041
5 12
78
10
36
14
614
75
8
10
3
314
12 8
38
7
7
5
5
7 1 4 108
4 1 10
71
410
8
78
8
78
1
8
7
1
36
12
14
53
6 12
12
3
12 214
4 1 10
110
1
212
112
1
2
23
A
8
1214
14
12
21,3
5
1
151
27
27
24,2
7,28
Sieb
enbü
rger
stra
ße
Fisc
herg
asse
Lief
erin
ger S
pitz
Saal
achs
tr.
Lank
essie
dlun
g
Ta
xham
Eur
opar
k
Schm
iedi
nger
str.
Frei
lass
ing
Alpi
ne P
ark
Freila
ssing
So
nnen
feld
Freila
ssing
Ru
pertu
skirch
e
Frei
lass
ing
Salz
burg
er P
latz
Sigg
erw
iese
nAn
ther
ing
Acha
rtin
g
Obe
rndo
rf Ba
hnho
fZi
egel
haid
en
Echin
g
Schl
acht
hof
Mun
tigl
Wei
twör
th-N
ußdo
rfO
icht
ensie
dlun
gO
bern
dorf
Stad
t
Arns
dorf
St. Geo
rgen b
. Sbg
.
Gut Wild
shut
Eiferd
ing
Rieders
bach
Pabi
ng
Zehm
emoo
s
Kirchb
erg
Frei
lass
ing
Fore
llenw
eg-
siedl
ung
Euge
n-M
ülle
r-Str.
Fasa
nerie
str.
Schu
leLe
hen
IKEA
K
leßh
eim
Eu
ropa
-
stra
ße
Kl
eßhe
imRe
d Bu
ll Are
na
Frie
dric
h-vo
n-W
alch
en-S
tr.
Rott
weg Sc
hmie
dkre
uzst
r.
Volk
ssch
ule
L
iefe
ring Ha
fner
müh
lweg
Gst
ötte
ngut
stra
ße
Kleß
heim
Mie
lest
r.
Kleß
heim
Kava
lierh
aus
Dr.-Gmeli
n-Stra
ße
Norbert
-Brüll-S
tr.
Siez
enhe
imLi
nden
weg
Am Römers
tein
Schwarz
enbe
rgkase
rne
Siezen
heim
Garten
str.
Vieh
haus
enAu
toba
hnw
eg
Viehha
usen
Ortsmitte
Him
mel
reic
hAn
drä-
Dopp
ler-W
eg
Viehha
usen
Schu
le
Kend
lersie
dlung
Eiche
tsied
lung
Schweiz
ersied
lung
Presse
zentru
m/Ku
glhof
Kloster
maier-
hofw
eg
Heimstr
.
Kuge
lhof
stra
ße
Rich
ard-
Knol
ler-S
tr.
Berg
erho
f
Högl
stra
ße
Kräu
tlerw
eg
Kröbe
nfelds
tr.
S.-Marc
us-St
r.
Gor
ians
traße
Schw
eden
str.
Nuß
dorfe
rstra
ße
Mör
kweg
Firm
ians
traße
Mül
ler-R
unde
gg-W
egVo
lkss
chul
e M
oos
Eich
etst
raße
Gla
nste
g
Jodo
k-Fi
nk-S
traße
Karo
linge
rstra
ße
Gse
nger
weg
Mar
ienb
ad
Send
lweg
Kräu
terh
ofw
egHo
chle
itner
Ham
mer
auer
Stra
ßeM
ayrb
achw
eg
Lehr
bauh
of
Mic
hael
-Wal
z-G
asse
Volks
schule
Maxgla
n
Julius
-Wels
er-Str
.
Körb
lleite
ngas
se
Girling
str.
Mar
tin-
Luth
er-P
latz
Otto-vo
n-
Lilien
thal-S
traße
Peter
-Pfen
ninge
r-Str.
Siezen
heim
Dr.-Han
s-Lech
ner-S
tr.
Obe
rst
Lepp
er-
ding
er-
Stra
ße
Freira
um M
axgla
n
Siezen
heim
Gewerb
egeb
iet
Radin
gerst
r.Nop
pinge
rgasse
Hans
-Sch
mid
-Pla
tzEt
richs
traße
Fürst
enbru
nn
G
lanstr
.
Fü
rsten
brunn
Volks
schule
Fürst
enbru
nn
Bus
kehre
Gla
negg
Ort
smitt
e
Glaneg
g
Schlo
ss
Fürst
enbru
nn
Glanrie
del
Fürst
enbru
nn
Schrot
erstr.
Gröd
igPfl
egers
tr.
Glaneg
gDop
plerst
r.
Gröd
igWeih
erweg G
rödig
Fisch
erweg G
rödig
Mostw
astl Grödig
Pfl
eger-
brüc
ke Gröd
ig Lo
hnerb
auer
Grö
dig
Mar
ktpl
atz
Sand
or-Ve
gh-St
r.
Niss
enst
r.
Sant
nerg
asse
Högl
wör
thw
eg
Doss
enw
eg
Eich
etho
fsie
dlun
g
Kom
mun
alfri
edho
f
Kons
tanze-
Weber-
Gasse
Wartbe
rgweg
Erza
bt-K
lotz
-Stra
ßeSe
nior
enhe
im N
onnt
al
König
-Ludw
ig- Straß
e
Zwies
elweg
Hofh
aym
er-A
llee
Beet
hove
nstr.
Dr.-F
ranz
-Reh
rl-Pl
atz
Unfa
llkra
nken
hausVo
lksg
arte
n
Fina
nzam
t
Wei
chse
lbau
m-
siedl
ung
Mar
ia-C
ebot
ari-S
tr.
Kain
dlw
eber
weg
Bocksb
erger-
straß
e
Pars
ch
Freisa
al-
w
eg
Gin
zkey
plat
z
Alpe
nsie
dlun
g
Alpe
nstr.
Abz
. Hel
lbru
nn
Gla
senb
ach
Ursu
linen
Aige
n S-
Bahn
Salz
achs
traße
Jose
f-Kau
t-Stra
ße
Valk
enau
erst
raße
Renn
bahn
str.
Erns
t-G
rein
-Stra
ße
Über
fuhr
stra
ße/
Diak
onie
zent
rum
Wäs
cher
-ga
sse
Spor
tzen
trum
Non
ntal
Mic
hael
-Pac
her-
Stra
ße
Jude
nber
g-al
m
Güt
erw
eg
Mitt
ereg
g
Zist
elal
m
Turn
erst
r.
Gni
gl S
-Bah
n
Bach
stra
ße
Zeisi
gstra
ße
Mai
erw
iesw
eg
May
rwie
sDa
xlue
gstra
ße
Reise
n-be
rger
str.
Borro
mäu
m-
stra
ßeMin
nes-
heim
str.
Ludw
ig-
Schm
eder
er-
Plat
z
May
rwie
s Ort
smitt
e
Hugo
-von
-Ho
fman
nsth
al-S
tr.
Schl
osss
tr.
Pars
cher
Str.
Volk
ssch
ule
Gni
gl
May
rwie
s Sch
uste
rweg
Fürb
erg-
str.
Gru
berfe
ld-
siedl
ung
Ger
sber
g-al
m
Jose
ph-M
essn
er-S
tr.
Voge
lweid
er-
straß
e
Ster
neck
-st
raße
Mer
ians
tr.
Bre
iten-
feld
erst
r.
Gni
gler
Stra
ße
Pos
chin
gers
tr.
Josef-
Waach
-Str.
Stei
nhau
sers
tr.
Viln
iuss
tr.
Schilli
ngho
f-
sie
dlung
Sam
S-B
ahn
Schlei
ferba
chweg
Lang
moosw
eg
Mauerm
ann-
straß
e
Maxstr
aße
Sied
lers
tr.
Aglassi
ngers
tr.G
ewer
beho
fstr.
Warwitz
str.
Hanna
kstr.
Stel
zham
erst
r.
Au
ersp
ergs
tr.
Baye
r-ha
mer
str.
Cana
valst
raße
Para
celsu
sstr.
Schw
arz-
p
ark
Wei
serh
ofst
r.
W.-D
ietri
ch-S
tr.
Hofw
irt
Robin
ig-
straß
e
Gril
lpar
zer-
stra
ße
Wirt
scha
fts-
kam
mer
Hayd
nstr.
Nel
böck
viad
ukt
St.-J
ulie
n-St
raße Ko
ngre
ssha
usEr
nest
-Thu
n-St
r.
Mira
bellp
latz
Mira
bell-
gart
enLa
ndes
-th
eate
r
Stru
berg
asse
Kirc
hens
tr.
Wer
kstä
tten
-st
r.Sc
hopp
erst
r.
Grüner
Wald
Aust
raße
Jako
b- H
arin
ger-S
tr.
Jägerw
irt
Berg
heim
Ort
smitt
e
Kase
rnBe
rghe
imFi
scha
ch
Hage
nau
Mar
ia P
lain
Plai
nbrü
cke
Erzh
erzo
g-Eu
gen-
Str.
Itzlin
g
Plai
n-sc
hule
Augu
st-
Gru
ber-S
tr .
Kreu
zstr.
Schere
rstr.
Enge
lber
t-W
eiß-
Weg
Plai
nbrü
cke
Zweig
str.
Goe
thes
tr.
Lenge
nfelde
n
Graf
enho
lzweg
Lengfe
lden
Orts
mitte
Land
str.
Lengfe
lden
Sie
dlung
Berghe
im
H
ande
ls-
z
entru
m
Lengfe
lden
Feue
rweh
r
Berghe
im
Plai
nbac
h-
straß
e
Werner-
v.-
Siemen
s-Plat
zN
eue
Mitt
eLe
hen
Rose
gger
-
str.
Jahn
str.
Gas
-w
erk-
gass
e
Aigl
hof/
LKH
Essh
aver
str.
Böhm
-Erm
olli-
Str.
Aigl
hof S
-Bah
nKu
en-
burg
str.
Stad
twer
kLe
hen
Mül
ln-A
ltsta
dt
Mön
chs-
berg
aufz
ugBäre
n-w
irt
Volk
ssch
ule
Mül
ln
Edua
rd-
Baum
gart
ner-
Stra
ße
W.-H
auth
aler
-Str.
Moo
sstr.
Bräu
-ha
usst
r.H.
-v.-
Kara
jan-
Plat
z
Augu
stin
erga
sse
Haup
tsch
ule
Max
glan
Röm
er-
gass
e
Reic
henh
alle
rSt
raße
Sinn
hubs
traße
FEST
UNG
SBAH
N
Fürs
ten-
brun
nstr.
Hübn
er-
gass
e
Pete
rsbr
unns
tr.
Rath
aus
Unip
ark/
Just
izge
bäud
eUl
rike-
G
schw
andt
ner-
Str.
Moz
arts
teg
Rott
Siez
enhe
im F
eilb
achs
tr.
Gug
gent
hal
Volk
ssch
ule
Lief
erin
g
Klei
ngm
ain
Flur
weg M
orzg
Schl
oss H
ellb
runn
Anif
Zoo
Salz
burg
Anif
Frie
sach
er
Gla
nzei
le
Arib
onen
str.
Mes
se
Salz
burg
aren
a
Peils
tein
erst
r.
Rasc
henb
ergs
tr.
Lauf
enst
r.
Hage
naus
tr.
Theo
dost
raße
Thom
as-
Bern
hard
-Str.
Diepolt
sdorf Tri
mmelkam
Kaig
asse
Kaje
tane
rpla
tz
Gröd
ig Holz
nerw
eg
Gröd
ig Gew
erbest
r.
Gröd
ig Fri
edho
f
Kühb
erg
St. Pa
ntaleo
n-Reit
h
Bern
ardi
gass
e
Walserf
eld Sc
hule
Kleß
heim
Sch
ule
6
3
4
5
4
525
27
24
27
22
87
1
35
23
232
23
840
34
4
151
2820
34
3521
S1
S11
Bürmoo
s
S11
Lam
prec
htsh
ause
n
6Pa
rsch
S11
S1
Berg
heim
Itzl
ing
Wes
t
Last
enst
r.
Lang
wie
d
Obe
rgni
gl
Itzl
ing
Pfla
nzm
ann
Lief
erin
g
Hau
ptba
hnho
f
May
rwie
s
Kleß
heim
S2S3
Chris
tian-
Dop
pler
-Klin
ik
Sam
Fürs
tenb
runn
Birk
ensi
edlu
ngSa
lzbu
rg S
üd
Frei
lass
ing
Salzb
urg Airp
ort
10
Uni
park
Non
ntal
20
Gai
sber
g-sp
itze15
1
22Sc
hallm
oos
14
A Bus-
term
inal
Non
ntal
1
Euro
park
112
2
3
2
Walserf
eld
10
10
7Sa
lzac
hsee
Bess
arab
iers
tr.
8
A
34
22Jo
sefia
u12
14
18
2724
28
28
Fading
erstr.
151
22
21110
Her
ausg
eber
: © S
alzb
urg
AG
Ges
taltu
ng u
nd G
rafik
: © 2
015,
dig
itale
Kar
togr
afie
F.Ru
ppen
thal
Gm
bH, K
arlsr
uhe
Stan
d: 2
8.10
.201
5 / P
erio
de 2
015
• Ä
nder
unge
n &
Dru
ckfe
hler
vor
beha
lten
BUS
Obu
s
S14 22
Lin
ien
net
zpla
n S
alzb
urg
Lin
e n
etw
ork
of
Salz
bu
rgLo
kalb
ahn/
S-Ba
hn
subu
rban
trai
n
Obu
slini
e t
rolle
ybus
line
Aut
obus
linie
bu
s lin
e
Tarif
zone
ngre
nze
far
e zo
ne li
mit
Kern
zone
fa
re z
one
Park
+ R
ide
Serv
ice
Cent
er /
Info
rmat
ion
nur H
aupt
verk
ehrs
zeit
rus
h ho
ur o
nly
an S
chul
tage
n sc
hool
day
s onl
y
an sc
hulfr
eien
Tag
en
scho
ol h
olid
ays o
nly
Bem
erku
ngen
:*1
Lin
ie 2
0 hä
lt n
icht
no
sto
p of
line
20
*2 L
inie
840
häl
t ni
cht
no
stop
of
line
840
*3 L
inie
151
häl
t ni
cht
no
stop
of
line
151
*4 L
inie
151
häl
t in
Ric
htun
g G
aisb
ergs
pitz
e ni
cht
no s
top
of li
ne 1
51 in
dir
ecti
on G
aisb
ergs
pitz
e*5
Lin
ie 1
51 h
ält
in R
icht
ung
Mir
abel
lpla
tz n
icht
no
stop
of
line
151
in d
irec
tion
Mir
abel
lpla
tz 1
H
alte
stel
le s
top
M.-R
einh
ardt
-Pla
tz 2
H
alte
stel
le s
top
Alte
r Mar
kt 3
H
alte
stel
le s
top
Moz
artp
latz
Alpe
nstr.
Fa. P
orsc
he*2
Akad
emie
stra
ße*2
Faist
auer
gass
e*2 He
rrna
u*2 Po
lizei
dire
ktio
n*2
Just
izge
bäud
e*1
Mak
artp
latz
/
T
heat
erga
sse
*2
Moz
arts
teg
Äuße
rer S
tein
*2
*2
Kies
el*2
*4
*4*4
*5
Land
es-
kran
kenh
aus
Ost
erm
ieth
ing
Sieze
nheim
Ortsmitt
eHim
melreic
h DOC
F.-Ha
nusc
h-Pl
atz
Mes
se
3
34
Restaurants nearby
Zentrum Herrnau
http://www.zentrumherrnau.at/
Alpbenstraÿe 48Mo.-Fr.: 09:00-19:00Sa.: 09:00-18:00SPAR (supermarket)Hofer (supermarket)dm (drugstore)pharmacycash machinewashing
Basic Bio Supermarket
Alpenstraÿe 75Mo.-Fr.: 08:00-19:00Sa.: 08:00-18:00
Merkur Supermarket
Otto-Holzbauer-Straÿe 1 (behind Motel One)Mo.-Thursday: 07:30-19:30Fr.: 07:00-20:00Sa.: 07:00-18:00
Die Shopping Arena
http://www.dieshoppingarena.at/
Alpenstraÿe 107Mo.-Fr.: 09:00-19:00Sa.: 09:00-18:00dm (drugstore)Eurospar (Supermarket)Libro (stationery)Müller (drugstore)cash mashine
35