Diophantine m-tuples, elliptic curves, Thue and index form equations



Grant no. 6422 supported by Croatian Science Foundation HRZZ (July 2014 - June 2018)


Abstract:

The proposed project will coordinate and support research activities of Croatian number theory group.

One of the main topics of research in this project will be Diophantine m-tuples (sets with the property that the product of its any two distinct elements increased by 1 is a perfect square) and their generalizations. The first Diophantine quadruple {1,3,8,120} was found by Fermat. Baker and Davenport proved that Fermat's set cannot be extended to a quintuple. This result motivated the famous conjecture that there does not exist a Diophantine quintuple. Dujella proved that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. Recently, Filipin, Fujita and Elsholtz significantly improved previously known upper bounds for the number of Diophantine quintuples, but the main conjecture still remains open. Various generalizations of Diophantine m-tuples will be studied, in particular, considering the problem in the rings of integers of number fields of small degree.

Diophantine triples were used by Dujella in construction of high-rank elliptic curves over Q and Q(T) and new results in that direction are expected. In determining exactly the Mordell-Weil group of several elliptic curves over Q(T), an important tool is recent algorithm by Gusić and Tadić for finding rationals at which the specialization map is injective. We plan to improve and extend this algorithm for elliptic curves over K(T) for general number fields K.

Members of the group will also work on Thue, Thue-Mahler and index forms equations, problems related to sums of divisors, polynomial root separation and applications of Diophantine approximations to cryptanalysis.

The results of the research will be published in respectable journals and presented on international conferences. Through this project we will intensify our scientific connections with international number-theorists and qualify young researchers for scientific research in number theory.


Principal investigator:

Andrej Dujella, Professor, Zagreb


Team members:

Ivica Gusić, Professor, Zagreb
Alan Filipin, Associate Professor, Zagreb
Borka Jadrijević, Associate Professor, Split
Zrinka Franušić, Associate Professor, Zagreb
Mirela Jukić Bokun, Assistant Professor, Osijek
Ana Jurasić, Assistant Professor, Rijeka
Petra Tadić, Assistant Professor, Pula
Tomislav Pejković, Assistant Professor, Zagreb
Ivan Soldo, Assistant Professor, Osijek
Sanda Bujačić Babić, postdoc, Rijeka
Nikola Adžaga, PhD student, Zagreb
Marija Bliznac Trebješanin, PhD student, Split
Vinko Petričević, postdoc, Zagreb


Publications:

  1. M. Bliznac Trebješanin and A. Filipin, Nonexistence of D(4)-quintuples, preprint, arXiv:1704.01874.

  2. Y. Bugeaud and T. Pejković, Explicit examples of p-adic numbers with prescribed irrationality exponent, preprint.

  3. S. Bujačić Babić, Two divisors of (n2+1)/2 summing up to δn+δ±2, δ even, preprint, arXiv:1707.00049.

  4. A. Dujella, M. Jukić Bokun and I. Soldo, A Pellian equation with primes and applications to D(-1)-quadruples, preprint, arXiv:1706.01959.

  5. A. Dujella and M. Kazalicki, Diophantine m-tuples in finite fields and modular forms, preprint, arXiv:1609.09356.

  6. A. Dujella and F. Luca, On fundamental units of class number one quadratic fields, preprint.

  7. A. Filipin and A. Jurasić, A polynomial variant of a problem of Diophantus and its consequences, preprint, arXiv:1705.09194.


  8. M. Bliznac Trebješanin, A. Filipin and A. Jurasić, On the polynomial quadruples with the property D(-1;1), Tokyo J. Math., to appear.

  9. Y. Bugeaud, A. Dujella, T. Pejković and B. Salvy, Absolute real root separation, Amer. Math. Monthly, to appear.

  10. Z. Franušić and B. Jadrijević, Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields, Publ. Math. Debrecen, to appear.

  11. T. Marošević and I. Soldo, Modified indices of political power - a case study of a few parliaments, CEJOR Cent. Eur. J. Oper. Res., to appear.


  12. N. Adžaga, A. Dujella, D. Kreso and P. Tadić, Triples which are D(n)-sets for several n's, J. Number Theory 184 (2018), 330-341.


  13. N. Adžaga, Automated conjecturing of Frobenius numbers via grammatical evolution, Experiment. Math. 26 (2017), 247-252.

  14. N. Adžaga and A. Filipin, On the extension of D(-8k2)-pair {8k2, 8k2 + 1}, Mosc. Math. J. 17 (2017), 165-174.

  15. A. Bayad, A. Dossavi-Yovo, A. Filipin and A. Togbé, On the extensibility of the D(4)-triple {k-2, k+2, 4k} over Gaussian integers, Notes Number Theory Discrete Math. 23 (2017), 1-26.

  16. S. Bujačić, A variation of a congruence of Subbarao for n=2α5β, α, β ≥ 0, Period. Math. Hungar. 75 (2017), 66-79.

  17. A. Dujella, M. Jukić Bokun and I. Soldo, On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 111 (2017), 1177-1185.

  18. A. Dujella and M. Kazalicki, More on Diophantine sextuples, in Number Theory - Diophantine problems, uniform distribution and applications, Festschrift in honour of Robert F. Tichy's 60th birthday (C. Elsholtz, P. Grabner, Eds.), Springer-Verlag, Berlin, 2017, pp. 227-235.

  19. A. Dujella, M. Kazalicki, M. Mikić and M. Szikszai, There are infinitely many rational Diophantine sextuples, Int. Math. Res. Not. IMRN 2017 (2) (2017), 490-508.

  20. A. Dujella and T. Pejković, Root separation for reducible monic polynomials of odd degree, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 21 (2017), 21-27.

  21. A. Filipin, The extension of some D(4)-pairs, Notes Number Theory Discrete Math. 23 (2017), 126-135.

  22. I. Gaál and B. Jadrijević, Determining elements of minimal index in an infinite family of totally real bicyclic biquadratic number fields, JP J. Algebra Number Theory Appl. 39 (2017), 307-326.


  23. Lj. Baćić Đuračković and A. Filipin, The extendibility of D(4)-pairs {F2k, F2k+6} and {P2k, P2k+4}, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 20 (2016), 27-35.

  24. A. Bayad, A. Filipin and A. Togbé, Extension of a parametric family of Diophantine triples in Gaussian integers, Acta Math. Hungar. 148 (2016), 312-327.

  25. A. Bérczes, A. Dujella, L. Hajdu and Sz. Tengely, Finiteness results for F-Diophantine sets, Monatsh. Math. 180 (2016), 469-484.

  26. M. Bliznac and A. Filipin, An upper bound for the number of Diophantine quintuples, Bull. Aust. Math. Soc. 94 (2016), 384-394.

  27. S. Bujačić and A. Filipin, Linear forms in logarithms, in Diophantine Analysis: Course Notes from a Summer School (J. Steuding, Ed.), Birkhäuser, Basel, ISBN 978-3-319-48816-5, 2016, pp. 1-59.

  28. M. Cipu, A. Filipin and Y. Fujita, Bounds for Diophantine quintuples II, Publ. Math. Debrecen 88 (2016), 59-78.

  29. A. Dujella, What is ... a Diophantine m-tuple?, Notices Amer. Math. Soc. 63 (2016), 772-774.

  30. A. Dujella, J. C. Peral and P. Tadić, Elliptic curves with torsion group Z/6Z, Glas. Mat. Ser. III 51 (2016), 321-333.

  31. A. Filipin and A. Jurasić, On the size of Diophantine m-tuples for linear polynomials, Miskolc Math. Notes 17 (2016), 861-876.

  32. C. Fuchs, A. Jurasić, R. Paulin, Elementary resolution of a family of quartic Thue equations over function fields, Monatsh. Math. 180 (2016), 205-211.

  33. T. Pejković, Quadratic Lagrange spectrum, Math. Z. 283 (2016), 861-869.

  34. T. Pejković, P-adic root separation for quadratic and cubic polynomials, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 20 (2016), 9-18.

  35. T. Pejković, An inequality for values of Koksma's functions of two algebraically dependent p-adic numbers, Period. Math. Hungar. 73 (2016), 251-258.

  36. I. Soldo, D(-1)-triples of the form {1, b, c} in the ring Z[√-t], t > 0, Bull. Malays. Math. Sci. Soc. (2) 39 (2016), 1201-1224.


  37. Y. Bugeaud and T. Pejković, Quadratic approximation in Qp, Int. J. Number Theory 11 (2015), 193-209.

  38. A. Dujella, I. Gusić and P. Tadić, The rank and generators of Kihara's elliptic curve with torsion Z/4Z over Q(t), Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 105-109.

  39. A. Dujella and J. C. Peral, Elliptic curves with torsion group Z/8Z or Z/2Z × Z/6Z, in Trends in Number Theory, Contemp. Math. 649 (2015), 47-62.

  40. A. Filipin, The extendibility of D(4)-pair {F2k, 5F2k}, Fibonacci Quart. 53 (2015), 124-129.

  41. I. Gusić and P. Tadić, Injectivity of the specialization homomorphism of elliptic curves, J. Number Theory 148 (2015), 137-152.

  42. B. Jadrijević, On elements with index of the form 2a3b in a parametric family of biquadratic fields, Glas. Mat. Ser. III 50 (2015), 43-63.

  43. B. Peker, A. Dujella and S. Cenberci, The non-extensibility of D(-2k+1)-triples {1, k2, k2+2k-1}, Miskolc Math. Notes 16 (2015), 385-390.


  44. J. Aguirre, A. Dujella, M. Jukić Bokun and J. C. Peral, High rank elliptic curves with prescribed torsion group over quadratic fields, Period. Math. Hungar. 68 (2014), 222-230.

  45. Lj. Baćić and A. Filipin, A note on the number of D(4)-quintuples, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18 (2014), 7-13.

  46. A. Bérczes, A. Dujella and L. Hajdu, Some Diophantine properties of the sequence of S-units, J. Number Theory 138 (2014), 48-68.

  47. J. Bosman, P. Bruin, A. Dujella and F. Najman, Ranks of elliptic curves with prescribed torsion over number fields, Int. Math. Res. Not. IMRN 2014 (11) (2014), 2885-2923.

  48. Y. Bugeaud and A. Dujella, Root separation for reducible integer polynomials, Acta Arith. 162 (2014), 393-403.

  49. S. Bujačić, Two divisors of (n2+1)/2 summing up to δn+ε, for δ and ε even, Miskolc Math. Notes 15 (2014), 333-344.

  50. A. Dujella, I. Gusić and L. Lasić, On quadratic twists of elliptic curves y2 = x(x - 1)(x - λ), Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18 (2014), 27-34.

  51. A. Dujella and M. Mikić, On the torsion group of elliptic curves induced by D(4)-triples, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 22 (2014), 79-90.

  52. A. Dujella, F. Najman, N. Saradha and T. N. Shorey, Products of three factorials, Publ. Math. Debrecen 85 (2014), 123-130.

  53. A. Dujella and J. C. Peral, High rank elliptic curves with torsion Z/2Z × Z/4Z induced by Diophantine triples, LMS J. Comput. Math. 17 (2014), 282-288.

  54. A. Dujella and J. C. Peral, Elliptic curves coming from Heron triangles, Rocky Mountain J. Math. 44 (2014), 1145-1160.

  55. A. Dujella and N. Saradha, Diophantine m-tuples with elements in arithmetic progressions, Indag. Math. (N.S.) 25 (2014), 131-136.

  56. C. Elsholtz, A. Filipin and Y. Fujita, On Diophantine quintuples and D(-1)-quadruples, Monatsh. Math. 175 (2014), 227-239.

  57. A. Filipin, Y. Fujita and A. Togbé, The extendibility of Diophantine pairs I: the general case, Glas. Mat. Ser. III 49 (2014), 25-36.

  58. A. Filipin, Y. Fujita and A. Togbé, The extendibility of Diophantine pairs II: examples, J. Number Theory 145 (2014), 604-631.

  59. Z. Franušić and I. Soldo, The problem of Diophantus for integers of Q(√-3), Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18 (2014), 15-25.

  60. A. S. Janfada, S. Salami, A. Dujella and J. C. Peral, On the high rank π/3 and 2π/3 - congruent number elliptic curves, Rocky Mountain J. Math. 44 (2014), 1867-1880.

Croatian Scientific Bibliography (CROSBI)


Conferences and events:

Torsion groups and Galois representations of elliptic curves, Zagreb, 25.-29.6.2018.
18th Central European Conference on Cryptology, Smolenice, Slovakia, 06.-08.06.2018. (A. Dujella)
Analytic Number Theory and Related Areas, Kyoto, Japan, 30.10.-01.11.2017. (A. Dujella)
Festkolloquium in honour of Robert F. Tichy's 60th birthday, Graz, Austria, 19.-20.10.2017. (A. Dujella, A. Filipin)
20th International Workshop for Young Mathematicians "Number Theory", Krakow, Poland, 17.-23.09.2017. (N. Adžaga, M. Bliznac Trebješanin)
Diophantine problems (DIOP) Manchester, UK, 11.-15.09.2017. (Z. Franušić, B. Jadrijević)
19th ÖMG Congress and Annual DMV Meeting, Salzburg, Austria, 11.-15.09.2017. (A. Filipin, A. Jurasić, T. Pejković)
23rd Czech and Slovak International Conference on Number Theory, Ostravice, Czech Republic, 28.08.-01.09.2017. (A. Dujella)
30th Journées Arithmétiques , Caen, France, 03.-07.07.2017. (M. Bliznac Trebješanin, B. Jadrijević, M. Jukić Bokun, I. Soldo)
17th Central European Conference on Cryptology, Warsaw, Poland, 28.-30.6.2017. (A. Dujella)
Diophantine Analysis and Related Fields 2017, Tokyo, Japan, 07.-09.01.2017. (A. Filipin)
1st Croatian Combinatorial Days, Zagreb, Croatia, 29.-30.09.2016. (M. Jukić Bokun, T. Pejković)
ELAZ2016, Strobl, Austria, 05.-09.09.2016. (N. Adžaga, A. Filipin)
5th International Conference on Uniform Distribution Theory (UDT 2016), Sopron, Hungary, 05.-08.07.2016. (A. Dujella)
17th International Conference on Fibonacci Numbers and their Applications, Caen, France, 27.06.-02.07.2016. (T. Pejković)
14th Meeting of the Canadian Number Theory Association Conference (CNTA XIV), Calgary, Canada, 20.-24.06.2016. (A. Filipin)
Sixth Croatian Mathematical Congress, Zagreb, Croatia, 14.-17.06.2016. (M. Bliznac, S. Bujačić, A. Dujella, A. Filipin, T. Pejković)
Computational Aspects of Diophantine Equations, Salzburg, Austria, 15.-19.02.2016. (A. Dujella, I. Soldo, P. Tadić)
Graz-Salzburg Workshop on Diophantine Problems, Graz, Austria, 26.-27.11.2015. (A. Filipin)
Novigrad Then and Now, Zadar, Novigrad, Croatia, 16.-17.10.2015. (A. Dujella)
The Geometry, Algebra and Analysis of Algebraic Numbers, Banff, Canada, 04.-09.10.2015. (A. Dujella)
Diophantine Equations, LMS-CMI graduate school, Hay-on-Wye, UK, 15.-19.09.2015. (P. Tadić)
French-German Summer School Galois Theory and Number Theory, Konstanz, Germany, 18.-24.07.2015. (M. Bliznac)
Diophantine Approximation and Related Topics, Aarhus, Denmark, 13.-17.07.2015. (S. Bujačić)
29th Journées Arithmétiques and Kálmán Győry 75 Symposium, Debrecen, Hungary, 06.-11.07.2015. (N. Adžaga, S. Bujačić, A. Dujella, A. Filipin, Z. Franušić, B. Jadrijević, A. Jurasić, T. Pejković)
Atelier PARI/GP 2015, Bordeaux, France, 12.-16.01.2015. (P. Tadić)
Workshop on Number Theory and Algebra (on the occasion of the 60th birthday of Ivica Gusić), Zagreb, Croatia, 26.-28.11.2014. (all members)
Conference on Diophantine m-tuples and related problems, Westville, USA, 13.-15.11.2014. (N. Adžaga, A. Filipin, Z. Franušić, B. Jadrijević, I. Soldo)
Advanced School and Workshop on L-functions and modular forms, Trieste, Italy, 01.-14.09.2014. (P. Tadić)
ELementare und Analytische Zahlentheorie (ELAZ), Hildesheim, Germany, 28.07.-01.08.2014. (S. Bujačić)
Summer school on Diophantine Analysis, Würzburg, Germany, 21.-26.07.2014. (S. Bujačić)
First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI, Bilbao, Spain, 30.06.-04.07.2014. (A. Dujella)


PhD Theses:

Ljubica Baćić: Sets in Which xy + 4 is Always a Square and Problem of the Extensibility of Some Parametric Diophantine Triples (Zagreb, 10.07.2014.)
(supervisor: Alan Filipin)

Miljen Mikić, Mordell-Weil Groups and Isogenies of the Families of Elliptic Curves (Zagreb, 2.12.2014.)
(supervisors: Andrej Dujella and Filip Najman)

Sanda Bujačić, Diophantine Problems With Sums of Divisors (Zagreb, 18.12.2014.)
(supervisor: Andrej Dujella)


Visitors:

Mahadi Ddamulira (Graz University of Technology), 05.-07.12.2017.
Anthony Scholl (University of Cambridge), 03.-06.12.2017.
Robert Tichy (Graz University of Technology), 05.-07.11.2017.
Clemens Heuberger (Alpen-Adria-Universität Klagenfurt), 24-30.09.2017.
Christian Elsholtz (Graz University of Technology), 09.-13.07.2017.
Dijana Kreso (University of British Columbia, Graz University of Technology), 03.-14.07.2017.
Volker Ziegler (University of Salzburg), 06.-09.06.2017.
Florian Luca (Wits University, Johannesburg; Max Planck Institute, Bonn; University of Ostrava), 19.-22.04.2017.
Bartosz Naskrecki (University of Bristol), 12.-16.12.2016.
Kwok Chi Chim (Graz University of Technology), 12.-15.12.2016.
Marc Munsch (Graz University of Technology), 04.-07.12.2016.
Dijana Kreso (Graz University of Technology), 15.11.-14.12.2016.
Kwok Chi Chim (Graz University of Technology), 03.-06.10.2016.
Stefan Planitzer (Graz University of Technology), 03.-06.10.2016.
Dijana Kreso (Graz University of Technology), 01.09.-15.10.2016.
Pallab Kanti Dey (Harish-Chandra Research Institute, Allahabad), 20.-26.06.2016.
Robert Tichy (Graz University of Technology), 16.-18.06.2016.
István Gaál (University of Debrecen), 15.-19.05.2016.
Yann Bugeaud (University of Strasbourg), 23.-27.03.2016.
Alain Togbé (Purdue University North Central, Westville), 09.-14.02.2016.
Mihai Cipu (Institute of Mathematics of the Romanian Academy, Bucharest), 07.-13.06.2015.
Márton Szikszai (University of Debrecen), 18.05.-13.06.2015.
Gökhan Soydan (Uludag University, Bursa and University of Debrecen), 01.05.-06.05.2015.
Dijana Kreso (Graz University of Technology), 20.04.-23.04.2015.
Daniel Krenn (Graz University of Technology), 20.04.-23.04.2015.
Maria Rita Iacò (Graz University of Technology), 20.04.-23.04.2015.
Kwok Chi Chim (Graz University of Technology), 20.04.-22.04.2015.
Dijana Kreso (Graz University of Technology), 30.03.-01.04.2015.
Jose Miguel Echarri Hernandez (University of the Basque Country, Bilbao), 21.03.-20.04.2015.
Attila Bérczes (University of Debrecen), 25.-28.11.2014.
Yann Bugeaud (Univesity of Strasbourg), 26.-30.11.2014.
Christian Elsholtz (Graz University of Technology), 26.-29.11.2014.
Clemens Fuchs (University of Salzburg), 25.-29.11.2014.
Enrique González Jiménez (Autonomous University of Madrid), 25.-30.11.2014.
Lajos Hajdu (University of Debrecen), 25.-28.11.2014.
Dijana Kreso (Graz University of Technology), 25.-29.11.2014.
Florian Luca (Wits University, Johannesburg), 26.-30.11.2014.
Juan Carlos Peral (University of the Basque Country, Bilbao), 26.-29.11.2014.
Attila Pethő (University of Debrecen), 25.-28.11.2014.
Ákos Pintér (University of Debrecen), 25.-29.11.2014.
Thomas Stoll (University of Lorraine, Nancy), 25.-28.11.2014.
Robert Tichy (Graz University of Technology), 26.-28.11.2014.
Attila Bérczes (University of Debrecen), 04.-10.09.2014.
Lajos Hajdu (University of Debrecen), 04.-10.09.2014.
Saradha Natarajan (Tata Institute of Fundamental Research, Mumbai), 04.-10.09.2014.


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