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

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

Abstract:

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:

Team members:

Publications:

1. N. Adžaga, On the size of Diophantine m-tuples in imaginary quadratic number rings, Bull. Math. Sci. 11(1) (2021) 1950020 (10 pages)

2. A. Dujella and M. Kazalicki, Diophantine m-tuples in finite fields and modular forms, Res. Number Theory 7 (2021), Article number 3, (24pp)

   
   

3. M. Cipu, A. Filipin and Y. Fujita, Diophantine pairs that induce certain Diophantine triples, J. Number Theory 210 (2020), 433-475.

4. M. Cipu, A. Filipin and Y. Fujita, An infinite two-parameter family of Diophantine triples, Bull. Malays. Math. Sci. Soc. 43 (2020), 481-498.

   
   

5. A. Bérczes, A. Dujella, L. Hajdu, N. Saradha and R. Tijdeman, Products of factorials which are powers, Acta Arith. 190 (2019), 339-350.

6. M. Bliznac Trebješanin and A. Filipin, Nonexistence of D(4)-quintuples, J. Number Theory 194 (2019), 170-217.

7. A. Dujella, M. Jukić Bokun and I. Soldo, A Pellian equation with primes and applications to D(-1)-quadruples, Bull. Malays. Math. Sci. Soc. 42 (2019), 2915-2926.

8. A. Dujella and F. Luca, On fundamental units of real quadratic fields of class number 1, Arch. Math. (Basel) 113 (2019), 349-353.

9. A. Dujella and J. C. Peral, Elliptic curves induced by Diophantine triples, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 113 (2019), 791-806.

10. A. Filipin and A. Jurasić, A polynomial variant of a problem of Diophantus and its consequences, Glas. Mat. Ser. III 54 (2019), 21-52.

11. Z. Franušić and B. Jadrijević, D(n)-quadruples in the ring of integers of Q(√2,√3), Math. Slovaca 69 (2019), 1263-1278.

12. I. Gaál, B. Jadrijević and L. Remete, Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields, Int. J. Number Theory 15 (2019), 11-27.

    
    

13. 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.

14. N. Adžaga, A. Dujella, D. Kreso, P. Tadić, On Diophantine m-tuples and D(n)-sets, RIMS Kokyuroku 2092 (2018), 130-137.

15. M. Bliznac Trebješanin, A. Filipin and A. Jurasić, On the polynomial quadruples with the property D(-1;1), Tokyo J. Math. 41 (2018), 527-540.

16. Y. Bugeaud and T. Pejković, Explicit examples of p-adic numbers with prescribed irrationality exponent, Integers 18A (2018), #A5 (15 pp)

17. S. Bujačić Babić, Two divisors of (n²+1)/2 summing up to δn + δ ± 2, δ even, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 22 (2018), 39-61.

18. A. Dujella, I. Gusić, V. Petričević and P. Tadić, Strong Eulerian triples, Glas. Mat. Ser. III 53 (2018), 33-42.

19. Z. Franušić and B. Jadrijević, Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields, Publ. Math. Debrecen 92 (2018), 293-315.

20. I. Gaál, B. Jadrijević and L. Remete, Totally real Thue inequalities over imaginary quadratic fields, Glas. Mat. Ser. III 53 (2018), 229-238.

21. T. Marošević and I. Soldo, Modified indices of political power: a case study of a few parliaments, CEJOR Cent. Eur. J. Oper. Res. 26 (2018), 645-657.

    
    

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

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

24. 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.

25. Y. Bugeaud, A. Dujella, T. Pejković and B. Salvy, Absolute real root separation, Amer. Math. Monthly 124 (2017), 930-936.

26. S. Bujačić, A variation of a congruence of Subbarao for n=2^α5^β, α, β ≥ 0, Period. Math. Hungar. 75 (2017), 66-79.

27. 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.

28. 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.

29. 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.

30. 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.

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

32. 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.

    
    

33. Lj. Baćić Đuračković and A. Filipin, The extendibility of D(4)-pairs {F_2k, F_2k+6} and {P_2k, P_2k+4}, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 20 (2016), 27-35.

34. 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.

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

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

37. 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.

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

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

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

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

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

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

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

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

46. 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.

    
    

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

48. 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.

49. 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.

50. A. Filipin, The extendibility of D(4)-pair {F_2k, 5F_2k}, Fibonacci Quart. 53 (2015), 124-129.

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

52. B. Jadrijević, On elements with index of the form 2^a3^b in a parametric family of biquadratic fields, Glas. Mat. Ser. III 50 (2015), 43-63.

53. 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.

    
    

54. 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.

55. 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.

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

57. 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.

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

59. 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.

60. 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.

61. 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.

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

63. 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.

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

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

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

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

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

69. 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.

70. 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.

Conferences and events:

PhD Theses:

Marija Bliznac Trebješanin: Diophantine D(4)-m-tuples and Related Problems (Zagreb, 25.1.2018.)
(supervisor: Alan Filipin)

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

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

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)

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