Glasnik Matematicki, Vol. 55, No. 1 (2020), 13-27.

DIOPHANTINE EQUATIONS CONNECTED TO THE KOMORNIK POLYNOMIALS

András Bazsó, Attila Bérczes, Ondřej Kolouch, István Pink and Jan Šustek

Institute of Mathematics, MTA-DE Research Group ``Equations Functions and Curves'', Hungarian Academy of Sciences and University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary
e-mail: bazsoa@science.unideb.hu

Institute of Mathematics, University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary
e-mail: berczesa@science.unideb.hu

Department of Mathematics, University of Ostrava, 30. dubna 22, Ostrava, 701 03, Czech Republic
e-mail: ondrej.kolouch@osu.cz

Institute of Mathematics, University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary
e-mail: pinki@science.unideb.hu

Department of Mathematics, University of Ostrava, 30. dubna 22, Ostrava, 701 03, Czech Republic
e-mail: jan.sustek@osu.cz

Abstract.   We investigate the power and polynomial values of the polynomials Pn(X) = ∏nk=0 (X2 · 3k - X3k - 1 ) for n ∈ ℕ. We prove various ineffective and effective finiteness results. In the case 0≤ n ≤ 3, we determine all pairs x,y of integers such that Pn(x)=y2 or Pn(x)=y3.

2010 Mathematics Subject Classification.   11D41, 11B83

Key words and phrases.   Diophantine equations, decomposition, polynomial values

Full text (PDF) (free access)

https://doi.org/10.3336/gm.55.1.02

References:

  1. A. Bazsó, Polynomial values of (alternating) power sums, Acta Math. Hungar. 146 (2015), 202-219.
    MathSciNet     CrossRef

  2. A. Bazsó, A. Bérczes, L. Hajdu and F. Luca, Polynomial values of sums of products of consecutive integers, Monatsh. Math. 187 (2018), 21-34.
    MathSciNet     CrossRef

  3. A. Bérczes, J.-H. Evertse and K. Győry, Effective results for hyper- and superelliptic equations over number fields, Publ. Math. Debrecen 82 (2013), 727-756.
    MathSciNet     CrossRef

  4. Y. F. Bilu, B. Brindza, P. Kirschenhofer, A. Pintér and R. F. Tichy, Diophantine equations and Bernoulli polynomials, Compositio Math. 131 (2002), 173-188, with an appendix by A. Schinzel.
    MathSciNet     CrossRef

  5. Y. F. Bilu and R. F. Tichy, The Diophantine equation f(x) = g(y), Acta Arith. 95 (2000), 261-288.
    MathSciNet     CrossRef

  6. A. Dujella and I. Gusić, Indecomposability of polynomials and related Diophantine equations, Q. J. Math. 57 (2006), 193-201.
    MathSciNet     CrossRef

  7. A. Dubickas and D. Kreso, Diophantine equations with truncated binomial polynomials, Indag. Math. (N.S.) 27 (2016), 392-405.
    MathSciNet     CrossRef

  8. P. Erdös and J. Surányi, Topics in the theory of numbers. (Translated from the second Hungarian edition by Barry Guiduli.) Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2003.
    MathSciNet     CrossRef

  9. J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.
    MathSciNet     CrossRef

  10. V. Komornik, M. Pedicini and A. Pethö, Multiple common expansions in non-integer bases, Acta Sci. Math. (Szeged) 83 (2017), 51-60.
    MathSciNet     CrossRef

  11. D. Kreso, On common values of lacunary polynomials at integer points, New York J. Math. 21 (2015), 987-1001.
    MathSciNet     CrossRef

  12. D. Kreso, Diophantine equations in separated variables and lacunary polynomials, Int. J. Number Theory, 13 (2017), 2055-2074.
    MathSciNet     CrossRef

  13. M. Kulkarni and B. Sury, On the Diophantine equation x(x+1)(x+2)∙∙∙(x+(m-1))=g(y), Indag. Math. (N.S.) 14 (2003), 35-44.
    MathSciNet    

  14. R. Lidl, G. Mullen and G. Turnwald, Dickson polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics 65, Longman Scientific & Technical, Harlow, 1993.
    MathSciNet     CrossRef

  15. H. London and R. Finkelstein, On Mordell's equation y2 - k = x3, Bowling Green State university, Bowling Green, OH, 1973.
    MathSciNet    

  16. Gy. Péter, A. Pintér and A. Schinzel, On equal values of trinomials, Monatsh. Math. 162 (2011), 313-320.
    MathSciNet     CrossRef

  17. Cs. Rakaczki, On the Diophantine equation Sm(x)=g(y), Publ. Math. Debrecen 65 (2004), 439-460.
    MathSciNet    

  18. A. Schinzel, Equal values of trinomials revisited, Tr. Mat. Inst. Steklova 276 (2012), 255-261.
    MathSciNet     CrossRef

  19. T. Stoll and R. F. Tichy, Diophantine equations involving general Meixner and Krawtchouk polynomials, Quaest. Math. 28 (2005), 105-115.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page