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Skolem’s conjecture confirmed for a family of exponential equations, II

Volume 197 / 2021

A. Bérczes, L. Hajdu, R. Tijdeman Acta Arithmetica 197 (2021), 129-136 MSC: 11D61, 11D79. DOI: 10.4064/aa191116-1-6 Published online: 8 October 2020

Abstract

According to Skolem’s conjecture, if an exponential Diophantine equation is not solvable, then it is not solvable modulo an appropriately chosen modulus. Besides several concrete equations, the conjecture has only been proved for rather special cases. In this paper we prove the conjecture for equations of the form $x^n-by_1^{k_1}\dots y_\ell ^{k_\ell }=\pm 1$, where $b,x,y_1,\dots ,y_\ell $ are fixed integers and $n,k_1,\dots ,k_\ell $ are non-negative integral unknowns. This result extends a recent theorem of Hajdu and Tijdeman.

Authors

  • A. BérczesInstitute of Mathematics
    University of Debrecen
    P.O. Box 12
    H-4010 Debrecen, Hungary
    e-mail
  • L. HajduInstitute of Mathematics
    University of Debrecen
    P.O. Box 12
    H-4010 Debrecen, Hungary
    e-mail
  • R. TijdemanMathematical Institute
    Leiden University
    Postbus 9512
    2300 RA Leiden, The Netherlands
    e-mail

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