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Skolem’s conjecture confirmed for a family of exponential equations

Volume 192 / 2020

L. Hajdu, R. Tijdeman Acta Arithmetica 192 (2020), 105-110 MSC: 11D61, 11D79. DOI: 10.4064/aa190114-25-2 Published online: 19 September 2019


Skolem’s conjecture states that if an exponential Diophantine equation is not solvable, then it is not solvable modulo an appropriately chosen modulus. Apart from many concrete equations, the conjecture has been proved only for rather special classes of equations. Here we show that the conjecture is valid for the Catalan equation $u^x-v^y=1$ provided that one of $u,v$ is a prime. This is the first instance where the conjecture is proved for a family of equations with more than one term on the left-hand side, of which the bases are multiplicatively independent.


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

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