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Products of factorials which are powers

Volume 190 / 2019

A. Bérczes, A. Dujella, L. Hajdu, N. Saradha, R. Tijdeman Acta Arithmetica 190 (2019), 339-350 MSC: 11D41, 11D85. DOI: 10.4064/aa171008-16-10 Published online: 29 July 2019

Abstract

Extending earlier research of Erdős and Graham, we consider the problem of products of factorials yielding perfect powers. On the one hand, we describe how the representability of $\ell$th powers behaves when the number of factorials is smaller than, equal to or larger than $\ell$, respectively. On the other hand, we investigate for which fixed $n=b_1$ it is possible to find integers $b_2,\dots,b_k$ at most $b_1$ (obeying certain conditions) such that $b_1!\cdots b_k!$ is a perfect power. Here we distinguish the cases where the factorials may be repeated or are distinct.

Authors

  • A. BérczesInstitute of Mathematics
    University of Debrecen
    P.O. Box 12
    H-4010 Debrecen, Hungary
    e-mail
  • A. DujellaDepartment of Mathematics
    Faculty of Science
    University of Zagreb
    Bijenička cesta 30
    10000 Zagreb, Croatia
    e-mail
  • L. HajduInstitute of Mathematics
    University of Debrecen
    P.O. Box 12, Hungary
    H-4010 Debrecen
    e-mail
  • N. SaradhaINSA Senior Scientist
    DAE - Center for Excellence in Basic Sciences
    Mumbai University
    Mumbai, India
    e-mail
  • R. TijdemanMathematical Institute
    Leiden University
    Postbus 9512
    2300 RA Leiden, The Netherlands
    e-mail

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