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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


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.


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

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