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On power values of pyramidal numbers, II

Volume 208 / 2023

Andrej Dujella, Kálmán Győry, Philippe Michaud-Jacobs, Ákos Pintér Acta Arithmetica 208 (2023), 199-213 MSC: Primary 11D41; Secondary 11D59, 11D61, 14G99. DOI: 10.4064/aa211213-27-7 Published online: 23 August 2023


For $m \geq 3$, we define the $m$th order pyramidal number by \[ \mathrm{Pyr}_m(x) = \tfrac{1}{6} x(x+1)((m-2)x+5-m). \] In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation $\mathrm{Pyr}_m(x) = y^2$ are found in positive integers $x$ and $y$, for $6 \leq m \leq 100$. In this paper, we consider the question of higher powers, and find all solutions to the equation $\mathrm{Pyr}_m(x) = y^n$ in positive integers $x$, $y$, and $n$, with $n \geq 3$, and $5 \leq m \leq 50$. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.


  • Andrej DujellaDepartment of Mathematics
    Faculty of Science
    University of Zagreb
    10000 Zagreb, Croatia
  • Kálmán GyőryInstitute of Mathematics
    University of Debrecen
    H-4032 Debrecen, Hungary
  • Philippe Michaud-JacobsMathematics Institute
    University of Warwick
    Coventry, CV4 7AL, UK
  • Ákos PintérInstitute of Mathematics
    University of Debrecen
    H-4032 Debrecen, Hungary
    MTA-DE Equations, Functions and
    Curves Research Group
    Eötvös Loránd Research Network (ELKH)

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