JournalspmVol. 67 / No. 1DOI 10.4171/pm/1855

Variants of the Diophantine equation <var>n</var>! + 1 = <var>y</var><sup>2</sup>

  • Omar Kihel

    Brock University, St. Catharines, Canada
  • Florian Luca

    UNAM, Campus Morelia, Michoacán, Mexico
  • Alain Togbé

    Purdue University North Central, Westville, USA
Variants of the Diophantine equation <var>n</var>! + 1 = <var>y</var><sup>2</sup>  cover

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In this note we study variants of the Brocard–Ramanujan Diophantine equation n! + 1 = y2. For example, Berend and Harmse [1] proved that the equation n! = yr(y + 1) has only finitely many positive integer solutions (n,y) when r ≥ 4 is a fixed integer. Here we find all the integer solutions of this equation when r = 2, 3 under the additional assumption that y + 1 is square-free or cube-free, respectively.