New bounds and conditions for the equation of Nagell–Ljunggren

The general case of the Nagell–Ljunggren equation isxp−1x−1=pe⋅yq,withx,y∈Z,e∈{0,1}, and p≠qp≠q odd primes. In this paper we derive a new bound q>f(p)q>f(p) for which there are no solutions. For small q   the problem is harder and we achieve a conditional result: q∤hp− for q