New bounds and conditions for the equation of Nagell–Ljunggren

Abstract: The general case of the Nagell–Ljunggren equation is and odd primes. In this paper we derive a new bound for which there are no solutions. For small q the problem is harder and we achieve a conditional result: for and some additional condition on must hold. Both functions are quadratic and they leave a small gap for , on which we have no general result. The picture is completed by some explicit, unconditional upper bounds for the absolute values , . [Copyright &y& Elsevier]