Summary: Let \((x, y, m, n)\) be a positive integer solution of the Nagell-Ljunggren equation \((x^m-1)/(x-1) = y^n\), \(x > 1, y > 1, m > 2\) and \(n > 1\). In this paper, we use the elementary method and the properties of the \(p\)-adic number to prove that if \(x^m>C\), where \(C\) is an effectively computable absolute constant, then \(n=\mathrm{ord}_x y\). This result basically solved a conjecture proposed by H. Edgar in [Rocky Mt. J. Math. 15, 327–329 (1985; Zbl 0583.10012)].