A note on the diophantine equation $(x^m-1)/(x-1) = y^n$

On the Diophantine equation (xn- 1)/(x-1) = yq

“…The following lemma is due to Saradha & Shorey [15] and originate in a work of Le [9]. Its proof uses Skolem's method.…”

Rational approximation to algebraic numbers of small height: the Diophantine equation |axn - byn|= 1

“…Our proof uses Theorem 1.1 in conjunction with work of Ljunggren [Lj1]. Part (b) is Theorem 1 of [Le1], but follows very easily from Theorem 1.1. Part (c) is a slight sharpening of work of Saradha and Shorey [SS], who deduced a like result under the hypothesis that z 32 or z e f2Y 3Y 4Y 8Y 9Y 16Y 27gX…”

On the Nagell-Ljunggren equation $(x^n - 1)/(x - 1) = y^q$

“…Besides the new upper bounds obtained in [10], [11], the main ingredient for the proof of Theorem 1 is a factorisation recalled in Lemma 1 below. It easily follows from Lemma 1 and from Theorem NL that, in order to prove ( * ) Actually, it is explained in [17], page 476, and in [5], Théorème 15, that inserting results from [7] and [1] in the same proof yields that (1) has no solution (x, y, n, q) with ω(n) > q − 2 that (1) has no solution outside (S), it is sufficient to solve (2) for any odd prime numbers p and q. We are able to considerably improve this assertion.…”

“…Alternatively, we can apply a result of Le [7], asserting that Equation (1) has no solution with x being a q-th power. Consequently, we have proved that if n is a power of q, then n = q.…”