On the diophantine equation $(x^m-1)/(x-1) = y^n$

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

“…First, we warn the reader that the proofs of the results obtained in Le [10] and in Yu & Le [19] are incorrect. Indeed, they all depend on Lemma 3 of [10], which is false, see the comment of Yuan Ping-Zhi [20].…”

Diophantine equations with Bernoulli polynomials

“…We have proved there that in this case there are effective finiteness results for x ∈ Z and y ∈ Q. Note that our results in this paper do not give effective results as in [2]. In [4], we have some results for general g.…”

“…One says that an equation f (x) = g(y) has infinitely many rational solutions with bounded denominators if there exists a positive integer λ such that f (x) = g(y) has infinitely many rational solutions x, y satisfying x, y ∈ 1 λ Z. Equations of the type f (x) = g(y) for f (x) = x(x+1) · · · (x+m−1) and various polynomials g(y) have been studied extensively during the last decade. Also the special case when g(y) = y n − r, where r is any rational number, was studied earlier in [2]. We have proved there that in this case there are effective finiteness results for x ∈ Z and y ∈ Q.…”

Recent developments in the preparation of nano-gold composite coatings

“…At a current density higher than 8 A/dm², the whole sample surface is bright. This effect of pattern formation is a frequently discussed phenomenon in the electroplating technology [3][4][5][6][7][8][9][10][11][12][13][14]. Several studies have been done for better understanding of pattern formation of inhomogeneous interfacial potential in electrodissolution reactions [15], electrocatalytic oxidation reactions and electrocatalysis [16].…”

“…The metal ions are then reduced by the addition of a reducing agent, e.g. NaBH 4 . For the synthesis of gold nanoparticles, the addition of a reducing agent is not necessary since an auto-reduction takes place.…”