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Abstract

Abstract. Given m; n X 2, we prove that, for suf¢ciently large y, the sum 1 n þ Á Á Á þ y n is not a product of m consecutive integers. We also prove that for m 6 ¼ n we have 1 m þ Á Á Á þ x m 6 ¼ 1 n þ Á Á Á þ y n , provided x; y are suf¢ciently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are 'almost' indecomposable, a result of independent interest.

Diophantine equations with Appell sequences

“…where P n/2 (x) is a polynomial of degree n/2 which is indecomposable for every n. These results from [6] and [11] suggest the following notion. We say that an Appell sequence {P n (x)} n≥0 is of special type if P n (x) is indecomposable over C for all odd n, and, for even n, every nontrivial decomposition of P n (x) is equivalent to a decomposition of the form…”

“…We note that Proposition 4 is a common generalization of Lemma 5.3 in [6], Lemma 2.4 in [3], and of the second statement of Lemma 12 in [11].…”

“…The decomposition of Bernoulli polynomials has been described by Bilu et al in [6]. Decomposition properties of Euler polynomials were recently investigated by Rakaczki and Kreso [11].…”

On the decomposability of linear combinations of Euler polynomials

“…For further results for the decomposability of an infinite family of polynomials we refer to [3], [6] and [5].…”

On the Indecomposability of Polynomials*

“…The criterium of BILU and TICHY has been already applied to several Diophantine equations of the form f n ðxÞ ¼ g m ðyÞ, where ð f n Þ and ðg n Þ are sequences of classical polynomials (see [1,2,5,7,[10][11][12]). In these results, the indecomposability of corresponding polynomials was usually proved using some analytical properties of these polynomials.…”