On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

The Bernoulli polynomials B n (x) are defined by the generating series The Bernoulli polynomials B n (x) are related to the sums of nth powers of natural numbers as follows. For any n ≥ 1, the sum 1 n + 2 n + · · · + k n is a polynomial function S n (k) of k andn + 1 .In this paper, for nonzero rational numbers a, b and rational polynomials C(y), we study the Diophantine equation aB m (x) = bB n (y) + C(y)

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“…Note that our results in this paper do not give effective results as in [2]. In [4], we have some results for general g.…”

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confidence: 55%

Diophantine equations with Bernoulli polynomials

Kulkarni

Sury

2005

Acta Arith.

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“…These equations have only been investigated in the literature in the case (a, b) = (1, 0). Rakaczki [20] and, independently, Kulkarni and Sury [19] characterized those pairs ( k, g(y) ) for which equation (13) has infinitely many integer solutions with (a, b) = (1, 0). Recently, Rakaczki and Kreso [22] proved an analogous result for equations (14) and (15).…”

Section: Introduction and New Resultsmentioning

confidence: 99%

“…Our goal is to extend the results of [19,20,22] to the general equations (13)- (15). To do this it will be useful to survey what is known about the decomposition properties of the polynomials involved in the equations under consideration.…”

Section: Introduction and New Resultsmentioning

confidence: 99%

Polynomial values of (alternating) power sums

Bazsó

2015

Acta Math. Hungar.

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“…Kulkarni and Sury [20] proved Corollary 5.2. If f is the polynomial on the left hand side of (5.3), then f (x) = f (x) + x n /n!, and f thus has only simple critical points.…”

Section: Corollary 52mentioning

confidence: 87%

“…To see that f has all distinct critical values it suffices to show that no two roots of f are such that their quotient is an n-th root of unity. It is shown in [20] that this holds by using the fact that the Galois groups of f and f are either symmetric or alternating, which is a well-known result of Schur. Note that Theorem 1.1 applies to equations of type f (x) = g(y), where f and g are any of the above mentioned polynomials.…”

Section: Corollary 52mentioning

confidence: 90%

Diophantine equations in separated variables

Kreso

Tichy

2017

Period Math Hung

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We study Diophantine equations of type f (x) = g(y), where both f and g have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials ( f n ) n are such that f n satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K of characteristic zero, and at least two distinct critical points and all distinct critical values, then the monodromy group of f is a doubly transitive permutation group. In particular, f cannot be represented as a composition of lower degree polynomials. Several authors have studied monodromy groups of polynomials with some similar properties. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if f (x) = g(h(x)) with g, h ∈ K [x] and deg g > 1, then either deg h ≤ 2, or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points.

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On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Cited by 21 publications

References 7 publications

Diophantine equations with Bernoulli polynomials

Diophantine equations with Bernoulli polynomials

Polynomial values of (alternating) power sums

Diophantine equations in separated variables

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