On the equation $f(1)1^k + f(2)2^k + ... + f(x)x^k + R(x)= by^z$

Urbanowicz, Jerzy

doi:10.4064/aa-51-4-349-368

Cited by 16 publications

(17 citation statements)

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“…Equations of the shape (2) S k (x) − S k (y) = z n have been considered by a number of authors, under the hypotheses that y = 0 (see e.g. [7], [15], [16], [20], [21], [23], [30], [31]), that y = [x/2] ( [34]) and that y = x − 3 ( [10], [34]). In the first two of these situations, the resulting polynomials on the left hand side of equation (2) have at least two distinct linear factors over Q, which allows the problem to be reduced to one of binomial Thue equations.…”

Section: Introductionmentioning

confidence: 99%

Superelliptic equations arising from sums of consecutive powers

2016

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Using only elementary arguments, Cassels solved the Diophantine equation (x − 1) 3 + x 3 + (x + 1) 3 = z 2 (with x, z ∈ Z). The generalization (x − 1) k + x k + (x + 1) k = z n (with x, z, n ∈ Z and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ {2, 3, 4} using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solution for k = 5 is x = z = 0, and that there are no solutions for k = 6. The chief innovation we employ is a computational one, which enables us to avoid the full computation of data about cuspidal newforms of high level.

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Section: Introductionmentioning

confidence: 99%

Superelliptic equations arising from sums of consecutive powers

2016

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“…In recent years there have been numerous papers on this topic (see [2], [3], [7], [10], [22]). The interested reader may wish to refer to the notes at the end of chapter 10 in [21].…”

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confidence: 99%

A computational approach for solving $y^2=1^k+2^k+\dotsb+x^k$

Jacobson

Walsh

2003

Math. Comp.

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Abstract. We present a computational approach for finding all integral solutions of the equation y 2 = 1 k + 2 k + · · · + x k for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for 2 ≤ k ≤ 70 assuming the Generalized Riemann Hypothesis, and for 2 ≤ k ≤ 58 unconditionally.

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“…Several generalizations of (4) have been considered, e.g. in the papers of Voorhoeve, Győry and Tijdeman [28], Brindza [10], Dilcher [11] and Urbanowicz [25][26][27]. Schäffer's conjecture has been confirmed only in a few cases: for n = 2 and k 58 by Jacobson, Pintér and Walsh [15]; and for n 2 and k 11 by Bennett, Győry and Pintér [5].…”

Section: Introduction and New Resultsmentioning

confidence: 99%