On some generalizations of the diophantine equation $1^k + 2^k + ... + x^k = y^z$

Brindza, Β.

doi:10.4064/aa-44-2-99-107

Cited by 23 publications

(34 citation statements)

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“…The main result of this work was a proof that the only positive integers (k ≥ 1, q > 1) for which this equation has infinitely many solutions are (k, q) ∈ {(1, 2), (3,2), (3,4), (5, 2)}.…”

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confidence: 99%

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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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confidence: 99%

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

Jacobson

Walsh

2003

Math. Comp.

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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].…”

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confidence: 99%

Polynomial values of (alternating) power sums

Bazsó

2015

Acta Math. Hungar.

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“…Techniques from algebraic number theory and Diophantine approximation have allowed the resolution of such equations with small exponents, as well as proofs of general theorems. This can be seen in the work of Brindza [5], Cassels [6], Győry, Tijdeman and Voorhoeve [8], Hajdu [9], Pintér [11], Schäffer [13], and Zhang and Bai [15] among many others.…”

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confidence: 96%