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Research Article | Open Access

On the number of integers which form perfect powers in the way of $x (y_{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2}) = z^{k}$

Tingting Wen(

)

School of Mathematics, Shandong University, Jinan 250100, China

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Abstract

Let $k ⩾ 2$ be an integer. We studied the number of integers which form perfect $k$ -th powers in the way of

$x (y_{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2}) = z^{k} .$

For $k ⩾ 4$ , we established a unified asymptotic formula with a power-saving error term for the number of such integers of bounded size under Lindelöf hypothesis, and we also gave an unconditional result for $k = 2$ .

Keywords

counting solutions of Diophantine equations perfect powers double Dirichlet series

CLC number: 11D45, 11N37

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AIMS Mathematics

Volume 9 Issue 4,
April 2024

Pages 8732-8748

DOI: 10.3934/math.2024423

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Cite this article:

Wen T. On the number of integers which form perfect powers in the way of

x (y_{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2}) = z^{k}

. AIMS Mathematics, 2024, 9(4): 8732-8748. https://doi.org/10.3934/math.2024423

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Received: 11 January 2024

Revised: 22 February 2024

Accepted: 26 February 2024

Published: 15 April 2024

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)

On the number of integers which form perfect powers in the way of x ( y 1 2 + y 2 2 + y 3 2 + y 4 2 ) = z k

Abstract

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On the number of integers which form perfect powers in the way of $x (y_{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2}) = z^{k}$