Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Quan-Hui Yang; Qing-Qing Zhao

doi:10.1016/j.crma.2017.11.015

Number theory

Powerful numbers in (1^ℓ + q^ℓ)(2^ℓ + q^ℓ)⋯(n^ℓ + q^ℓ)

Presented by: the Editorial Board

Quan-Hui Yang ¹; Qing-Qing Zhao ²

¹ School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
² Jincheng College, Nanjing University of Aeronautics and Astronautics, Nanjing 211156, China

Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 13-16.

Abstract
Résumé

Let q be a positive integer. Recently, Niu and Liu proved that, if $n \geq \max {q, 1198 - q}$ , then the product $(1^{3} + q^{3}) (2^{3} + q^{3}) \dots (n^{3} + q^{3})$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ℓ and $n \geq \max {q, 11 - q}$ , the product $(1^{ℓ} + q^{ℓ}) (2^{ℓ} + q^{ℓ}) \dots (n^{ℓ} + q^{ℓ})$ is not a powerful number, and (2) that, for any positive odd integer ℓ, there exists an integer $N_{q, ℓ}$ such that, for any positive integer $n \geq N_{q, ℓ}$ , the product $(1^{ℓ} + q^{ℓ}) (2^{ℓ} + q^{ℓ}) \dots (n^{ℓ} + q^{ℓ})$ is not a powerful number.

Soit q un entier positif. Récemment, Niu et Liu ont montré que, si $n \geq \max (q, 1198 - q)$ , alors le produit $(1^{3} + q^{3}) (2^{3} + q^{3}) \dots (n^{3} + q^{3})$ n'est pas un nombre puissant. Dans cette Note, nous montrons : (1) que le produit $(1^{ℓ} + q^{ℓ}) (2^{ℓ} + q^{ℓ}) \dots (n^{ℓ} + q^{ℓ})$ n'est pas un nombre puissant pour toute puissance ℓ d'un nombre premier impair et $n \geq \max (q, 11 - q)$ ; (2) que, pour tout nombre impair positif ℓ, il existe un entier $N_{q, ℓ}$ tel que pour tout entier $n \geq N_{q, ℓ}$ , le produit $(1^{ℓ} + q^{ℓ}) (2^{ℓ} + q^{ℓ}) \dots (n^{ℓ} + q^{ℓ})$ ne soit pas un nombre puissant.

Received: 2017-06-14
Accepted: 2017-11-21
Published online: 2017-12-06

DOI: 10.1016/j.crma.2017.11.015

Author's affiliations:

Quan-Hui Yang ¹; Qing-Qing Zhao ²

¹ School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
² Jincheng College, Nanjing University of Aeronautics and Astronautics, Nanjing 211156, China

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     author = {Quan-Hui Yang and Qing-Qing Zhao},
     title = {Powerful numbers in (1\protect\textsuperscript{\protect\emph{\ensuremath{\ell}}}\,+\,\protect\emph{q}\protect\textsuperscript{\protect\emph{\ensuremath{\ell}}})(2\protect\textsuperscript{\protect\emph{\ensuremath{\ell}}}\,+\,\protect\emph{q}\protect\textsuperscript{\protect\emph{\ensuremath{\ell}}})\ensuremath{\cdots}(\protect\emph{n}\protect\textsuperscript{\protect\emph{\ensuremath{\ell}}}\,+\,\protect\emph{q}\protect\textsuperscript{\protect\emph{\ensuremath{\ell}}})},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {13--16},
     publisher = {Elsevier},
     volume = {356},
     number = {1},
     year = {2018},
     doi = {10.1016/j.crma.2017.11.015},
     language = {en},
}

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Quan-Hui Yang; Qing-Qing Zhao. Powerful numbers in (1^ℓ + q^ℓ)(2^ℓ + q^ℓ)⋯(n^ℓ + q^ℓ). Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 13-16. doi : 10.1016/j.crma.2017.11.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.11.015/

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