Soit q un entier positif. Récemment, Niu et Liu ont montré que, si , alors le produit n'est pas un nombre puissant. Dans cette Note, nous montrons : (1) que le produit n'est pas un nombre puissant pour toute puissance ℓ d'un nombre premier impair et ; (2) que, pour tout nombre impair positif ℓ, il existe un entier tel que pour tout entier , le produit ne soit pas un nombre puissant.
Let q be a positive integer. Recently, Niu and Liu proved that, if , then the product is not a powerful number. In this note, we prove (1) that, for any odd prime power ℓ and , the product is not a powerful number, and (2) that, for any positive odd integer ℓ, there exists an integer such that, for any positive integer , the product is not a powerful number.
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@article{CRMATH_2018__356_1_13_0, author = {Yang, Quan-Hui and Zhao, Qing-Qing}, 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}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2017.11.015/} }
TY - JOUR AU - Yang, Quan-Hui AU - Zhao, Qing-Qing TI - Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ) JO - Comptes Rendus. Mathématique PY - 2018 SP - 13 EP - 16 VL - 356 IS - 1 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2017.11.015/ DO - 10.1016/j.crma.2017.11.015 LA - en ID - CRMATH_2018__356_1_13_0 ER -
%0 Journal Article %A Yang, Quan-Hui %A Zhao, Qing-Qing %T Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ) %J Comptes Rendus. Mathématique %D 2018 %P 13-16 %V 356 %N 1 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2017.11.015/ %R 10.1016/j.crma.2017.11.015 %G en %F CRMATH_2018__356_1_13_0
Yang, Quan-Hui; Zhao, Qing-Qing. Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ). Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 13-16. doi : 10.1016/j.crma.2017.11.015. http://archive.numdam.org/articles/10.1016/j.crma.2017.11.015/
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