Number theory
Powerful numbers in (1 + q)(2 + q)⋯(n + q)
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 13-16.

Let q be a positive integer. Recently, Niu and Liu proved that, if nmax{q,1198q}, then the product (13+q3)(23+q3)(n3+q3) is not a powerful number. In this note, we prove (1) that, for any odd prime power and nmax{q,11q}, the product (1+q)(2+q)(n+q) is not a powerful number, and (2) that, for any positive odd integer , there exists an integer Nq, such that, for any positive integer nNq,, the product (1+q)(2+q)(n+q) is not a powerful number.

Soit q un entier positif. Récemment, Niu et Liu ont montré que, si nmax(q,1198q), alors le produit (13+q3)(23+q3)(n3+q3) n'est pas un nombre puissant. Dans cette Note, nous montrons : (1) que le produit (1+q)(2+q)(n+q) n'est pas un nombre puissant pour toute puissance d'un nombre premier impair et nmax(q,11q); (2) que, pour tout nombre impair positif , il existe un entier Nq, tel que pour tout entier nNq,, le produit (1+q)(2+q)(n+q) ne soit pas un nombre puissant.

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DOI: 10.1016/j.crma.2017.11.015
Yang, Quan-Hui 1; Zhao, Qing-Qing 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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Yang, Quan-Hui; Zhao, Qing-Qing. Powerful numbers in (1 + q)(2 + q)⋯(n + q). Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 13-16. doi : 10.1016/j.crma.2017.11.015. http://www.numdam.org/articles/10.1016/j.crma.2017.11.015/

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