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)
[Nombres de la forme (1^ℓ + q^ℓ)(2^ℓ + q^ℓ)…(n^ℓ + q^ℓ) qui ne sont pas puissants]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2017-06-14
Accepté le : 2017-11-21
Publié le : 2017-12-06

DOI : 10.1016/j.crma.2017.11.015

Affiliations des auteurs :

Quan-Hui Yang ¹ ; Qing-Qing Zhao ²

@article{CRMATH_2018__356_1_13_0,
     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},
}

TY  - JOUR
AU  - Quan-Hui Yang
AU  - Qing-Qing Zhao
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
DO  - 10.1016/j.crma.2017.11.015
LA  - en
ID  - CRMATH_2018__356_1_13_0
ER  -

%0 Journal Article
%A Quan-Hui Yang
%A Qing-Qing Zhao
%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
%R 10.1016/j.crma.2017.11.015
%G en
%F CRMATH_2018__356_1_13_0

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/

Bibliographie
Cité par

[1] T. Amdeberhan; L.A. Medina; V.H. Moll Arithmetical properties of a sequence arising from an arctangent sum, J. Number Theory, Volume 128 (2008), pp. 1807-1846

[2] Y.-G. Chen; M.-L. Gong On the products $(1^{ℓ} + 1) (2^{ℓ} + 1) \dots (n^{ℓ} + 1)$ II, J. Number Theory, Volume 144 (2014), pp. 176-187

[3] Y.-G. Chen; M.-L. Gong; X.-Z. Ren On the products $(1^{ℓ} + 1) (2^{ℓ} + 1) \dots (n^{ℓ} + 1)$ , J. Number Theory, Volume 133 (2013), pp. 2470-2474

[4] J. Cilleruelo Squares in $(1^{2} + 1) (2^{2} + 1) \dots (n^{2} + 1)$ , J. Number Theory, Volume 128 (2008), pp. 2488-2491

[5] J. Cilleruelo; F. Luca; A. Quirós; I.E. Shparlinski On squares in polynomial products, Monatshefte Math., Volume 159 (2010), pp. 215-223

[6] S. Cuellar; J.A. Samper A nice and tricky lemma (lifting the exponent), Math. Reflec., Volume 2007 (2007) no. 3

[7] J.-H. Fang Neither $\prod_{k = 1}^{n} (4 k^{2} + 1)$ nor $\prod_{k = 1}^{n} (2 k (k - 1) + 1)$ is a perfect square, Integers, Volume 9 (2009), pp. 177-180

[8] S.W. Golomb Powerful numbers, Amer. Math. Mon., Volume 77 (1970), pp. 848-852

[9] E. Gürel On the occurrence of perfect squares among values of certain polynomial products, Amer. Math. Mon., Volume 123 (2016), pp. 597-599

[10] E. Gürel A note on the products $({(m + 1)}^{2} + 1) \dots (n^{2} + 1)$ and $({(m + 1)}^{3} + 1) \dots (n^{3} + 1)$ , Math. Commun., Volume 21 (2016), pp. 109-114

[11] E. Gürel; A.U.O. Kisisel A note on the products $(1^{u} + 1) (2^{u} + 1) \dots (n^{u} + 1)$ , J. Number Theory, Volume 130 (2010), pp. 187-191

[12] S.-F. Hong; X. Liu Squares in $(2^{2} - 1) \dots (n^{2} - 1)$ and p-adic valuation, Asian-Eur. J. Math., Volume 3 (2010), pp. 329-333

[13] C.-Z. Niu; W.-X. Liu On the products $(1^{3} + q^{3}) (2^{3} + q^{3}) \dots (n^{3} + q^{3})$ , J. Number Theory, Volume 180 (2017), pp. 403-409

[14] J. Sándor; D.S. Mitrinović; B. Crstici Handbook of Number Theory I, Springer, The Netherlands, 2006

[15] P. Spiegelhalter; J. Vandehey Squares in polynomial product sequences | arXiv

[16] S.-C. Yang; A. Togbé; B. He Diophantine equations with products of consecutive values of a quadratic polynomial, J. Number Theory, Volume 131 (2011), pp. 1840-1851

[17] Z.-F. Zhang Powers in $\prod_{k = 1}^{n} (a k^{2^{l} \cdot 3^{m}} + b)$ , Funct. Approx. Comment. Math., Volume 46 (2012), pp. 7-13

[18] W.-P. Zhang; T.-T. Wang Powerful numbers in $(1^{k} + 1) (2^{k} + 1) \dots (n^{k} + 1)$ , J. Number Theory, Volume 132 (2012), pp. 2630-2635

Cité par Sources :

Commentaires - Politique