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Terms of Lucas sequences having a large smooth divisor

Part of: Sequences and sets

Published online by Cambridge University Press:  25 March 2022

Nikhil Balaji  and
Florian Luca Open the ORCID record for Florian Luca [Opens in a new window]
Nikhil Balaji*
Affiliation:
Department of Computer Science and Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India
Florian Luca
Affiliation:
School of Maths Wits University, 1 Jan Smuts, Braamfontein, Johannesburg 2000, South Africa Research Group in Algebric Structures and Applications, King Abdulaziz University, Abdulah Sulayman, Jeddah 22254, Saudi Arabia Centro de Ciencias Matemáticas UNAM, Morelia, Mexico e-mail: florian.luca@wits.ac.za
*
e-mail: nbalaji@cse.iitd.ac.in
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Abstract

We show that the $Kn$ –smooth part of $a^n-1$ for an integer $a>1$ is $a^{o(n)}$ for most positive integers n.

Keywords

Linearly recurrent sequencesLucas sequencesABC conjecture

MSC classification

Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
Secondary: 11B37: Recurrences 11B65: Binomial coefficients; factorials; $q$-identities
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 1 , March 2023 , pp. 225 - 231
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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