The Library
On power values of pyramidal numbers, II
Tools
Dujella, Andrej , Győry, Kálmán, Michaud-Jacobs, Philippe and Pintér, Ákos (2023) On power values of pyramidal numbers, II. Acta Arithmetica, 208 . pp. 199-213. doi:10.4064/aa211213-27-7 ISSN 0065-1036.
|
PDF
WRAP-power-values-pyramidal-numbers-II-Michaud-Jacobs-2023.pdf - Accepted Version - Requires a PDF viewer. Available under License Creative Commons Attribution 4.0. Download (531Kb) | Preview |
Official URL: https://doi.org/10.4064/aa211213-27-7
Abstract
form ≥ 3, we define themth order pyramidal number by
p r _m( x ) =16x ( x + 1 ) ( ( m − 2 ) x + 5 − m ) .
In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equationp r _m( x ) =y2are found in positive integersxandy, for6 ≤ m ≤ 100. In this paper, we consider the question of higher powers, and find all solutions to the equationp r _m( x ) =ynin positive integersx,y, andn, withn ≥ 3, and5 ≤ m ≤ 50. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.
Item Type: | Journal Article | ||||||
---|---|---|---|---|---|---|---|
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Journal or Publication Title: | Acta Arithmetica | ||||||
Publisher: | Polska Akademia Nauk: Instytut Matematyczny | ||||||
ISSN: | 0065-1036 | ||||||
Official Date: | 23 August 2023 | ||||||
Dates: |
|
||||||
Volume: | 208 | ||||||
Page Range: | pp. 199-213 | ||||||
DOI: | 10.4064/aa211213-27-7 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Free Access (unspecified licence, 'bronze OA') | ||||||
Copyright Holders: | © Instytut Matematyczny PAN, 2023 | ||||||
Date of first compliant deposit: | 2 August 2023 | ||||||
Date of first compliant Open Access: | 24 August 2023 | ||||||
Related URLs: |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year