Skip to main content Accessibility help
×
Home
Hostname: page-component-846f6c7c4f-jk8t6 Total loading time: 0.22 Render date: 2022-07-07T10:54:12.155Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION

Published online by Cambridge University Press:  20 February 2006

M. A. BENNETT
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2 Canadabennett@math.ubc.ca
N. BRUIN
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6 Canadanbruin@sfu.ca
K. GYÖRY
Affiliation:
Number Theory Research Group of the Hungarian Academy of Sciences, Institute of Mathematics, University of Debrecen, P.O. Box 12, 4010 Debrecen, Hungary, gyory@math.klte.hu, hajdul@math.klte.hu
L. HAJDU
Affiliation:
Number Theory Research Group of the Hungarian Academy of Sciences, Institute of Mathematics, University of Debrecen, P.O. Box 12, 4010 Debrecen, Hungary, gyory@math.klte.hu, hajdul@math.klte.hu
Get access

Abstract

We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

Keywords

Type
Research Article
Copyright
2006 London Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported in part by grants from NSERC (M.A.B. and N.B.), the Erwin Schrödinger Institute in Vienna (M.A.B. and K.G.), the Netherlands Organization for Scientific Research (NWO) (K.G. and L.H.), the Hungarian Academy of Sciences (K.G. and L.H.), by FKFP grant 3272-13/066/2001 (L.H.) and by grants T29330, T42985 (K.G. and L.H.), T38225 (K.G.) and F34981 (L.H.) of the Hungarian National Foundation for Scientific Research.
23
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *