Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Another $ S $-unit variant of Diophantine tuples


Authors: Clemens Fuchs and Sebastian Heintze
Journal: Proc. Amer. Math. Soc. 149 (2021), 27-35
MSC (2010): Primary 11D61
DOI: https://doi.org/10.1090/proc/15193
Published electronically: October 16, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that there are only finitely many triples of integers $ 0 < a < b < c $ such that the product of any two of them is the value of a given polynomial with integer coefficients evaluated at an $ S $-unit that is also a positive integer. The proof is based on a result of Corvaja and Zannier and thus is ultimately a consequence of the Schmidt subspace theorem.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11D61

Retrieve articles in all journals with MSC (2010): 11D61


Additional Information

Clemens Fuchs
Affiliation: Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, Austria
Email: clemens.fuchs@sbg.ac.at

Sebastian Heintze
Affiliation: Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, Austria
Email: sebastian.heintze@sbg.ac.at

DOI: https://doi.org/10.1090/proc/15193
Keywords: Diophantine tuples, $S$-units, gcd in number fields
Received by editor(s): October 21, 2019
Received by editor(s) in revised form: April 20, 2020
Published electronically: October 16, 2020
Additional Notes: This research was supported by Austrian Science Fund (FWF): I4406.
Communicated by: Rachel Pries
Article copyright: © Copyright 2020 American Mathematical Society