On the Diophantine equation : Higher-order recurrences

Authors:
Clemens Fuchs, Attila Petho and Robert F. Tichy

Journal:
Trans. Amer. Math. Soc. **355** (2003), 4657-4681

MSC (2000):
Primary 11D45; Secondary 11D04, 11D61, 11B37

DOI:
https://doi.org/10.1090/S0002-9947-03-03325-7

Published electronically:
June 10, 2003

MathSciNet review:
1990766

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a field of characteristic and let be a linear recurring sequence of degree in defined by the initial terms and by the difference equation

with . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation

has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

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Additional Information

**Clemens Fuchs**

Affiliation:
Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria

Email:
clemens.fuchs@tugraz.at

**Attila Petho**

Affiliation:
Institute for Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, PO Box 12, Hungary

Email:
pethoe@math.klte.hu

**Robert F. Tichy**

Affiliation:
Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria

Email:
tichy@tugraz.at

DOI:
https://doi.org/10.1090/S0002-9947-03-03325-7

Keywords:
Diophantine equations,
linear recurring sequences,
$S$-unit equations

Received by editor(s):
October 18, 2002

Received by editor(s) in revised form:
February 7, 2003

Published electronically:
June 10, 2003

Additional Notes:
This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT

The second author was supported by the Hungarian National Foundation for Scientific Research, Grant Nos. 29330 and 38225

Dedicated:
Dedicated to Wolfgang M. Schmidt on the occasion of his 70th birthday.

Article copyright:
© Copyright 2003
American Mathematical Society