Pure powers in recurrence sequences and some related diophantine equations

We prove that there are only finitely many terms of a non-degenerate linear recurrence sequence which are qth powers of an integer subject to certain simple conditions on the roots of the associated characteristic polynomial of the recurrence sequence. Further we show by similar arguments that the D...

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Bibliographic Details
Published inJournal of number theory Vol. 27; no. 3; pp. 324 - 352
Main Authors Shorey, T.N., Stewart, C.L.
Format Journal Article
LanguageEnglish
Published New York, NY Elsevier Inc 01.11.1987
Academic Press
Subjects
Difference and functional equations, recurrence relations
Exact sciences and technology
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Sciences and techniques of general use
Recurrence
Existence theorem
Number theory
Diophantine equation
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Summary:We prove that there are only finitely many terms of a non-degenerate linear recurrence sequence which are qth powers of an integer subject to certain simple conditions on the roots of the associated characteristic polynomial of the recurrence sequence. Further we show by similar arguments that the Diophantine equation ax 2 t + bx t y + cy 2 + dx t + ey + f = 0 has only finitely many solutions in integers x, y, and t subject to the appropriate restrictions, and we also treat some related simultaneous Diophantine equations.
ISSN:0022-314X
1096-1658
DOI:10.1016/0022-314X(87)90071-0