×

Pure powers in recurrence sequences and some related diophantine equations. (English) Zbl 0624.10009

Authors’ summary: “We prove that there are only finitely many terms of a nondegenerate linear recurrence sequence which are q-th 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 \(ax^{2t}+bx^ ty+cy^ 2+dx^ t+ey+f=0\) 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.”
Reviewer: P.Kiss

MSC:

11B37 Recurrences
11D04 Linear Diophantine equations
11D41 Higher degree equations; Fermat’s equation
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Baker, A., Bounds for the solutions of the hyperelliptic equation, (), 439-444 · Zbl 0174.33803
[2] Baker, A., A sharpening of the bounds for linear forms in logarithms II, Acta arith., 24, 33-36, (1973) · Zbl 0261.10025
[3] Baker, A., ()
[4] Brindza, B., On S-integral solutions of the equation ym = f(x), Acta math. hungar., 44, 133-139, (1984) · Zbl 0552.10009
[5] {\scP. Kiss}, Differences of the terms of linear recurrences, Studia Sci. Math. Hungar., to appear. · Zbl 0628.10008
[6] Lang, S., ()
[7] Lewis, D.J., Diophantine equations: p-adic methods, () · Zbl 0218.10035
[8] Mahler, K., Eine arithmetische eigenschaft der Taylor-koeffizienten rationaler funktionen, (), 50-60 · JFM 61.0176.02
[9] Mordell, L.J., ()
[10] Nemes, I.; Pethö, A., Polynomial values in linear recurrences, Publ. math., 31, 229-233, (1984) · Zbl 0557.10010
[11] {\scI. Nemes and A. Pethö}, Polynomial values in linear recurrences II, J. Number Theory, to appear.
[12] Pethö, A., Perfect powers in second order linear recurrences, J. number theory, 15, 5-13, (1982) · Zbl 0488.10009
[13] Van der Poorten, A.J., ()
[14] Schmidt, W.M., Diophantine approximation, () · Zbl 0529.10032
[15] Shorey, T.N.; Stewart, C.L., On the Diophantine equation ax2t + bxty + cy2 = d and pure powers in recurrence sequences, Math. scand., 52, 24-36, (1983) · Zbl 0491.10016
[16] Stewart, C.L., On some Diophantine equations and related linear recurrence sequences, () · Zbl 0491.10015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.