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

Shorey, T. N.; Stewart, C. L.

J. Number Theory 27, 324-352 (1987).

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

Keywords:

perfect powers in sequences; linear recurrence sequence

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