Pure powers in recurrence sequences and some related diophantine equations

Abstract

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^2t + bx^ty + cy² + 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.