On the denominators of equivalent algebraic numbers

This article does not have an abstract

Pure powers in recurrence sequences and some related diophantine equations

AbstractWe 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 ax2t + bxty + cy2 + dxt + 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

Some Methods Of Erdős Applied To Finite Arithmetic Progressions

Since 1934 Erdős has introduced various methods to derive arithmetic properties of blocks of consecutive integers. This research culminated in 1975 when Erdős and Selfridge [ES] established the old conjecture that the product of two or more consecutive positive integers is never a perfect power. It is very likely that the product of the terms of a finite arithmetic progression of length at least four is never a perfect power. In the present paper it is shown how Erdős' methods have been extended to obtain results for arithmetic progressions