ARITHMETICAL PROPERTIES OF POWERS OF ALGEBRAIC NUMBERS

Abstract

We consider the sequences of fractional parts $\{\xi \alpha^n\}, n \,{=}\,1,2,3,\dots,$ and of integer parts $[\xi \alpha^n], n \,{=}\,1,2,3,\dots,$ where $\xi$ is an arbitrary positive number and $\alpha\,{>}\,1$ is an algebraic number. We obtain an inequality for the difference between the largest and the smallest limit points of the first sequence. Such an inequality was earlier known for rational $\alpha$ only. It is also shown that for roots of some irreducible trinomials the sequence of integer parts contains infinitely many numbers divisible by either 2 or 3. This is proved, for instance, for $[\xi((\sqrt {13}-1)/2)^n], n\,{=}\,1,2,3,\dots.$ The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known.