ARITHMETICAL PROPERTIES OF POWERS OF ALGEBRAIC NUMBERS

Abstract

We consider the sequences of fractional parts {ξαn}, n = 1, 2, 3,…, and of integer parts [ξαn], n = 1, 2, 3,…, where ξ is an arbitrary positive number and α > 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 α 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 [ ξ ( ( 13 − 1 ) / 2 ) n ] , n = 1, 2, 3,…. 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. 2000 Mathematics Subject Classification 11J71, 11R04, 11R06, 11A41.