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Arithmetical Properties of Powers of Algebraic Numbers
Author(s) -
Dubickas Artūras
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305017728
Subject(s) - arithmetic function , mathematics , algebraic number , algebra over a field , arithmetic , pure mathematics , discrete mathematics , mathematical analysis
We consider the sequences of fractional parts {ξα n }, n = 1, 2, 3,…, and of integer parts [ξα n ], 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.