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Artūras Dubickas 《Proceedings Mathematical Sciences》2005,115(4):391-397
Suppose that α > 1 is an algebraic number and ξ > 0 is a real number. We prove that the sequence of fractional partsξα
n
, n = 1, 2, 3, …, has infinitely many limit points except when α is a PV-number and ξ ∈ ℚ(α). For ξ = 1 and α being a rational
non-integer number, this result was proved by Vijayaraghavan. 相似文献
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Arithmetical Properties of Powers of Algebraic Numbers 总被引:2,自引:0,他引:2
We consider the sequences of fractional parts {n}, n = 1, 2,3,..., and of integer parts [n], n = 1, 2, 3,..., where isan arbitrary positive number and > 1 is an algebraic number.We obtain an inequality for the difference between the largestand the smallest limit points of the first sequence. Such aninequality was earlier known for rational only. It is alsoshown that for roots of some irreducible trinomials the sequenceof integer parts contains infinitely many numbers divisibleby either 2 or 3. This is proved, for instance, for , n = 1, 2, 3,.... The fact that thereare infinitely many composite numbers in the sequence of integerparts of powers was proved earlier for Pisot numbers, Salemnumbers and the three rational numbers 3/2, 4/3, 5/4, but nosuch algebraic number having several conjugates outside theunit circle was known. 2000 Mathematics Subject Classification11J71, 11R04, 11R06, 11A41. 相似文献
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A. Dubickas 《Lithuanian Mathematical Journal》1995,35(4):328-332
We give a lower bound for |α−1|, where α is an algebraic number, and also an upper bound for the number of real zeros of a
polynomial. A lower bound for the maximal modulus of conjugates of a totally real algebraic integer is also obtained.
Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 4, pp.
421–431, October–December, 1995. 相似文献
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Artūras Dubickas 《Czechoslovak Mathematical Journal》2006,56(3):949-956
We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic
numbers lying in E. This extends the result of Schinzel who proved the same statement for every real quadratic field E. A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure. 相似文献
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Arturas Dubickas 《Siberian Mathematical Journal》2006,47(5):879-882
Let ξ ≠ = 0 and α > 1 be reals. We prove that the fractional parts {ξ αn}, n = 1, 2, 3, ..., take every value only finitely many times except for the case when α is the root of an integer: α = q 1/d, where q ≥ 2 and d ≥ 1 are integers and ξ is a rational factor of a nonnegative integer power of α. 相似文献