Arithmetical Properties of Powers of Algebraic Numbers
Arithmetical Properties of Powers of Algebraic Numbers
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代数数幂的算术性质
DOI：
10.1112/s0024609305017728
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发表时间：
2006
影响因子：
0.9
通讯作者：
A. Dubickas
中科院分区：
文献类型：
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作者：
A. Dubickas
我们考虑分数部分序列{<$αn}，n = 1，2，3，.，和整数部分序列[<$αn]，n = 1，2，3，.，其中<$是任意正数，α > 1是代数数。得到了第一序列最大极限点与最小极限点之差的一个不等式。这样的不等式以前只在有理数α下才知道。它还表明，根的一些不可约三项式的序列的整数部分包含无限多个数字可除的2或3。例如，对于[n（（13−1）/2）n]，n = 1，2，3，.，这是可以证明的。事实上，在幂的整数部分的序列中有无穷多个合数，这一点在前面已经为皮索数、塞勒姆数和三个有理数3/2、4/3、5/4证明了，但是没有这样的代数数在单位圆之外有几个共轭。2000年数学学科分类11 J71、11 R 04、11 R 06、11 A41。
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.