Prime and composite numbers as integer parts of powers

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• 作者：Giedrius Alkauskas and Arturas Dubickas
• 关键词：Pisot numbers ; transcendental numbers ; prime and composite numbers
• 刊名：Acta Mathematica Hungarica
• 出版年：2004
• 出版时间：September 2004
• 年：2004
• 卷：105
• 期：3
• 页码：249-256
• 全文大小：231 KB

文摘

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξn !], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζn], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression.