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Prime and composite numbers as integer parts of powers. (English) Zbl 1102.11004
Summary: 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 \(\xi>1\) for which the numbers \([\xi^{n!}]\), \(n =2,3, \dots\), 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 \(\zeta\) (obtained as the limit of a certain recurrent sequence) for which the sequence \([\zeta^n]\), \(n =1,2,\dots\), has infinitely many elements in an arbitrary integer arithmetical progression.

MSC:
11A41 Primes
11J81 Transcendence (general theory)
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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