Integer parts of powers of Pisot and Salem numbers
Let \(\alpha\) be a Pisot number or a Salem number. The author proves that the integer parts of its powers \([\alpha^n]\) are composite for infinitely many integers \(n\). In fact, in the case of Salem numbers, inifinitely many of the \([\alpha^n]\) are even. The result for Pisot numbers generalizes a result of \textit{D. Cass} [Proc. Am. Math. Soc. 101, 610-612 (1987; Zbl 0642.12003)], who proved the same result if \(\alpha > 1\) is a unit in a real quadratic number field. The well-known paper of \textit{W. Forman} and \textit{H. N. Shapiro} [Commun. Pure. Appl. Math. 20, 561-573 (1967; Zbl 0149.29302)] proves the same result for \(\alpha = 3/2\) and \(\alpha = 4/3\).
scientific article; zbMATH DE number 1837453