Divisibility properties of certain recurrent sequences

A. Dubickas

Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 76-82

Résumé

Let $g$ and $m$ be two positive integers, and let $F$ be a polynomial with integer coefficients. We show that the recurrent sequence $x_0=g$, $x_n=x_{n-1}^n+F(n)$, $n=1,2,3,\dots$, is periodic modulo $m$. Then a special case, with $F(z)=1$ and with $m=p>2$ being a prime number, is considered. We show, for instance, that the sequence $x_0=2$, $x_n=x_{n-1}^n+1$, $n=1,2,3,\dots$, has infinitely many elements divisible by every prime number $p$ which is less than or equal to 211 except for three prime numbers $p=23, 47, 167$ that do not divide $x_n$. These recurrent sequences are related to the construction of transcendental numbers $\zeta$ for which the sequences $[\zeta^{n!}]$, $n=1,2,3,\dots$, have some nice divisibility properties.

Bibliographie

[1] S. D. Adhikari, P. Rath, N. Saradha, On the sets of uniqueness of the distribution function of $\{\xi(p/q)^n\}$, submitted

[2] S. Akiyama, C. Frougny, J. Sakarovitch, On number representation in a rational base, submitted

[3] G. Alkauskas, A. Dubickas, “Prime and composite numbers as integer parts of powers”, Acta Math. Hung., 105 (2004), 249–256 | DOI | MR | Zbl

[4] R. C. Baker, G. Harman, “Primes of the form $[c^p]$”, Math. Zeitschr., 221 (1996), 73–81 | DOI | MR | Zbl

[5] Y. Bugeaud, “Linear mod one transformations and the distribution of fractional parts $\{\xi(p/q)^n\}$”, Acta Arith., 114 (2004), 301–311 | DOI | MR | Zbl

[6] D. Cass, “Integer parts of powers of quadratic units”, Proc. Amer. Math. Soc., 101 (1987), 610–612 | DOI | MR | Zbl

[7] A. Dubickas, “Integer parts of powers of Pisot and Salem numbers”, Archiv der Math., 79 (2002), 252–257 | DOI | MR | Zbl

[8] A. Dubickas, “Sequences with infinitely many composite numbers”, Analytic and Probabilistic Methods in Number Theory (Palanga, 2001), eds. A. Dubickas et al., TEV, Vilnius, 2002, 57–60 | MR | Zbl

[9] A. Dubickas, “Arithmetical properties of powers of algebraic numbers”, Bull. London Math. Soc. (to appear) | MR

[10] A. Dubickas, A. Novikas, “Integer parts of powers of rational numbers”, Math. Zeitschr. (to appear)

[11] L. Flatto, J. C. Lagarias, A. D. Pollington, “On the range of fractional parts $\{\xi(p/q)^n\}$”, Acta Arith., 70 (1995), 125–147 | MR | Zbl

[12] W. Forman, H. N. Shapiro, “An arithmetic property of certain rational powers”, Comm. Pure Appl. Math., 20 (1967), 561–573 | DOI | MR | Zbl

[13] R. K. Guy, Unsolved Problems in Number Theory, Springer–Verlag, New York, 1994 | MR

[14] J. F. Koksma, “Ein mengen-theoretischer Satz über Gleichverteilung modulo eins”, Compositio Math., 2 (1935), 250–258 | MR

[15] K. Mahler, “An unsolved problem on the powers of 3/2”, J. Austral. Math. Soc., 8 (1968), 313–321 | DOI | MR | Zbl

[16] H. W. Mills, “A prime representing function”, Bull. Amer. Math. Soc., 53 (1947), 604 | DOI | MR | Zbl

[17] T. Vijayaraghavan, “On the fractional parts of the powers of a number”, J. London Math. Soc., 15 (1940), 159–160 | DOI | MR

[18] E. M. Wright, “A prime representing function”, Amer. Math. Monthly, 58 (1951), 616–618 | DOI | MR | Zbl