# Article

Keywords:
linear recurrence sequence; period modulo $p$; polynomial splitting in $\mathbb F_p[z]$
Summary:
Let $a_{d-1},\dots ,a_0 \in \mathbb Z$, where $d \in \mathbb N$ and $a_0 \neq 0$, and let $X=(x_n)_{n=1}^{\infty }$ be a sequence of integers given by the linear recurrence $x_{n+d}=a_{d-1}x_{n+d-1}+\dots +a_0x_{n}$ for $n=1,2,3,\dots$. We show that there are a prime number $p$ and $d$ integers $x_1,\dots ,x_d$ such that no element of the sequence $X=(x_n)_{n=1}^{\infty }$ defined by the above linear recurrence is divisible by $p$. Furthermore, for any nonnegative integer $s$ there is a prime number $p \geq 3$ and $d$ integers $x_1,\dots ,x_d$ such that every element of the sequence $X=(x_n)_{n=1}^{\infty }$ defined as above modulo $p$ belongs to the set $\{s+1,s+2,\dots ,p-s-1\}$.
References:
 Adleman, L. M., Odlyzko, A. M.: Irreducibility testing and factorization of polynomials. Math. Comput. 41 (1983), 699-709. DOI 10.1090/S0025-5718-1983-0717715-6 | MR 0717715 | Zbl 0527.12002
 Carroll, D., Jacobson, E., Somer, L.: Distribution of two-term recurrence sequences mod $p^e$. Fibonacci Q. 32 (1994), 260-265. MR 1285757
 Dubickas, A.: Distribution of some quadratic linear recurrence sequences modulo 1. Carpathian J. Math. 30 (2014), 79-86. MR 3244094
 Dubickas, A.: Arithmetical properties of powers of algebraic numbers. Bull. Lond. Math. Soc. 38 (2006), 70-80. DOI 10.1112/S0024609305017728 | MR 2201605 | Zbl 1164.11025
 Dubickas, A.: On the distance from a rational power to the nearest integer. J. Number Theory 117 (2006), 222-239. DOI 10.1016/j.jnt.2005.07.004 | MR 2204744 | Zbl 1097.11035
 Everest, G., Poorten, A. van der, Shparlinski, I., Ward, T.: Recurrence Sequences. Mathematical Surveys and Monographs 104 American Mathematical Society, Providence (2003). DOI 10.1090/surv/104/06 | MR 1990179
 Kaneko, H.: Limit points of fractional parts of geometric sequences. Unif. Distrib. Theory 4 (2009), 1-37. MR 2557644 | Zbl 1249.11066
 Kaneko, H.: Distribution of geometric sequences modulo $1$. Result. Math. 52 (2008), 91-109. DOI 10.1007/s00025-008-0287-3 | MR 2430415 | Zbl 1177.11060
 Lagarias, J. C., Odlyzko, A. M.: Effective versions of the Chebotarev density theorem. Algebraic Number Fields: $L$-Functions and Galois Properties Proc. Symp., Durham, 1975 A. Fröhlich Academic Press, London (1977), 409-464. MR 0447191 | Zbl 0362.12011
 Laksov, D.: Linear recurring sequences over finite fields. Math. Scand. 16 (1965), 181-196. MR 0194349 | Zbl 0151.01502
 Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press Cambridge (1994). MR 1294139 | Zbl 0820.11072
 Neukirch, J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften 322 Springer, Berlin (1999). DOI 10.1007/978-3-662-03983-0 | MR 1697859 | Zbl 0956.11021
 Niederreiter, H., Schinzel, A., Somer, L.: Maximal frequencies of elements in second-order linear recurring sequences over a finite field. Elem. Math. 46 (1991), 139-143. MR 1119645 | Zbl 0747.11062
 Ribenboim, P., Walsh, G.: The ABC conjecture and the powerful part of terms in binary recurring sequences. J. Number Theory 74 (1999), 134-147. DOI 10.1006/jnth.1998.2315 | MR 1670556 | Zbl 0923.11033
 Schinzel, A.: Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications 77 Cambridge University Press, Cambridge (2000). MR 1770638 | Zbl 0956.12001
 Schinzel, A.: Special Lucas sequences, including the Fibonacci sequence, modulo a prime. A Tribute to Paul Erdős A. Baker, et al. Cambridge University Press Cambridge (1990), 349-357. MR 1117027 | Zbl 0716.11009
 Somer, L.: Distribution of residues of certain second-order linear recurrences modulo $p$. Applications of Fibonacci Numbers, Vol. 3 G. E. Bergum, et al. Proc. 3rd Int. Conf., Pisa, 1988 Kluwer Academic Publishers Group, Dordrecht (1990), 311-324. MR 1125803 | Zbl 0722.11008
 Somer, L.: Primes having an incomplete system of residues for a class of second-order recurrences. Applications of Fibonacci Numbers, Proc. 2nd Int. Conf. A. N. Philippou, et al. Kluwer Academic Publishers Dordrecht (1988), 113-141. MR 0951911 | Zbl 0653.10005
 P. Stevenhagen, H. W. Lenstra, Jr.: Chebotarëv and his density theorem. Math. Intell. 18 (1996), 26-37. DOI 10.1007/BF03027290 | MR 1395088 | Zbl 0885.11005
 Voloch, J. F.: Chebyshev's method for number fields. J. Théor. Nombres Bordx. 12 (2000), 81-85. DOI 10.5802/jtnb.266 | Zbl 1007.11069
 Zaïmi, T.: An arithmetical property of powers of Salem numbers. J. Number Theory 120 (2006), 179-191. DOI 10.1016/j.jnt.2005.11.012 | MR 2256803 | Zbl 1147.11037
 Zheng, Q.-X., Qi, W.-F., Tian, T.: On the distinctness of modular reductions of primitive sequences over ${\mathbb Z}/(2^{32}-1)$. Des. Codes Cryptography 70 (2014), 359-368. DOI 10.1007/s10623-012-9698-y | MR 3160735
 Zhuravleva, V.: Diophantine approximations with Fibonacci numbers. J. Théor. Nombres Bordx. 25 (2013), 499-520. DOI 10.5802/jtnb.846 | MR 3228318 | Zbl 1283.11102
 Zhuravleva, V.: On the two smallest Pisot numbers. Math. Notes 94 (2013), 820-823 translation from Mat. Zametki 94 784-787 (2013). DOI 10.1134/S0001434613110163 | MR 3227021 | Zbl 1284.11106