Linear recurrence sequences without zeros

Let ad−1,⋯, a0 ∈ ℤ, where d ∈ ℕ and a0 ≠ 0, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X = ({x_n})_{n = 1}^\infty $$\end{document} be a sequence of integers given by the linear recurrence xn+d = ad−1xn+d−1 + ⋯ + a0xn for n = 1, 2, 3, ⋯. We show that there are a prime number p and d integers x1, ⋯, xd such that no element of the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X = ({x_n})_{n = 1}^\infty $$\end{document} defined by the above linear recurrence is divisible by p. Furthermore, for any nonnegative integer s there is a prime number p ⩾ 3 and d integers x1, ⋯, xd such that every element of the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X = ({x_n})_{n = 1}^\infty $$\end{document} defined as above modulo p belongs to the set {s + 1, s + 2, ⋯, p − s − 1}.