LINEAR RECURRENCE SEQUENCES WITHOUT ZEROS

Let a(d-1), ... ,a(0) is an element of Z, where d is an element of N and a(0) not equal 0, and let X = (x(n))(n=1)(infinity) be a sequence of integers given by the linear recurrence x(n+d) = a(d-1) x(n+1+d-1)+ ... +a(0)x for n = 1, 2, 3, ... We show that there are a prime number p and d integers x(1), ... , x(d) such that no element of the sequence X - (x(n))(n=1)(infinity) defined by the above linear recurrence is divisible by p. Furthermore, for any nonnegative integer 8 there is a prime number p >= 3 and d integers x(1), ... , x(d) such that every element of the sequence X = (x(n))(n-1)(infinity) defined as above modulo p belongs to the set {s+1, s+2, ... ,p - s - 1}.