Intervals without primes near an iterated linear recurrence sequence

Let [Formula: see text] be a fixed positive integer. Let [Formula: see text] be a linear recurrence sequence for every [Formula: see text], and we set [Formula: see text], where [Formula: see text]. In this paper, we obtain sufficient conditions on [Formula: see text] so that the intervals [Formula: see text] do not contain any prime numbers for infinitely many integers [Formula: see text], where [Formula: see text] is an explicit positive constant depending only on the orders of [Formula: see text]. As a corollary, we show that if for each [Formula: see text], the sequence [Formula: see text] is positive, strictly increasing, and the constant term of its characteristic polynomial is [Formula: see text], then for every Pisot or Salem number [Formula: see text], the numbers [Formula: see text] are composite for infinitely many integers [Formula: see text].