LAUSR

Let (fn)n=1∞ be an unbounded sequence of integers satisfying a linear recurrence relation with integer coefficients. We show that for any k ∈ ℕ there exist infinitely many n ∈ ℕ for which 2k + 1 consecutive integers fn − k,…,fn,…,fn + k are all divisible by certain primes. Moreover, if the sequence of integers (fn)n=1∞ satisfying a linear recurrence relation is unbounded and non-degenerate then for some constant c > 0 the intervals (|fn|− clog n,|fn| + clog n) do not contain prime numbers for infinitely many n ∈ ℕ. Applying this argument to sequences of integer parts of powers of Pisot and Salem numbers α we derive a similar result for those sequences as well which implies, for instance, that the shifted integer parts ⌊αn⌋ + l, where l = −k,…,k and n runs through some infinite arithmetic progression of positive integers, are all composite.