On the greatest square free factor of terms of a linear recurrence sequence

For any integer m let P (m) denote the greatest prime factor of m and let Q(m) denote the greatest square free factor of m with the convention that P (0) = P (±1) = 1 = Q(±1) = Q(0). Thus, if m = p1 1 · · · p hr r with p1, . . . , pr distinct primes and h1, . . . , hr positive integers, then Q(m) = p1 · · · pr. van der Poorten and Schlickewei [6] and Evertse [1] proved, by means of a p-adic version of Schmidt’s Subspace Theorem due to Schlickewei [8], that if (un) ∞ n=0 is a non-degenerate linear recurrence sequence then P (un) → ∞ as n → ∞. (4)