Additive Diophantine Equations with Binary Recurrences, $${{\mathcal {S}}}$$-Units and Several Factorials

Abstract There are many results in the literature concerning linear combinations of factorials among terms recurrence sequences. Recently, Grossman and Luca provided effective bounds for such binary In this paper we show that under certain conditions, even greatest prime divisor $$u_n-a_1m_1!-\dots -a_km_k!$$ u n - a 1 m ! ⋯ k tends to infinity, an way. We give some applications result, as well.