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