Linear combinations of factorials and S -units in a binary recurrence sequence

In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an S-unit whose coefficients are bounded. In particular, we find the largest member of the Fibonacci sequence which can be written as a sum or a difference between a factorial and an S-unit associated to the set of primes { 2 , 3 , 5 , 7 }.