  In [10] the third author of this paper presented two conjectures on the
additive decomposability of the sequence of ''smooth'' (or ''friable'')
numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second
(less demanding) conjecture. The goal of this paper is to extend and sharpen
their result in three directions by using a different approach (based on the
theory of $S$-unit equations).
