  A sequence of integers $ \{ s_n \}_{n \in \mathbb{N}} $ is called a
T-sequence if there exists a Hausdorff group topology on $ \mathbb{Z} $ such
that $ \{ s_n \}_{n \in \mathbb{N}} $ converges to zero. For every finite set
of primes $ S $ we build a Hausdorff group topology on $ \mathbb{Z} $ such that
every growing sequence of $ S $-integers converges to zero. As a corollary, we
solve in the affirmative an open problem by I.V. Protasov and E.G. Zelenuk
asking if $ \{ 2^n + 3^n \}_{n \in \mathbb{N}} $ is a T-sequence. Our results
rely on a nontrivial number-theoretic fact about $ S $-unit equations.
