  We investigate the Membership Problem for hypergeometric sequences: given a
hypergeometric sequence $\langle u_n \rangle_{n=0}^\infty$ of rational numbers
and a target $t \in \mathbb{Q}$, decide whether $t$ occurs in the sequence. We
show decidability of this problem under the assumption that in the defining
recurrence $p(n)u_{n}=q(n)u_{n-1}$, the roots of the polynomials $p(x)$ and
$q(x)$ are all rational numbers. Our proof relies on bounds on the density of
primes in arithmetic progressions. We also observe a relationship between the
decidability of the Membership problem (and variants) and the Rohrlich-Lang
conjecture in transcendence theory.
