  While the twin prime conjecture is still famously open, it holds true in the
setting of finite fields: There are infinitely many pairs of monic irreducible
polynomials over $\mathbb{F}_q$ that differ by a fixed constant, for each $q
\geq 3$. Elementary, constructive proofs were given for different cases by Hall
and Pollack. In the same spirit, we discuss the construction of a further
infinite family of twin prime tuples of odd degree, and its relations to the
existence of certain Wieferich primes and to arithmetic properties of the
combinatorial Bell numbers.
