  It is well known that the Bernoulli polynomials $\mathbf{B}_n(x)$ have
nonintegral coefficients for $n \geq 1$. However, ten cases are known so far in
which the derivative $\mathbf{B}'_n(x)$ has only integral coefficients. One may
assume that the number of those derivatives is finite. We can link this
conjecture to a recent conjecture about the properties of a product of primes
satisfying certain $p$-adic conditions. Using a related result of Bordell\`es,
Luca, Moree, and Shparlinski, we then show that the number of those derivatives
is indeed finite. Furthermore, we derive other characterizations of the primary
conjecture. Subsequently, we extend the results to higher derivatives of the
Bernoulli polynomials. This provides a product formula for these denominators,
and we show similar finiteness results.
