We show that \begin{equation*} \gcd(2!S(2n+1,2),…,(2n+1)!S(2n+1,2n+1))=\text{denominator of $B_{2n}$}, \end{equation*} where $S(n,k)$ is the Stirling number of the second kind and $B_{n}$ is the Bernoulli number.

We consider certain local-global principles related with some splitting problems for connected linear algebraic groups over global fields. The main tools are certain reciprocity results due to Prasad and Rapinchuk, Harder’s Hasse principle for homogeneous projective spaces of reductive groups for number fields and their extensions to global function fields.