From the text: Suppose that \(a_n \in \mathbb Z\), \(T \in \mathbb N\), and \(a_{n + T} = a_n\) for \(n = 0, 1, \ldots \). We consider the arithmetic properties of series of the form
\[ \sum_{n=0}^\infty a_n n! \tag{1} \]
which converge in all fields \(\mathbb Q_p\) of \(p\)-adic numbers and are elements of the ring of polyadic integers. In what follows, we propose a classification of polyadic numbers and, in particular, state an assertion about the transcendence properties of the polyadic number \(\gamma = \sum_{n=0}^\infty n!\).