Polyadic numbers and its applications

The talk is devoted to the integral polyadic numbers represented canonically by the series of the type
$$ \sum_{n=1}^{+\infty}a_n n!,\quad a_n=0,1,\dots,n-1, $$
that converge in all fields of $p$ -adic numbers. We suggest some classification of its arithmetical properties and point out to its connection with a notion of “global relation” introduced by E. Bombieri. In the talk, we consider some applications and formulate some new problems.