The talk concerns the following "polynomial moment problem" which arose recently in connection with Poincare's center-focus problem for polynomial vector fields. For a given polynomial $P(z)$ to describe polynomials $Q(z)$ orthogonal to all powers of $P(z)$ on a segment $\left[a,b\right]$. In the talk we describe the main stages of a solution of this problem with the emphasis to the ones connected to the number theory. In more details, first, using some combinatorial considerations, similar to the ones appearing in the "Dessins d'enfants" theory, we reduce this problem to the description of algebraic functions of the form $Q(P^{-1}(z))$ the branches of which satisfy a certain system of linear equations over $\mathbb{Q}$. Then we show that in order to obtain such a description it is enough to describe irreducible permutation representations over $\mathbb{Q}$ of the groups containing a full cycle. Finally, on the base of the Schur rings theory, we provide a classification of such representations.