It has been shown that any prime number $p>5$ divides the repunit number $R_{p-1}$. The question of whether there are composite numbers $n$ such that $n \vert R_{n-1}$ has been answered ($n=91$ is the first such number). We investigate the distribution of these composite numbers, called deceptive primes, for $n \le 2 \cdot 10^7$, and the conjectures and questions that arise from our search.

This paper investigates the properties of eigenvalues and eigenvectors of the problem $Jy + \lambda Ry = (1/\lambda ) Cy$, where $J$ is a "tridiagonal'' real symmetric matrix and $R$ and $C$ are positive diagonal matrices. The results obtained are used to solve the corresponding system of differential equations with boundary and initial conditions.