Generalizations of Arnold's version of Euler's Theorem for matrices

In this talk I will go over my very recent paper with Marcin Mazur. We have proved that for a square matrix $A$ with integer entries, a prime number $p$, and a positive integer $k$, one has that the characteristic polynomials of the matrices $A^{p^k}$ and $A^{p^{k-1}}$ are congruent modulo $p^k$. Therefore, the traces of these two matrices are congruent modulo $p^k$. V.I. Arnold conjectured this latter result in 2004, and he proved it for $k = 1,2,3$. In 2006, A.V. Zarelua proved it for an arbitrary positive integer $k$. Arnold has remarked in the same paper that there are 2-by-2 integer matrices $A$ such that the traces of $A^6$ and $A^4$ are not congruent modulo 6, and therefore he has not suggested an extension of Euler's Theorem to matrices for a modulus which is not a power of a prime number. In our paper we have found a generalization of Euler's Theorem for 2-by-2 integer matrices and an arbitrary modulus $n>1$. If $n = p^k$ is a power of a prime number $p$, then our generalization reduces to the statement in the second sentence of this abstract where A is now a 2-by-2 integer matrix.