Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})$$\end{document}. We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo pk then the characteristic polynomials of Ap and Bp are congruent modulo pk+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of AΦ(n) and AΦ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod_{i=1}^{l} p_i^{\alpha_i}$$\end{document} is a prime factorization of n and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$$\end{document}.