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

Posted in
Speaker:
Bogdan Petrenko
Affiliation:
SUNY at Brockport/MPI
Date:
Wed, 2010-08-11 14:15 - 15:15
Location:
MPIM Lecture Hall
Parent event:
Number theory lunch seminar

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.

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