Japanese Journal of Mathematics

, Volume 5, Issue 2, pp 183–189

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

Article

DOI: 10.1007/s11537-010-1023-9

Mazur, M. & Petrenko, B.V. Jpn. J. Math. (2010) 5: 183. doi:10.1007/s11537-010-1023-9

## Abstract.

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 $${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})$$. We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices AB 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, $$\prod_{i=1}^{l} p_i^{\alpha_i}$$ is a prime factorization of n and $$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$$.

### Keywords and Phrases:

Euler congruences Euler’s theorem Fermat’s little theorem congruences for traces

### Mathematics Subject Classification (2010):

05A10 11A07 11C20