We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Generalizations of Arnold's version of Euler's theorem for matrices.
- Authors
Mazur, Marcin; Petrenko, Bogdan
- Abstract
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 A, B are congruent modulo p then the characteristic polynomials of A and B are congruent modulo p, 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 A and A 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 $$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$$.
- Subjects
EULER'S numbers; MATRICES (Mathematics); POLYNOMIALS; PRIME numbers; FACTOR tables
- Publication
Japanese Journal of Mathematics, 2010, Vol 5, Issue 2, p183
- ISSN
0289-2316
- Publication type
Article
- DOI
10.1007/s11537-010-1023-9