, Volume 5, Issue 2, pp 183-189
Date: 25 Dec 2010

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

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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 p k then the characteristic polynomials of A p and B p are congruent modulo p k+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 A Φ(n) and A Φ(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 \(\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2\) .

Communicated by: Takeshi Saito
To the memory of Vladimir Igorevich Arnold (1937–2010), for his vision and inspiration.