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 A, B 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 \(\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2\).

中文翻译：

---

矩阵的欧拉定理的Arnold版本的推广

由Arnold猜想并由Zarelua证明的最新结果表明，对于素数p，正整数k和具有整数项的方阵A，其\（{\ textrm tr}（A ^ {p ^ k} ）\ equiv {\ textrm tr}（A ^ {p ^ {k-1}}）（{\ textrm mod} {p ^ k}）\）。我们给出一个更一般的结果的一个简短证明，其中指出，如果两个积分矩阵的特征多项式甲，乙是全等的模p ^ķ然后的特征多项式甲 ^p和乙 ^p是全等的模p ^{ķ 1}，然后我们证明，阿诺德的猜想很容易从中得出。利用这一结果，我们证明欧拉定理的下面概括为任何2×2积分矩阵甲：的特征多项式甲 ^Φ（Ñ）和甲 ^{Φ（Ñ）-φ（Ñ）}是一致的模Ñ。这里ϕ是欧拉函数，\（\ prod_ {i = 1} ^ {l} p_i ^ {\ alpha_i} \）是n的素因式分解，而\（\ Phi（n）=（\ phi（n）+ \ prod_ {i = 1} ^ {l} p_i ^ {\ alpha_i-1}（p_i + 1））/ 2 \）。