Japan. J. Math. 1, 1--24 (2006)

On the matricial version of Fermat-Euler congruences

V. I. Arnold


Abstract: The congruences modulo the primary numbers $ n=p^a$ are studied for the traces of the matrices $ A^n$ and $ A^{n-\varphi (n)}$, where $ A$ is an integer matrix and $ \varphi (n)$ is the number of residues modulo $ n$, relatively prime to $ n$.
  We present an algorithm to decide whether these congruences hold for all the integer matrices $ A$, when the prime number $ p$ is fixed. The algorithm is explicitly applied for many values of $ p$, and the congruences are thus proved, for instance, for all the primes $ p\leq 7$ (being untrue for the non-primary modulus $ n=6$).
  We prove many auxiliary congruences and formulate many conjectures and problems, which can be used independently.