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.