On the matricial version of Fermat-Euler congruences
V. I. Arnold
Abstract:
The congruences modulo the primary numbers are
studied for the traces of the matrices and , where is an integer matrix and is the number of residues modulo , relatively prime to .
We present an algorithm to decide whether these congruences hold for all
the integer matrices , when the prime number is fixed. The algorithm is explicitly applied for many values of , and the
congruences are thus proved, for instance, for all the primes (being untrue for the non-primary modulus ).
We prove many auxiliary congruences and formulate many conjectures and
problems, which can be used independently.