Abstract (EN)

The congruences modulo the primary numbers n=p a are studied for the traces of the matrices A n and A n-φ(n), where A is an integer matrix and φ(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 ≤ 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.