The congruences modulo the primary numbers n=pa are studied for the traces of the matrices An and An-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.
Keywords and phrases.
Young diagram Newton–Girard formula multinomial coefficients Cesaro averaging symmetric functions finite Lobachevsky plane Vieta mapping Euler zeta function Euler group little Fermat Theorem geometric progression arithmetical turbulence