Number-theoretical turbulence in Fermat-Euler arithmetics and large young diagrams geometry statistics

Many stochastic phenomena in deterministic mathematics had been discovered recently by the experimental way, imitating the Kolmogorov's semi-empirical methods of discovery of the turbulence laws. From the deductive mathematics point of view most of these results are not theorems, being only descriptions of several millions of particular observations. However, I hope that they are even more important than the formal deductions from the formal axioms, providing new points of view on difficult problems where no other approaches are that efficient. I shall describe below two such examples: the Fermat-Euler statistics of the residues (modulo an integer number) of geometric progressions and the Young diagrams statistics describing the integer numbers partitions into integer summands and the symmetric groups representations.