On Wieferich and Non-Wieferich Primes with Prime Bases

We study the matter of the density and the distribution of Wieferich and non-Wieferich primes and especially the issues related to bases that are prime numbers. The existence of special partitions of the set of all prime numbers in two sets of which one has density equal to one and the other is maximal and is free of Wieferich pairs, but which contains hybrid Wieferich pairs with prime base and modulus belonging to one or the other is proved.