Fermat-Euler dynamical systems and the statistics of arithmetics of geometric progressions (Q1400106)

The integer \(n\) is said to belong to the set \((N+)\) if \(N\) is the largest integer such that \(a^{\varphi(n)/N}\equiv+1\pmod n\), where \(a\) is prime to \(n\). Similarly the set \((M-)\) is that for which \(M\) is the largest integer such that \(a^{\varphi(n)/M}\equiv -1\pmod n\). In what follows only \(a=2\) is considered. Various properties of the sets \((N+)\) and \((M-)\) are proved and some problems are posed.