Solving many problems of the modern theory of prime numbers allows, on the one hand, to deepen the understanding of how to develop the fundamental foundations of mathematics, and on the other - to create more effective arithmetic methods for constructing fast algorithms or discrete orthogonal transformations in the analysis and processing of complex data. One of the problems of modern mathematics in combination with cryptography is the problem of finding primitive roots. This article considers the problem of calculating the set of all primitive roots of an arbitrary prime number p. In addition, the importance of this task in the modern world is described, in particular, the use of the theory of primitive roots in cryptography. An algorithm for checking the natural number n for the property of being a primitive root of a given prime number is constructed. During the work it was found that there are non-specific recursive cycles, the properties of the structures of recursive cycles of primitive roots were investigated. It is proved that all primitive roots of any prime number form pairs in which the recursive cycle of one is an inversion of the recursive cycle of the other element of the pair. Examples of primitive roots and their inner cycles, as well as inversion pairs are given. This property of primitive roots has not been noted before in the literature. In the course of the work, the possibilities of representing recursive cycles in two-dimensional space are also investigated. The results are presented in the form of graphs of inversion pairs of primitive roots of prime numbers. It is shown that recursive cycles form dynamic processes. It is proved that dynamic processes have a chaotic character, the study of which is an important task of the theory of dynamical systems. In the future, it is planned to study in detail the structure of internal cycles for pairs of numbers. The analysis of such structures is a step towards solving complex theoretical and mathematical problems and cryptography problems where primitive roots are used.