The paper presents a fractal analysis of interval series, the members of which are consecutive deviations of natural prime numbers. Fractal analysis, which has been in full force since the end of the last century, has made it possible to identify new, unusual properties of geometric and physical objects and processes, including predicting the behavior of time and spatial series. The combination of two structural blocks - the spatial interval series of increasing power and the fractal set made it possible to apply the fractal technique to the study of the sequence of intervals. With its help, the idea of the phenomenon of intervals of prime numbers as a structure that does not contradict the nature of most natural phenomena is expanded. Using the Hurst method and scaling it is established that the appearance of intervals of prime numbers is not random. With restrictions on the available computer memory, criteria-based estimates of interval series of different capacities were carried out and it was found that they have the properties of scale invariance, multifractality and self-similarity. The performed estimates confirm that the continuum of primes at all scale levels belongs to fractal sets.