Hierarchical Model-based Prediction on the Maximum Gap between Consecutive Primes below an Arbitrary Number

The distribution of the primes in natural number sequence has no simple law, which seems to show a certain degree of randomness. In this paper, we found a hierarchical progression pattern that the prime intervals are gradually tend to be evenly distributed on different orbits belonging
to different levels, meanwhile, the prime numbers tend to give priority to spread all over each orbit of lower-energy level, and then gradually fill the orbits of higher-energy level with the expansion of prime number range. Moreover, by analyzing the count of prime numbers in different energy levels and different orbits based on the established hierarchical progression model, we investigated that the count of prime numbers in different energy levels decreases exponentially with the increase of energy levels, and its natural logarithm is approximately linear with the change of energy level. Based on these findings, we propose two kinds of strategies to estimate the value of maximum gap between consecutive primes lessen than a given number. Our developed results reveal that the prime distribution shows a layered increasing law, and give a reliable prediction guide to understand the upper limit of maximum interval below an arbitrary number.