I came up with a crazy idea... is there a pattern in the distribution of consecutive prime numbers? So I decided to conduct an experiment. At first, a dataset with the first 5.000.000 prime numbers was loaded (see the acknowledgements for more details). Then, iteratively, subsets were created of that set, containing consecutive prime numbers and for this subset, computed a distance matrix S[i,j] = (p[i] - p[j])^2 was computed, where p[i] and p[j] are the i-th and j-th prime of the subset mentioned earlier. Finally, for the distance matrix S the eigenvalues were computed and with a heuristic method (knee detection) the number of potential clusters was computed.
What is a remarkable observation is that for datasets with more than 100 consecutive prime numbers (no matter which are this prime numbers - large or small), their differences can be grouped in 3 (in a few cases) or 4 clusters! Moreover, the related eigenvectors preserve >80% of the variance of the variables!
Files:
- functions.py: Utility functions I used in my experiments.
- main1.py: The main code of the experiment 1 (random group of consecutive primes).
- main2.py: The main code of the experiment 2 (consecutive groups of consecutive primes).
- main3.py: The main code of the experiment 3 (start from a prime and progressively check concecutive numbers).
The list of primes I used was extracted from files downloaded from [https://primes.utm.edu/lists/small/millions/]https://primes.utm.edu/lists/small/millions/. I wish to thank Dr. [https://www.utm.edu/staff/caldwell/] Chris K. Caldwell for maintaining the repository with files with list of prime numbers.