Pseudo Random test of prime numbers

TL;DR

This paper applies cryptographic statistical tests to prime number sequences, revealing their partial randomness, self-similarity, and chaotic characteristics, thus providing new insights into prime distribution.

Contribution

It introduces the use of cryptographic randomness tests on prime-derived sequences and identifies their partial randomness and self-similar properties.

Findings

Prime sequences exhibit partial randomness according to FIPS standards.

Self-similarity observed in the second difference sequence of primes.

Prime numbers show characteristics of a chaotic system.

Abstract

The prime numbers look like a randomly chosen sequence of natural numbers, but there is still no strict theory to determine 'Randomness'. In these years, cryptography has developed a battery of statistical tests for randomness. In this paper, we just apply these methods to study the distribution of primes. Here the binary sequence constructed by second difference of primes is used as samples. We find this sequence can't reach all the 'random standard' of FIPS 140-1/2, but still show obvious random feature. The interesting self-similarity is also observed in this sequence. These results add the evidence that prime numbers is a chaos system.