[Paper Review] Pseudo Random test of prime numbers

Liang Wang, Huang Yan|arXiv (Cornell University)|Mar 18, 2006

Chaos-based Image/Signal Encryption10 references3 citations

TL;DR

This paper applies statistical randomness tests from cryptography—specifically FIPS 140-1/2—to the second difference sequence of prime numbers. Although the sequence fails to meet all FIPS randomness standards, it exhibits strong random-like features and self-similarity, suggesting prime numbers may behave as 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.

Motivation & Objective

• To investigate whether prime numbers exhibit statistical randomness by analyzing derived sequences.
• To test the hypothesis that prime numbers, despite deterministic generation, display random-like behavior.
• To assess the suitability of prime number sequences as pseudo-random sources using established cryptographic tests.
• To explore self-similarity and structural patterns in the second difference sequence of primes.
• To contribute evidence toward the idea that prime numbers may be governed by chaotic dynamics.

Proposed method

• Construct a binary sequence from the second differences of consecutive prime numbers.
• Apply a suite of statistical randomness tests from FIPS 140-1/2 to evaluate the sequence’s compliance with randomness criteria.
• Use visual and quantitative analysis to detect self-similarity in the sequence.
• Compare test results against standard randomness thresholds to determine pass/fail status.
• Analyze the distribution and correlation structure of the binary sequence to assess random-like features.
• Utilize computational methods to generate and process large segments of the prime sequence for testing.

Experimental results

Research questions

• RQ1 Does the second difference sequence of prime numbers pass standard statistical randomness tests?
• RQ2 To what extent does the prime number sequence resemble a pseudo-random sequence in statistical behavior?
• RQ3 Are there detectable self-similar patterns in the second difference sequence of primes?
• RQ4 Can the observed statistical features of primes be interpreted as evidence of underlying chaotic dynamics?
• RQ5 Why does the sequence fail some FIPS 140-2 tests despite showing strong random-like characteristics?

Key findings

• The second difference sequence of primes does not fully satisfy all FIPS 140-1/2 randomness criteria, indicating partial failure in standard statistical tests.
• Despite failing some tests, the sequence displays strong random-like features, suggesting non-trivial statistical randomness.
• Self-similarity is observed in the sequence, indicating structural regularity at multiple scales.
• The results support the hypothesis that prime numbers may be governed by chaotic dynamics rather than purely deterministic or random processes.
• The sequence shows measurable correlation and distribution patterns consistent with pseudo-random behavior, though not perfect.
• The study provides empirical evidence linking number-theoretic sequences to concepts in chaos theory and statistical randomness.