[Paper Review] On Randomness of Goldbach Sequences

Krishnama Raju Kanchu, Subhash Kak|arXiv (Cornell University)|Jul 26, 2012

Analytic Number Theory Research7 references3 citations

TL;DR

This paper investigates the randomness properties of Goldbach sequences—representations of even numbers as sums of two primes—using autocorrelation analysis and introduces 'Goldbach ellipses' as structured sequences with strong randomness. It demonstrates that both standard Goldbach partitions and these ellipse-based sequences exhibit excellent statistical randomness, particularly at multiples of primorials, making them suitable for cryptographic applications.

ABSTRACT

We consider the use of Goldbach numbers as random sequences. The randomness is analyzed in terms of the autocorrelation function of the sequence of number of partitions. The distinct representations of an even number n as the sum of two primes is a local maximum for multiples of the product of the consecutive smallest primes less than the number. Specific partitions, which we call Goldbach ellipses, are examined. It is shown that such ellipse sequences also have excellent randomness property.

Motivation & Objective

• To evaluate the randomness of Goldbach sequences derived from the number of prime partitions of even numbers.
• To investigate whether structured subsets of Goldbach partitions, termed 'Goldbach ellipses,' maintain strong randomness properties.
• To determine if the number of Goldbach partitions reaches local maxima at multiples of primorials (products of the smallest consecutive primes).
• To explore the cryptographic potential of these sequences based on their statistical randomness.
• To provide a quantitative analysis of autocorrelation behavior in Goldbach partition sequences.

Proposed method

• Analyzes the autocorrelation function of the sequence of Goldbach partition counts for even integers.
• Identifies that the number of representations of an even number as a sum of two primes peaks at multiples of primorials.
• Introduces 'Goldbach ellipses' as a geometrically structured subset of Goldbach partitions with enhanced regularity.
• Applies statistical analysis to assess randomness, focusing on autocorrelation and distribution patterns.
• Uses numerical data and graphical representations (figures and tables) to validate the randomness properties.
• Compares the statistical behavior of Goldbach sequences at primorial multiples to general even numbers.

Experimental results

Research questions

• RQ1 Do Goldbach partition counts exhibit local maxima at multiples of primorials?
• RQ2 How does the autocorrelation of Goldbach partition sequences behave, and does it indicate randomness?
• RQ3 Do Goldbach ellipses—structured subsets of partitions—display superior randomness compared to general sequences?
• RQ4 Can Goldbach sequences be considered cryptographically secure due to their statistical randomness?
• RQ5 What is the relationship between the primorial structure and the distribution of Goldbach partitions?

Key findings

• The number of Goldbach partitions for an even number n reaches a local maximum when n is a multiple of the product of the smallest consecutive primes (i.e., a primorial).
• Goldbach sequences exhibit strong randomness as indicated by low and flat autocorrelation values, especially at primorial multiples.
• Goldbach ellipses, defined as specific structured subsets of partitions, also demonstrate excellent randomness properties.
• The statistical behavior of Goldbach sequences at primorial multiples suggests potential utility in cryptographic applications.
• The analysis confirms that the distribution of partitions is not uniform but shows predictable peaks at primorial values, which enhances randomness in specific contexts.
• The results support the use of Goldbach sequences as a source of pseudorandom sequences in cryptography due to their favorable statistical properties.