On Randomness of Goldbach Sequences

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.

💡 Research Summary

The paper investigates whether sequences derived from Goldbach partitions can serve as high‑quality random sources. Starting from the Goldbach conjecture, the authors define G(n) as the number of distinct ways an even integer n can be expressed as the sum of two primes. By listing G(4), G(6), G(8), … they obtain a deterministic integer sequence S. The central question is whether S exhibits statistical randomness comparable to white noise, which would make it useful for cryptographic or simulation purposes.

To answer this, the authors adopt the autocorrelation function R(k)=E