Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes

TL;DR

This paper explores Goldbach-like conjectures across various algebraic number systems, providing numerical evidence and formulating new questions about sums of primes in these complex domains.

Contribution

It introduces new Goldbach-type conjectures for Gaussian, Hurwitz, Octavian, and Eisenstein primes, extending classical ideas to complex and quaternionic integers.

Findings

Gaussian integers with a>2, b>2 are sums of two Gaussian primes

Eisenstein integers with a>3, b>3 are sums of two Eisenstein primes

Hurwitz quaternions with positive entries are sums of two Hurwitz primes

Abstract

We formulate Goldbach type questions for Gaussian, Hurwitz, Octavian and Eisenstein primes. They are different from Goldbach type statements by Takayoshi Mitsui from 1960 for number fields or C.A. Holben and James Jordan from 1968 for Gaussian integers. Here is what we meeasure: 1) Every even Gaussian integer a+ib satisfying a>2, b>2 is a sum of two Gaussian primes with positive coefficients. 2) Every Eisenstein integer a+bw with a>3,b>3 and w=(1+sqrt(-3))/2 is the sum of two Eisenstein primes with positive coefficients. Note that no evenness condition is asked in the Eisenstein case. 3) Every Lipschitz integer quaternion with positive entries is the sum of two Hurwitz primes. 4) There exists a constant K such that every Octavian integer with coefficients larger than K is the sum of two Octavian primes. Except in the Octonion case, where the fewest experiments were done, the statements can be linked to difficult questions like Landau or Bunyakovsky conjectures. We therefore look also more closely and numerically at some Hardy-Littlewood constants following early computations from Daniel Shanks and Marvin Wunderlich from the 50ies and 70ies.