2016/06/20 by Oliver Knill, Knill, Oliver
Mathematics · #11A41 #11P32 #11R52 #Analytic Number Theory Research #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1606.05958
openalex publication_date 2016/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.