Divisor Exclusion in Goldbach's Conjecture

Abstract (English)

We establish structural constraints on Goldbach representations of even integers by analyzing how prime divisors of the odd root impose categorical exclusions on potential addends. We prove that if an odd prime $p$ divides the odd root $d$ of an even number $n = 2^k d$, then $p$ cannot appear as an addend in any Goldbach representation of $n$, except in the base case $n = 2p$. This constraint is uniform across entire towers $W_d = \{2^k d : k \geq 1\}$, enabling a direct computational optimization: for highly composite odd roots with many distinct prime factors, we can precompute the excluded set $\mathcal{P}(d)$ once and eliminate these candidates from the search space across all elements of the tower, reducing computational overhead in Goldbach verification algorithms. We analyze the speedup potential and discuss implementation strategies for practical verification of Goldbach's conjecture on large integers. From a theoretical perspective, these results delineate the structural (arithmetic) component of Goldbach from the stochastic (analytic) component, clarifying where modern computational and analytic methods apply.