A "Vertical" Generalization of the binary Goldbach's Conjecture as applied on primes with prime indexes of any order i (i-primes)

TL;DR

This paper surveys a new meta-conjecture called the Vertical Goldbach's Conjecture, which generalizes the binary Goldbach's Conjecture to primes with recursive prime indexes, suggesting an infinite class of stronger conjectures.

Contribution

It introduces and discusses the Vertical Goldbach's Conjecture, a novel meta-conjecture extending Goldbach's Conjecture to iterative primes with recursive indexes, including new results and computational verifications.

Findings

VGC is a meta-conjecture proposing infinitely many stronger Goldbach-like conjectures.

VGC exhibits self-similar properties in prime distribution.

Computational verifications support the plausibility of VGC.

Abstract

This article is a survey based on our earlier paper ("The 'Vertical' Generalization of the Binary Goldbach's Conjecture as Applied on 'Iterative' Primes with (Recursive) Prime Indexes (i-primeths)" [11]), a paper in which we have proposed a new generalization of the binary/"strong" Goldbach's Conjecture (GC) briefly called "the Vertical Goldbach's Conjecture" (VGC), which is essentially a meta-conjecture, as VGC states an in finite number of Goldbach-like conjectures stronger than GC, which all apply on "iterative" primes with recursive prime indexes (named "i-primes"). VGC was discovered by the author of this paper in 2007, after which it was improved and extended (by computational verifications) until the present (2020). VGC distinguishes as a very important "meta-conjecture" of primes because it states a new class containing an infinite number of conjectures stronger/stricter than GC. VGC has great potential importance in the optimization of the GC experimental verification (including other possible theoretical and practical applications in mathematics and physics). VGC can be also regarded as a very special self-similar property of the distribution of the primes. This present survey contains some new results on VGC.