**Authors:** Andrei Lucian Dragoi

This article proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC), briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), discovered in 2007[1] and perfected until 2016[2] by using the arrays (S_p and S_i,p) of Matrix of Goldbach index-partitions (GIPs) (simple M_p,n and recursive M_i,p,n, with iteration order i ≥ 0), which are a useful tool in studying BGC by focusing on prime indexes (as the function P_n that numbers the primes is a bijection). Simple M (M_p,n) and recursive M (M_i,p,n) are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project”), which is the generalization with iteration order i≥0 of the known “higher-order prime numbers” (alias “superprime numbers”, “super-prime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes (iPx is the x-th o-primeth, with iteration order i ≥ 0 as explained later on). The author of this article also brings in a S-M-synthesis of some Goldbach-like conjectures (GLC) (including those which are “stronger” than BGC) and a new class of GLCs “stronger” than BGC, from which VBGC (which is essentially a variant of BGC applied on a serial array of subsets of primeths with a general iteration order i ≥ 0) distinguishes as a very important conjecture of primes (with great importance in the optimization of the BGC experimental verification and other potential useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar propriety of the primes subset of (noted/abbreviated as or as explained later on in this article). Keywords: Prime (number), primes with prime indexes, the i-primeths (with iteration order i≥0), the Binary Goldbach Conjecture (BGC), the Ternary Goldbach Conjecture (TGC), Goldbach index-partition (GIP), fractal patterns of the number and distribution of Goldbach index-partitions, Goldbach-like conjectures (GLC), the Vertical Binary Goldbach Conjecture (VBGC) and Vertical Ternary Goldbach Conjecture (VTGC) the as applied on i-primeths

**Comments:** 21 Pages.

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[v1] 2017-01-25 02:44:01

[v2] 2017-01-31 05:09:49

[v3] 2017-02-02 03:50:49

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