**Authors:** Andrei-Lucian Dragoi

This paper proposes the generalization of the both binary (strong) and ternary (weak) Goldbach’s Conjectures (BGC and TGC) [1,2,3,4] [5,6,7], briefly called “the Vertical Goldbach’s Conjectures” (VBGC and VTGC), which are essentially meta-conjectures (as VBGC states an infinite number of conjectures stronger than BGC). VBGC was 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 (VBGC 1.5e - the conjecture only - 23.02.2017 - 21 pages) The "Vertical" (generalization of) the Binary Goldbach's Conjecture (VBGC) as applied on “iterative” primes with (recursive) prime indexes (i-primeths). Available from: https://www.researchgate.net/publication/313038562_VBGC_15e_-_the_conjecture_only_-_23022017_-_21_pages_The_Vertical_generalization_of_the_Binary_Goldbach%27s_Conjecture_VBGC_as_applied_on_iterative_primes_with_recursive_prime_indexes_i-primeths [accessed Apr 14, 2017].

**Comments:** 30 Pages.

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

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

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