**Author:**

Dragoi, Andrei-Lucian M.

**Category:**

Research Papers

**Date Published:**

February 3, 2017

**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 Goldba

**Abstract:**

This article 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), discovered in 2007[1] and perfected until 2016[3] by using the arrays ( p S and ,ip S ) of Matrix of Goldbach index-partitions (GIPs) (simple , pnM and recursive ,, i p nM , with iteration order 0 i ), which are a useful tool in studying BGC by focusing on prime indexes (as the function n P that numbers the primes is a bijection). Simple M , pnM and recursive M ,, i p nM are related to the concept of generalized “primeths” (a term first used by Fernandez N. in his “The Exploring Primeness Project” [8]), which is the generalization with iteration order 0 i of the known “higher-order prime numbers” (alias “super-prime 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 (i x P is the x-th i-primeth, with iteration order 0 i as explained later on). The author of this article also

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