The “Vertical” Generalization of the Binary Goldbach’s Conjecture as Applied on “Iterative” Primes by Ioan Degău

Journal of Advances in Mathematics and Computer Science 25(2): 1-32, 2017; Article no.JAMCS.36895 ISSN: 2456-9968 (Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-0851)

The “Vertical” Generalization of the Binary Goldbach’s Conjecture as Applied on “Iterative” Primes with (Recursive) Prime Indexes (i-primeths) Andrei-Lucian Drăgoi1* 1

Independent Researcher in Mathematics, Physics and Biology, Street Aleea Arinii Dornei, nr. 11, bl. I9, Sc. A, ap. 4, et. 2, Bucuresti, Sector 6, Romania. Author’s contribution The sole author designed, analyzed and interpreted and prepared the manuscript. Article Information

DOI: 10.9734/JAMCS/2017/36895 Editor(s): (1) Francisco Welington de Sousa Lima, Professor, Dietrich Stauffer Laboratory for Computational Physics, Departamento de Física, Universidade Federal do Piaui, Teresina, Brazil. Reviewers: (1) Weijing Zhao, Civil Aviation University of China, China. (2) A. N. Chavan, Shivaji University, India. (3) Bilge Peker, Necmettin Erbakan University, Turkey. Complete Peer review History: http://www.sciencedomain.org/review-history/21625

Received: 21st September 2017 Accepted: 19th October 2017 Published: 28th October 2017

Original Research Article _______________________________________________________________________________

Abstract This article proposes a synthesized classification of some Goldbach-like conjectures, including those which are “stronger” than the Binary Goldbach’s Conjecture (BGC) and launches a new generalization of BGC briefly called “the Vertical Binary Goldbach’s Conjecture” (VBGC), which is essentially a metaconjecture, as VBGC states an infinite number of conjectures stronger than BGC, which all apply on “iterative” primes with recursive prime indexes (i-primeths). VBGC was discovered by the author of this paper in 2007 and perfected (by computational verifications) until 2017 by using the arrays of matrices of Goldbach index-partitions, which are a useful tool in studying BGC by focusing on prime indexes. VBGC distinguishes as a very important conjecture of primes, with potential importance in the optimization of the BGC experimental verification (including other possible theoretical and practical applications in mathematics and physics) and a very special self-similar property of the primes set.

Keywords: Primes with prime indexes; i-primeths; the Binary Goldbach Conjecture; Goldbach-like conjectures; the Vertical Binary Goldbach Conjecture.

_____________________________________

*Corresponding author: E-mail: dr.dragoi@yahoo.com;

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

2010 mathematics subject classification: 11N05 (Distribution of primes, URL: http://www.ams.org/msc/msc2010.html?t=11N05&btn=Current)

1 Introduction This paper proposes the generalization of the binary (strong) Goldbach’s Conjectures (BGC) [1-7], briefly called “the Vertical Binary Goldbach’s Conjecture” (VBGC), which is essentially a meta-conjecture, as VBGC states an infinite number of conjectures stronger than BGC, which all apply on “iterative” primes with recursive prime indexes named “i-primeths” in this article, as derived from the concept of generalized “primeths”, a term first introduced in 1995 by N. J. A. Sloane and Robert G. Wilson in their “primeth recurrence” concept in their array of integers indexed as A007097 (formerly M0734) [8] in The Online Encyclopedia of Integer Sequences (Oeis.org); the term “primeth” was then used from 1999 by Neil Fernandez in his “The Exploring Primeness Project” [9]). The “i-primeth” concept is the generalization with iteration order i ≥ 0 of the known “higher-order prime numbers” (alias “super-prime numbers”, “superprime numbers”, ”super-primes”, ” super-primes” or “prime-indexed primes[PIPs]”) as a subset of (simple or recursive) primes with (also) prime indexes, with order i ≥ 0 , as noted in this paper and explained later on.

iP x being the x-th i-primeth, with iteration

VBGC was discovered in 2007 and perfected until 2017 by using the arrays ( S p and (M) of Goldbach index-partitions (GIPs) (simple

Si, p ) of Matrices

M p,n and recursive M i , p ,n , with iteration order i ≥ 0

, also related to the concept of “i-primeths”), which are a useful tool in studying BGC/VBGC by focusing on prime indexes (as the function

that numbers the primes is bijective).

There are a number of (relative recently discovered) GLCs stronger than BGC (and implicitly stronger than TGC), that can also be synthesized using M p,n concept: these stronger GLCs (as VBGC also is) are tools that can inspire new strategies of finding a formal proof for BGC, as I shall try to argue in this paper. Additionally, there are some arguments that Twin Prime Conjecture (TPC) may be also (indirectly) related to BGC as part of a more extended and profound conjecture, so that any new clue for BGC formal proof may also help in TPC (formal) demonstration. 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 i-primeths with a general iteration order i ≥ 0 ) distinguishes as a very important conjecture of primes (with potential importance in the optimization of the BGC experimental verification and other possible useful theoretical and practical applications in mathematics [including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very special self-similar property of the primes subset of N (noted/abbreviated as℘ or ℘* as explained later on in this paper). Primes (which are considered natural numbers [positive integers] >1 that each has no positive divisors other than 1 and itself (like 2, 3, 5, 7, 11 etc) by the latest modern conventional definition, as number 1 is a special case [10,11] which is considered neither prime nor composite, but the unit of all integers) are conjectured (by BGC) to have a sufficiently dense and (sufficiently) uniform distribution in N , so that: (1) Any natural even number

2 n, with n > 1 can be splitted in at least one Goldbach

partition/pair(GP)corresponding to at least one Goldbach index-partition (GIP) [12].

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Or (2) Any positive integer n > 1 can be expressed as the arithmentic average of at least one pair of primes. BGC is specifically reformulated by the author of this article in order to emphasize the importance of studying the Primes Distribution (PD) [13,14,15,16] defined by a global and local density and uniformity with multiple interesting fractal patterns [17]: BGC is in fact an auto-recursive fractal property of PD in N alias the Goldbach Distribution of Primes (GDP) (as the author will try to argue later on in this article), but also a property of ℘ , a property which is indirectly expressed as BGC, using the subset of even naturals).

2 The Array S p of the Simple Matrix of Goldbach Index-Partitions

( M p ,n )

Definition

℘*

℘.

and

may

define

the

prime

subset

℘* =

{P1 ( = 2 ) , P2 ( = 3) , P3 ( = 5) ,..., Px ,..., Py ,...P∞ } , with x, y ∈ N * and 0 < x < y , with Px ( Py ) being the x-th (y-th) primes of ℘ * and P∞ marking the already proved fact that ℘ * has an

infinite number of (natural) elements (Euclid's 2nd theorem [18]). The numbering function of primes is a bijection that interconnects ℘ * with element of

( Pn )

N * so that each element of ℘ * corresponds to only (just) one N * and vice versa: 1 ↔ P1 ( = 2 ) , 2 ↔ P2 ( = 3) , ..., x ↔ Px (the x-th prime), y ↔ Py

(the y-th prime), …,

∞ ↔ P∞ .

Originally, Goldbach considered that number 1 was the first prime:

although still debated until present, today the mainstream considers that number 1 is neither prime nor composite, but the unity of all the other integers. However, in respect to the first "ternary" formulation of GC (TGC) (which was re-formulated by Euler as the BGC and also demonstrated by the same Euler to be

P0 = 1 (the ℘= and

stronger than TGC, as TGC is a consequence of BGC), the author of this article also defines unity

all

integers,

implicitly

the

unity

all

primes)

{P0 ( = 1) , P1 ( = 2 ) , P2 ( = 3) , P3 ( = 5) ,..., Px ,..., Py ,...P∞ } , with x, y ∈ N and 0 ≤ x < y , although

only

℘* = ℘− {P0 ( = 1)}

shall be used in this paper (as it is used in the mainstream of modern

mathematics). The 1st formulation of BGC. For any even integer n > 2 , it will always exist at least one pair of (other two) integers x, y ∈ N * with x ≤ y so that

( )

Px + Py = n , with Px Py being the x-th (y-th) primes of

℘ * . Important observation: The author considers that analyzing those “homogeneous” triplets of three naturals ( n, x, y ) (no matter if primes or composites) is more convenient and has more “analytical” potential than analyzing (relatively) “inhomogeneous” triplets of type

( n, Px , Py ) : that’s why the author

proposes Goldbach index partitions (GIPs) as an alternative to the standard Goldbach partitions (GPs) proposed by Oliveira e Silva. The existence of (at least) a triplet

( n, x, y )

for each even integer n > 2

(as BGC claims) may suggest that BGC is profoundly connected to the generic primality (of any

and

Py ) and, more specifically, argues that GC is in fact a property of PD in N (and a property of ℘ * as 3

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

composed of indexed/numbered elements). The most important property of Primes and PD and is that

Px → x ⋅ ln ( x) ⇔ Px / x → ln ( x) , for x →∞

Px ≅ x ⋅ ln ( x ) , for any progressivelylarge x

(which is the alternative [linearithmic] expression of the Prime Number Theorem [19], as if ℘ * is a result of an apparently random quantized linearithmization of conclusion: For any even integer

N * −{1}

so that

Pn → n ⋅ ln ( n ) . In

n > 2 , at least one GIP exists (BGC – 1st condensed formulation).

The 2nd formulation of BGC using the Matrix of Goldbach index-partitions (M-GIP or M). [1] Let us consider an infinite string of matrices

being composed of lines made by GIPs

 xn,1  M  M n =  xn, j M   xn,m n 

S = {M 1, M 2 , M 3 ,..., M n ,...M ∞ } , with each generic

( x, y ) , such as:

yn,1   M  yn, j  , with Px + Py = n, ∀ j ∈ [1, mn ] n, j n, j  M  yn,mn 

M n , j ≥ 1 and j ≤ mn ) ( mn is the total maximum number of j-indexed lines of M n )

( j is the index of any chosen line of (xn,i,yn,i ∈ N * , xn,i < xn,i+1 for

mn ≥ 2 , ∀ i ∈ [1, mn ] )

[2] Let us also consider the function that counts the lines of any (that numbers the lines of a GM) is classically named as

M n , such as: l ( n ) = mn

. This function

r (n ) = l (n ) = mn (“r” stands for the number

of “rows”). [3] An empty/null matrix Using 1.

S , M , M∅

( M ∅ ) is defined as a matrix with zero rows and/or columns.

and r (n ) as previously defined, BGC has two formulations sub-variants:

M n ≠ M ∅ (OR S doesn’t contain any M ∅ ) for any even integer n > 2 or shortly:

∀ even integer n > 2 ⇔ M n ≠ M ∅ (the 2nd formulation of BGC – 1st sub-variant). 2.

For any even integer nd

n > 2 , r ( n ) > 0 or shortly: ∀ even integer n > 2 ⇔ r (n) > 0 (the

formulation of BGC – 2nd sub-variant).

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The 3rd formulation of BGC using the generalization of

( M p,n )

S (S p )

and the generalization of

for GIPs matrix containing more than 2 columns (as based on GIPs with more than 2

elements). [1] Let us consider an infinite set OF infinite strings OF matrix:

{

}

S 2 = M 2,1 , M 2,2 , M 2,3 ,..., M 2,n ,... M 2,∞ (the generic M 2,n of S2 has 2 columns based

on [binary] GIPs with 2 elements);

{

}

S3 = M 3,1, M 3,2 , M 3,3 ,..., M 3,n ,...M 3,∞ (the generic M 3,n of S3 has 3 columns based on

b) c)

[ternary] GIPs with 3 elements); …;

S p = M p ,1 , M p ,2 , M p ,3 ,..., M p ,n ,... M p ,∞

based on [p-nary] GIPs with p elements and natural p>3); …,

S ∞ = M ∞ ,1 , M ∞ ,2 , M ∞ ,3 ,..., M ∞,n ,... M ∞ ,∞

{

infinite ( ∞ ) number of columns based on

} (the generic } (the generic

∞ − nary

number of elements) g) With each generic M p ,n being composed of

M p ,n of S p has p columns

M ∞,n of S∞ has potentially

GIPs with a potentially infinite

( ∞)

m p ,n lines and p columns made by p-nary GIPs

with p elements, such as:

 xn,1   M  M p,n =  xn, j  M   xn,m p,n 

...

xn,k

...

xn,k + j

  M M  ... xn, p+ j  , with P xn , j + ... + Pxn , j +k + ... + Pxn , p + j = n,   M M  ... xn, p+m p,n   ...

... xn,k +m p ,n

xn, p

∀ j ∈ 1, m p,n  and ∀ k ∈ [1, p ] ,

j is the index of any chosen line of M p,n , j ≥ 1 and j ≤ m p,n and m p ,n is the total maximum number of j-indexed lines of M p,n ) (

( k is the index of any chosen column of and (

is the total number of

M p,n , k ≥ 1 and k ≤ p

k -indexed columns of M p,n )

xn,k + j ∈ N * , xn, j ≤ xn, j +1 for m p,n ≥ 2 , ∀ j ∈ 1, m p ,n  and ∀ k ∈ [1, p ] )

[2]

Let

also consider the

function that

counts

the

lines

any

M p,n , such as:

r ( p, n ) = l ( p, n ) = m p,n .

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Using 1.

S p , M p,n , M ∅ and r ( p , n ) as previously defined, BGC has two formulations sub-variants:

M 2,n ≠ M ∅

(OR

S2 doesn’t contain any M ∅ ) for any even integer n > 2 or shortly:

∀ even integer n > 2 ⇔ M 2,n ≠ M ∅ (the 3rd formulation of BGC – 1st sub-variant). 2.

For any even integer rd

(the 3

n > 2 , r (2, n ) > 0

or shortly:

∀ even integer n > 2 ⇔ r (2, n ) > 0

formulation of BGC –2 sub-variant).

3 A Synthesis and A/B Classification of the Main Known Goldbach-like Conjectures (GLCs) Using the M p,n Concept 3.1 The Goldbach-like conjectures (GLCs) category/class GLCs definition. A GLC may be defined as any additional special (observed/conjectured) property of and its elements

M p,n other that GC (with n > 2 ), with possibly other inferior limits a ≥ 2 , with

n > a ≥ 2 ). GLCs classification. GLCs may be classified in two major classes using a double criterion such as: 1.

Type A GLCs (A-GLCs) are those GLCs that claim: [1] Not only that all

M p,n ≠ M ∅ for a

chosen p>1 and for any / any odd / any even integer n > a ≥ 2 (with a being any finite natural established by that A-GLC and n > a ) BUT ALSO [2] any other non-trivial(nt) accessory

( ≠ M ∅ ) of S p . A specific A-GLC is considered authentic if the other non-trivial accessory property/properties of all M p,n ( ≠ M ∅ ) (claimed by that A-GLC)

property/properties of all M p,n

isn’t/aren’t a consequence of the 1st claim (of the same A-GLC). Authentic (at least conjectured as such) A-GLCs are (have the potential to be) “stronger” than GC as they claim “more” than GC does. 2.

Type B GLCs (B-GLCs) are those GLCs that claim: no matter if all

M p,n ≠ M ∅ or just some

M p,n ≠ M ∅ for a chosen p>1 and for some / some odd / some even integer n > a ≥ 2 (with a being any finite natural established by that B-GLC and

n > a ), all those M p,n

that are yet non-

M ∅ (for n > a ) have (an)other non-trivial accessory property/properties. A specific B-GLC is

(

)

considered authentic if the other non-trivial accessory property/properties of all M p,n ≠ M ∅ (claimed by that B-GLC for n > a ) isn’t/aren’t a consequence of the fact that some

M p,n ≠ M ∅ for n > a . Authentic (at least conjectured as such) B-GLCs are “neutral” to GC

(uncertainly “stronger” or “weaker” conjectures) as they claim “more” but also “less” than GC does (although they may be globally weaker and easier to formally prove than GC).

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Other variants of the (generic) Goldbach Conjecture (GC) and GLCs include the statements that: 1.

“[…] Every [integer] number that is greater than 2 is the sum of three primes” (Goldbach's original conjecture formulated in 1742, sometimes called the "Ternary" Goldbach conjecture (TGC), written in a June 7, 1742 letter to Euler) (which is equivalent to: “every integer >2 is the sum of at least one triad of primes*”, *with the specification that number 1 was also considered a prime by the majority of mathematicians contemporary to Goldbach, which is no longer the case now]”). This (first) variant of GC (TGC) can be formulated using (ternary) M 3,n (based on GIPs with 3 elements) such as: a.

Type A formulation variant as applied to ℘( not just to℘*) : “

b. 2.

∀ integer n > 2 ⇔ M 3,n ≠ M ∅ (with xn, j ,k ≥ 0 and Px ∈℘ )” n , j ,k

Type B (neutral) formulation variant: not supported.

“Every even integer n > 4 is the sum of 2 odd primes.” (Euler’s binary reformulation of the original GC, which was initially expressed by Goldbach in a ternary form as previously explained). Since BGC (as originally reformulated by Euler) contains the obvious triviality that there are infinite many even positive integers of form 2 p = p + p (with p being any prime), the nontrivial BGC (ntBGC) sub-variant that shall be treated in this paper (alias “BGC” or “ntBGC”) is that: “every even integer n > 6 is the sum of at least one pair of distinct odd primes” [20,21] (which is equivalent to: “every even integer m > 3 is the arithmetic average of at least one pair of distinct odd primes”). Please note that ntBGC doesn’t support the definition of a GLC, as 2 p = p + p is a trivial property of some even integers implying the complementary relative triviality that: 2c ≠ 2 p ≠ p + p (with

being any composite natural number and

prime). ntBGC can be formulated using (binary) a.

Type A formulation variant: “

M 2,n ( M n )

being any

M 2,n (based on GIPs with 2 elements) such as:

∀ even integer n > 6 , M 2,n ( M n ) ≠ M ∅ AND

contains at least one line with both elements (GIPs)≠1 (as

P1 = 2

is the only

even prime) AND distinct to each other (as distinct GIPs means distinct primes as based on the bijection of the prime numbering function)” b.

Type B (neutral) formulation variant: “

∀ even integer n > 6 , all M 2,n ( M n ) that are

S p may also contain empty M 2,n ( M n ) = M ∅ for some specific [but still unfound] n values ) will contain at least one line with both elements (GIPs)≠1 (as P1 = 2 is non-empty (as

the only even prime) AND distinct to each other (as distinct GIPs means distinct primes as based on the bijection of the prime numbering function)”. 3.

“

∀ odd integer n > 5 , n is the sum of 3 (possibly identical) primes.” [22] (the [weak] Ternary

Goldbach's conjecture/theorem [TGC/TGT] formally proved by Harald Helfgott in 2013 [23,24,25], so that TGC is very probably [but not surely however] a proved theorem (as TGT), and no longer a “conjecture”) (which is equivalent to: “

∀ odd integer n > 5 , n is the sum of at

least one triad of [possibly identical] primes”). TGC can be formulated using (ternary)

M 3,n

(based on GIPs with 3 elements) such as:

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∀ odd integer n > 5 ⇔ M 3,n ≠ M ∅ ”

Type A formulation variant: “

Type B (neutral) formulation variant: not supported.

“

∀ integer n > 17 , n is the sum of exactly 3 distinct primes.” (cited as “Conjecture 3.2” by

Pakianathan and Winfree in their article, which is equivalent to: “

∀ integer n > 17 , n is the

sum of at least one triad of distinct primes”) (this is a conjecture stronger than TGC, but weaker than BGC as it is implied by BGC). This stronger version of TGC (sTGC) can also be formulated using (ternary) M 3,n (based on GIPs with 3 elements) such as: a.

Type A formulation variant: “

∀ integer n > 17 ⇒ M 3,n ≠ M ∅ AND M 3,n contains

at least one line with all 3 elements (GIPs) distinct from each other” b.

Type B (neutral) formulation variant: “

∀ integer n > 17 ⇒ those M 3,n which are

≠ M ∅ will contain at least one line with all 3 elements (GIPs) distinct from each other” 5.

“

∀ odd integer n > 5 , n is the sum of a prime and a doubled prime [which is twice of any

prime].” (Lemoine’s conjecture [LC] [26,27] which was erroneously attributed by MathWorld to Levy H. who pondered it in 1963 [28,29]. LC is stronger than TGC, but weaker than BGC. LC also has an extension formulated by Kiltinen J. and Young P. (alias the "refined Lemoine conjecture" [30]), which is stronger than LC, but weaker than BGC and won’t be discussed in this article (as this paper mainly focuses on those GLCs stronger than BGC). LC can be formulated using (ternary, not binary) M 3,n (based on GIPs with 3 elements) such as: a.

Type A formulation variant: “

∀ odd integer n > 5 ⇒ M 3,n ≠ M ∅ AND M 3,n

contains at least one line with at least 2 identical elements (GIPs)” b.

Type B (neutral) formulation variant: “

∀ odd integer n > 5 ⇒ those M 3,n which are

≠ M ∅ will contain at least one line with at least 2 identical elements (GIPs)” 6.

There are also a few original conjectures on partitions of integers as summations of primes published by Smarandache F. [31] that won’t be discussed in this article, as these conjectures depart from VBGC (as VBGC presentation is the main purpose of this article).

There are also a number of (relative recently discovered) GLCs stronger than BGC (and implicitly stronger than TGC), that can also be synthesized using M p,n concept: these stronger GLCs (as VBGC also is) are tools that can inspire new strategies of finding a formal proof for BGC, as I shall try to argue next. Additionally, there are some arguments that Twin Prime Conjecture (TPC) [32] (which states that “there is an infinite number of twin prime (p) pairs of form

( p, p + 2 ) “ ) may be also (indirectly) related to BGC as

part of a more extended and profound conjecture [33,34,35], so that any new clue for BGC formal proof may also help in TPC (formal) demonstration. Moreover, TPC may be weaker (and possibly easier to proof) than BGC (at least regarding the efforts towards the final formal proof) as the superior limit of the primes gap was recently “pushed“ to be ≤246 [36], but the Chen's Theorem I (that ”every sufficiently large even number can be written as the sum of either 2 primes, OR a prime and a semiprime [the product of just 2 primes]” [37,38,39] ) has not been improved since a long time (at least by the set of proofs that are accepted

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in the present by the mainstream) except Cai’s new proved theorem published in 2002 (“There exists a natural number N such that every even integer n larger than N is a sum of a prime ≤ n0.95 and a semi-prime” [40,41] , a theorem which is a similar but a weaker statement than LC that hasn’t a formal proof yet). 1.

The Goldbach-Knjzek conjecture [GKC]

[42] (which is stronger than BGC):

“

∀ even integer n > 4 , there is at least one prime number p [so that] n < p ≤ n / 2 and q = n − p is also prime [with n = p + q implicitly]”. GKC can also be reformulated as: “every even integer n > 4 is the sum of at least one pair of primes with at least one element in the semin , n / 2  ”. GKC can be also formulated using (binary) M 2,n (based on GIPs open interval

(

with 2 elements) such as: Type A formulation variant: “

M 2,n ( M n ) b.

contains at least one line with at least one element

Type B (neutral) formulation variant: “

Px ∈

(

x , so that Px ∈ ( n , n / 2  .” 

∀ even integer n > 4 ⇒ those M 2,n ( M n )

≠ M ∅ will contain at least one line with at least one element x , so that

which are

∀ even integer n > 4 ⇒ M 2,n ( M n ) ≠ M∅ AND

n , n / 2  .”

The Goldbach-Knjzek-Rivera conjecture [GKRC] [43] (which is obviously stronger than BGC, but also stronger than GKC for number

[so that]

n ≥ 64 ): “ ∀ even integer n > 4

n < p < 4 n and q = n − p is also prime [with n = p + q

implicitly]”. GKRC can also be reformulated as: “

∀ even integer n > 4 , n is the sum of at

least one pair of primes with one element in the double-open interval formulated using (binary) a.

)

n , 4 n ”. GKRC can be

∀ even integer n > 4 ⇒ M 2,n ( M n ) ≠ M ∅ AND

contains at least one line with one element

Type B (neutral) formulation variant: “ which are

(

M 2,n (based on GIPs with 2 elements) such as:

Type A formulation variant: “

M 2,n ( M n )

, there is at least one prime

x , so that Px ∈

(

)

n , 4 n .”

∀ even integer n > 4 ⇒ those M 2,n ( M n )

≠ M ∅ will contain at least one line with one element x , so that Px ∈ ( n , 4 n ) .”

Any other GLC that establishes an additional inferior limit

a>0

for

r ( 2, n ) so that

r ( 2, n ) ≥ a > 0 (like Woon’s GLC [44]) can also be considered stronger that BGC, as BGC only suggests

r ( 2, n ) > 0 for any even integer n > 6 (which implies a greater average number of n than the more selective Woon’s GLC does).

GIPs per each

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There is also a remarkable set of original conjectures (many of them stronger than BGC and/or TPC) originally proposed by Sun Zhi-Wei [1,2] [45,46], a set from which I shall cite [3] (by rephrasing) some of those conjectures that have an important element in common with the first special case of VBGC: the recursive PP function in which x

next cited conjectures as 1.

is the x-th prime and PP is the x

Px -th prime (which is denoted in the

Pq which is the q-th prime, with q being also a prime number).

Conjecture 3.1 (Unification of GC and TPC, 29 Jan. 2014). For any integer

)

(

n>2

there is at

 (1 < q < 2n − 1) , ( 2n − q ) , P  q+2 + 2  (Sun’s Conjecture 3.1  8 [SC3.1 or U-GC-TPC], which is obviously stronger than BGC and was tested up to n = 2 × 10 )

least one triad of primes

Conjecture 3.2 (Super TPC [SPTC], 5 Feb. 2014). For any integer

(

)(

)

n>2

there is at least one

( 0 < k < n ) , P + 2 = prime , P  Pn −k + 2 = prime  (Sun’s Conjecture 3.2 [SC3.2 k  9 or SPTC], which is obviously stronger than TPC and was tested up to n = 10 ) [4,5]

triad

Conjecture 3.3 (28 Jan. 2014). For any integer

n>2

(

there is at least one pentad

)(

)

 ( 0 < k < n − 1) , ( 6k − 1 = prime ) , ( 6k + 1 = prime ) , P  n −k = prime , Pn −k + 2 = prime   (Sun’s Conjecture 3.3 [SC3.3], which is obviously stronger than TPC as it implies TPC; SC3.3 was tested up to n = 2 × 10 ) 7

Conjecture 3.7-i (1 Dec. 2013). There are infinite many positive even integers associated with a hexad of

n>3

which are primes

 ( n + 1) , ( n − 1) , ( Pn + n ) , ( Pn − n ) , ( nPn + 1) , ( nPn − 1) (Sun’s Conjecture 3.7-1 [SC3.7i], which is obviously stronger than TPC as it implies TPC; n = 22 110 is the first/smallest value

of 5.

predicted by SC3.7-I)

Conjecture 3.12-i (5 Dec. 2013). All positive integers

(

)

n>7

have at least one associated pair

( k < n − 1) , 2 + P = prime  (Sun’s Conjecture 3.12-i [SC3.12-i]) n −k   6.

Conjecture 3.12-ii (6 Dec. 2013). All positive integers

(

)

n>3

have at least one associated pair

( k < n − 1) , k !+ P = prime  (Sun’s Conjecture 3.12-ii [SC3.12-ii]) n −k   7.

Remark 3.19 (which is an implication of the Conjecture 3.19 not cited in this article). There is an infinite number of triads of primes

(

)

( q > 1) , r = Pq − q + 1 , ( Pr − r + 1)  (Sun’s Remark  

on Sun’s Conjecture 3.19 [SRC3.19]) [1] Wikipedia page about Sun Zhi-Wei: https://en.wikipedia.org/wiki/Sun_Zhiwei [2] The personal page of Sun Zhi-Wei: http://maths.nju.edu.cn/~zwsun/ [3] See also Sun’s Z-W. personal web page on which all conjectures are presented in detail. URL: http://math.nju.edu.cn/~zwsun [4] See also the first announcement of this conjecture made by Sun Z-W. himself on 6 Feb 2014). URL: https://listserv.nodak.edu/cgibin/wa.exe?A2=NMBRTHRY;b81b9aa9.1402 [5] See also the sequence A218829 on OEIS.org proposed by Sun Z-W. URLs: http://oeis.org/A218829; http://oeis.org/A218829/graph;

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

Conjecture 3.21-i (6 Mar. 2014). For any integer

(

n > 5 there will always exist at least one triad

)

( 0 < k < n ) , ( 2k + 1 = prime ) , P + k ⋅ n = prime  (Sun’s Conjecture 3.21-i [SC3.21k⋅n   i]) 9.

Conjecture 3.23-i (1 Feb. 2014). For any integer

(

)

n > 13 there is at least one triad of primes

(1 < q < n ) , ( q + 2 ) , Pn−q + q + 1  (the Sun’s Conjecture 3.23-i [SC3.23-i]).  

4 The ‘i-primeths’ ( i℘* ) Definition The definition of the generalized “i-primeths” concept ℘* . This paper chooses to use the term “primeth(s)” because this is the shortest and also the most suggestive of all the alternatives [6] used until now (as the “th” suffix includes, by abbreviation, the idea of “index of primes”). “Primeths” were originally defined as a subset of primes with (also) prime indexes (with the numbering of the elements of ℘ * starting

from

P1 = 2 ). As primes are in fact

those positive integers with a prime index (the “prime index” being

non-tautological defined as a positive integer >1 that has only 2 distinct divisors: 1 and itself), all the standard primes may be considered primeths with iteration order i=0 (or shortly: 0-primeths) NOT with i=1 (as Fernandez first considered) (as the i=0 marks the genesis of ℘* from the ordinary N ⊃℘* and cannot be considered an iteration on ℘ * ). This new alternative definition (and notation) of i-primeths ( containing

iP x elements with i ≥ 0 and x ∈ N * ) has three advantages, with an accent strictly on the

number (i) of P-on-P iterations and NOT on the general standard definition (and notation) of iterated functions like 1.

P1 ( x ) ≡ Po1 ( x ) = P ( x )

and

  : Pi ( x ) ≡ Poi ( x ) = P  P.. P( x )  (i −1) nested functions P   

The iteration order i is also the number of (“vertical”) iterations for producing the i-primeths from the 0-primeths

( 0℘* =℘*)

(as in the original primeths definition, the standard primes were

considered 1-primeths not 0-primeths, as if they were produced from but N doesn’t contain just primes, as ℘* ≠ N ); a. 2.

using 1 vertical iteration,

These iterations numbered by order i are easy to follow when implemented in different algorithms using a programming language on a computer; The concept of primes can be generalized as “i-primeths” i℘* , with i℘* also including ℘* as the special case of 0-primeths

( 0℘* =℘*) ⊂ i℘* ;

This definition clearly separates ℘* from the ordinary

℘*

( ℘*) and considering 0

as a

( − 1)

N using 0 (not 1) as a starting order (i) for

℘ * (a “bulky”

( −1)

℘* “contaminated” with composite

positive integers that can be considered “(-1)-primeths” convertible to 0-primeths by different sieves of primes.

[6] Alternative terms for “primeths”: “higher-order prime numbers”, “superprime numbers”, “super-prime numbers”, ”superprimes”, ” superprimes” or “prime-indexed primes[PIPs]”. URL (OEIS page): http://oeis.org/wiki/Higher-order_prime_numbers

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

℘*

inevitably “contains”

by its indexes , in the sense that

℘* contains all the generic

Px elements with indexes x ∈ N * (an index x that “scrolls” all N * ). The same prime may

be part of more than one i-primeths subset i℘* , as

is not necessarily a prime.

This slightly different definition of the i-primeths ( i℘* containing generic

iP x elements with

i ≥ 0 and x ∈ N * , as explained previously) is NOT a new “anomaly” and it was also used by Smarandache F. as cited by Murthy A. [47] and also by Seleacu V. and Bălăcenoiu I. [48]. The elements of the generalized set of i-primeths

i℘* :

℘* =℘* = { 0 P1 ( = P1 = 2 ) , 0 P2 ( = P2 = 3 ) , 0 P3 ( = P3 = 5 ) ,...0 Px ( = Px ) ,...} (alias 0-primeths)

{

} (alias 1-primeths

℘* = 1 P1 ( = Pp1 = P2 = 3 ) , 1P2 ( = Pp2 = P3 = 5 ) ,...1 Px ( = Ppx ) ,...

℘* =

{ P (P 2

pP1

}

)

[7]

)

= P3 = 5 , 2 P2 ( PpP 2 = P5 = 11) ,...2 Px ( = PpPx ) ,... ,

…   ℘* =  i P1 = P  

, iP

P ... P

i iterations of P

P ... P 2 i iterations of P

, ..., i

Px = P

P ... P x i iterations of

  , , ...   P 

with

x ∈ N * −{1,2}

5 The Meta-conjecture VBGC - The Extension and Generalization of BGC as Applied on i-primeths ( i℘*) Meta-conjecture VBGC – main co-statements:

Alternatively

defining

i-primeths

as:

      i P = P  P P ...P x  , with P ( x ) being the x-th prime in the … Px = P  P ( P ( x ) )  ( ( ))  x   [2 iterations  [ ( i ≥0 ) iterations ]   of P on P ]     

(

set of standard primes (usually denoted as P ( x ) or and the generic

    1 Px = P  P ( x )  ,  [1 iteration   of P on P ]   

   , Px = P  x iterations   [0  of P on P] 

)

and equivalent to Px alias “0-primeths”)

iP x being named the generic set of i-primeths (with” i” being

the

“iterative”/recursive order of that i-primeth which measures the number of P-on-P iterations associated with that specific i-primeth subset). The inductive variant of (the meta-conjecture) VBGC (iVBGC) proposed in this paper states that: “All even positive integers

2m ≥ 2 ⋅ fx ( a , b ) AND (also) 2m ≥ 2 ⋅ fx2 ( a , b ) , can be

[7] Primes subset (3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, …), also known as sequence A006450 in OEIS. URL (OEIS page): https://oeis.org/A006450

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

written as the sum of at least one pair of DISTINCT odd i-primeths

( a , b ) , with a ≥ b ≥ 0

positive integers pair

a P > bP x y , with the

defining the (recursive) orders of each of those

i-primeths pair AND the pair of distinct positive integers

( x , y ) , with x >

y > 1 defining

the indexes of each of those i-primeths pair, with

2( a +1)(b+1)( a +b+2)   fx ( a, b ) = 2[( a +1)(b+1)( a +b+3)/a ]−a  ( a +1)(b+1)( a +b+2)−( a +b−2) 2

for ( a = b = 0 ) for ( a = b ) AND ( a > 0) for ( a ≠ b ) AND ( a > 0 ) OR ( b > 0 ) 

and

2(a +1)(b+1)( a +b+2)   fx2 ( a, b ) = 2[( a +1)(b+1)( a +b+3)/a ]−2a  (a +1)(b+1)( a +b+2)−(a +b−2) 2 a.

for ( a = b = 0 ) for ( a = b ) AND ( a > 0 )

.”

for ( a ≠ b ) AND ( a > 0 ) OR ( b > 0) 

A secondary inductive (form of) (the meta-conjecture) VBGC (siVBGC[a,0]) proposed in

2m ≥ 2 ⋅ int  fy ( a )  , with

this paper states that: “All even positive integers

fy ( a ) = e 4a , can be written as the sum of at least one pair of DISTINCT odd iprimeths

a P > 0P x y , with the positive integers pair

(recursive) orders of the i-primeths pair pair 3.

( x , y ) , with x >

( a , 0 ) , with a > 0

defining the

( a P , 0P ) AND the distinct positive integers x

y > 1 defining the indexes of each of those i-primeths.”

The analytical variant of (the meta-conjecture) VBGC (aVBGC) proposed in this paper (from which the previous inductive VBGC was derived) states that: “For any pair of finite positive integers

( a , b ) , with a ≥ b ≥ 0

( bP) ,

primeth respectively

(na,b = nb,a ) ≥ 3

defining the (recursive) orders of an a-primeth

there will always exist a single finite positive integer

so that, for any positive integer

one pair of finite distinct positive integers

primeths)

a P + b P = 2m x y

that:

(

( a P ) and a b-

m > na,b it will always exist at least

( x , y ) , with x > and

y > 1 (indexes of distinct odd i-

a P > bP x y

and

the

function

)

f ( a , b ) = f ( b, a ) = na ,b = nb,a ≥ 3 has a finite positive integer value for any

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

combination of finite positive integers any

( a, b )

, without any catastrophic-like infinities for

pair of finites positive integers.“

Important note. I have chosen the additional conditions

(a ≥ b ≥ 0) ∧ ( x >

y > 1) ⇔

a P > bP x y so that to lower the no. of lines per each Goldbach Matrix (GM) and to simplify the algorithm of searching

( a P , bP ) pairs, as the set a P is much less dense that the set bP x

a a for a > b AND the sieve using P (which searches an P starting from 2 m to 3 ) finds a a P , bP b x y pair much more quicker than a sieve using P (if a > b ).

(

)

( )

f ( 0, 0 ) = n0,0 = 3

f (1, 0 ) = f ( 0,1) = n1,0 = n0,1 = 3 ; f (1, 0 ) < f ( 2, 0 ) = 2564 ;

( ) f ( 2, 0 ) = f ( 0, 2 ) = ( n2,0 = n0,2 ) = 2 564 ; f ( 2,0) <

f (1,1) ; the conjectured sequence

of all even integers that cannot be expressed as the sum of two distinct 2-primeth and 0-primeth

2 P > 0P [8] x y was also submitted to OEIS, reviewed and approved as A282251 e.

( )

f (1,1) = n1,1 = 40 306 ;

f (1,1) > f ( 2, 0 ) = 2564 ,

also

predicted

fx (1,1) > fx ( 2, 0 ) ; f.

(

)

f ( 3, 0 ) = f ( 0, 3) = n3,0 = n0,3 = 125 771 ; predicted by

f ( 3, 0 ) > f ( 2, 0 ) = 2564 , as also

fx ( 3,0 ) > fx ( 2,0 ) ; f ( 3, 0 ) < f ( 2,1) = 1 765 126 , as also predicted by

fx ( 3,0 ) < fx ( 2,1) ; f ( 3, 0 ) > f (1,1) =40 306 , as also predicted by fx ( 3,0 ) > fx (1,1) g.

(

)

f ( 2,1) = f (1, 2 ) = n2,1 = n1,2 = 1 765 126 ; f ( 2,1) > f ( 3, 0 ) = 125 771 , as also predicted by

fx ( 2,1) > fx ( 3, 0 ) ; f ( 2,1) < f ( 2, 2 ) = 161 352 166 , as also predicted by

fx ( 2,1) < fx ( 2, 2 ) ; h.

(

)

f ( 4, 0 ) = f ( 0, 4 ) = n4,0 = n0,4 = 6 204 163 ; f ( 4,0) > f ( 3, 0 ) = 125 771 , as also predicted

predicted by

fx ( 4,0 ) > fx ( 3,0 ) ; f ( 4,0 ) < f ( 2, 2 ) = 161 352 166 ,

which

also

fx ( 4, 0 ) < fx ( 2, 2 ) ;

[8] Official page at URL: https://oeis.org/A282251; Complete review at URL: https://oeis.org/draft/A282251; Review history at URL: https://oeis.org/history?seq=A282251

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

(

)

f ( 3,1) = f (1,3) = n3,1 = n1,3 = 32 050 472 ; f ( 3,1) > f ( 2,1) = 1 765 126 , as also predicted by

fx ( 3,1) > fx ( 2,1) ; f ( 3,1) > f ( 4, 0 ) = 6 204 163 as also predicted by

fx ( 3,1) > fx ( 4, 0 ) ;

fx ( 3,1) > fx ( 2, 2 )

erroneously

predicts

that

fx ( 3,1) ≅ 7.04 × 10 overestimates the computed f ( 3,1) ;

f ( 3,1) ;

( )

f ( 2, 2 ) = n2,2 = 161 352 166 ; f ( 2,2 ) > f ( 2,1) = 1 765 126 , as also predicted by

fx ( 2, 2 ) > fx ( 2,1) ;

f ( 2, 2 )

f ( 4, 0 ) = 6 204 163 ,

also

predicted

fx ( 2, 2 ) > fx ( 4,0 ) ; f ( 2, 2 ) < f ( 5,0 ) = 260 535 479 , which is also predicted by fx ( 2, 2 ) < fx ( 5,0 ) ; k.

(

)

f ( 5, 0 ) = f ( 0,5 ) = n5,0 = n0,5 = 260 535 479 ; f ( 5, 0 ) > f ( 4, 0 ) = 6 204 163 , as also predicted by

fx ( 5,0 ) > fx ( 4,0 ) ; however, fx ( 5, 0 ) ≅ 5.5 × 1011 overestimates

f ( 5, 0 ) over 2 m = 1010 , as also in the case of fx ( 3,1) ≅ 7.04 × 1013 overestimating f ( 3,1) ;

(

)

f ( 4,1) = f (1, 4 ) = n4,1 = n1,4 = ? (computing in progress); f ( 4,1) is expected to be smaller than

f ( 3, 2 ) according to the prediction fx ( 4,1) < fx ( 3, 2 ) ; obviously f ( 4,1) is

expected to be larger than

f ( 3,1) as also according to the prediction fx ( 4,1) > fx ( 3,1) ;

f ( 4,1) is ALSO expected to be larger than f ( 3, 3) as according to the prediction

fx ( 4,1) > fx ( 3,3) ; however, fx ( 4,1) ≅ 1.5 × 1020 surely overestimates f ( 4,1) ;

(

surely overestimates n.

)

f ( 3, 2 ) = f ( 2,3) = n3,2 = n2,3 = ? (computing in progress);

fx ( 3, 2 ) ≅ 2.4 × 1024

f ( 3, 2 ) ;

( )

f ( 3, 3 ) = n3,3 = ? (computing in progress); fx ( 3, 3) ≅ 3.5 × 1013 surely overestimates

f ( 3, 3) over 2 m = 1010 ; o.

…[working progress on other higher indexes function values]

The 2D matrix/array of the finite values

f ( a , b ) is a conjectured meta-sequence of integers

and was also proposed to OEIS, BUT rejected in the meantime, with the main argument that OEIS doesn’t accept conjectured meta-sequences (the sequence of f values was considered “too ambitious”) and that it hadn’t an “appropriate form”, although OEIS doesn’t mention this (main) exclusion-criterion (applied to VBGC f[a,b] meta-sequence) explicitly in their publishing policy [9] .

[9] Review history at URL: https://oeis.org/history?seq=A281929&start=50 (last page URL); Review history in pdf downloadable format at URL: http://dragoii.com/VBGC_A281929_OEIS_rejection_history.pdf

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

Interestingly,

f (a, b)

applied on

a ∈ [0,5]

and

b∈ [0,5]

( 3 ) , ( 2 564 ) , ( 40 306 ) , (125 771) , (1 765 126 ) ,  F=  ( 6 204 163 ) , ( 32 050 472 ) , (161 352 166 ) , ( 260 535 479 ) 

(relatively)

compact

has its values in the set

which

has

pattern

exponential

such

≅

EF = {(1.1) , ( 7.8 ) , (10.6 ) , ( 11.7 ) , (14.4 ) , ( 15.6 ) , (17.3 ) , (18.9 ) , (19.4 )} ,

with

as: a

relatively

1.2 ± 0.15 ) between its last 7 elements so that

constant geometric progression (of about

( 19.4 / 18.9) ≅ ( 18.9 / 17.3) ≅ ( 17.3 / 15.6) ≅ ( ≅15.6 / ≅14.4 ) ≅   ≅ (1.2 ± 0.15) ( ≅14.4 / ≅11.7 ) ≅ ( ≅11.7 / ≅10.6) ≅ ( ≅10.6/ ≅ 7.8)    . ≅ ≅ The single exception of this rule is the gap between the exponents 1.1 and 7.8 . See the next figures. ≅

≅

f(a,b)=f(b,a)

f(a,b)=f(b,a) 1.80E+09 1.60E+09 1.40E+09 1.20E+09 1.00E+09 8.00E+08 6.00E+08 4.00E+08 2.00E+08 0.00E+00

Expon. (f(a,b)=f(b,a))

y = 14.9e2.0537x R² = 0.9138

Fig. D-1. The exponential pattern-1 of the

f ( a, b ) values for a ∈ [ 0,5] and b∈ [0,5] ln[f(a,b)]

ln[f(a,b)]

Linear (ln[f(a,b)])

25 20

y = 2.0537x + 2.7013 R² = 0.9138

15 10 5 0 0

Fig. D-2. The exponential pattern-2 of the

f ( a, b ) values for a ∈ [ 0,5] and b∈ [0,5]

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

( 3 ) , ( 2 564 ) , ( 40 306 ) , (125 771) , (1 765 126 ) ,  has ALSO a correspondent matrix in F=  , 6 204 163 , 32 050 472 161 352 166 , 260 535 479 ( ) ( ) ( ) ( )  

which

is a column index and

( )

b is a line index

(

) ( ) ( ) ( n4,0 = n0,4 ) ) ( ) ( ) ( ) ( n4,1 = n1,4 ) ) ( ) ( ) ( ) ( n4,2 = n2,4 ) ) ( ) ( ) ( ) ( n4,3 = n3,4 ) ) ( ) ( ) ( ) ( n4,4 ) ) ( ) ( ) ( ) ( n4,5 = n5,4 )

 n0,0 n1,0 = n0,1 n2,0 = n0,2 n3,0 = n0,3   n =n n1,1 n2,1 = n1,2 n3,1 = n1,3  0,1 1,0  n =n n1,2 = n2,1 n2,2 n3,2 = n2,3 0,2 2,0 M = f (a,b)  n = n n1,3 = n3,1 n2,3 = n3,2 n3,3  0,3 3,0  n =n n1,4 = n4,1 n2,4 = n4,2 n3,4 = n4,3  0,4 4,0  n1,5 = n5,1 n2,5 = n5,2 n3,5 = n5,3  n0,5 = n5,0  ( 3) ( 3) ( 2564) (125 771)  3 40 306 1 765 126 32 ( ) ( ) ( ) ( 050 472)   ( 2564) (1 765 126) (161 352 166) ( ?) = ( 32 050 472) ( ?) ( ?)  (125 771)  ( 6 204 163) ( ?) ( ?) ( ?)  ( ?) ( ?) ( ?) ( 260 535 479)

( ( ( ( (

matrix of exponents in which

     ME = LN  f (a,b)      

( n0,0 )

( n0,1 = n1,0 ) ( n0,2 = n2,0 ) ( n0,3 = n3,0 ) ( n0,4 = n4,0 ) ( n0,5 = n5,0 )

 (1.1)   (1.1) ≅  ( 7.85 ) ME = f (a ,b)  (11.74 )   (15.64 )   (19.38 ) b.

f (a ,b)

( n1,0 = n0,1) ( n1,1) ( n1,2 = n2,1) ( n1,3 = n3,1) ( n1,4 = n4,1) ( n1,5 = n5,1)

( 6 204 163) ( 260 535 479)  ( ?) ( ?)   ( ?) ( ?)  ( ?) ( ?)   ( ?) ( ?)  ( ?) ( ?) 

is also a column index and

( n2,0 = n0,2 ) ( n2,1 = n1,2 ) ( n2,2 ) ( n2,3 = n3,2 ) ( n2,4 = n4,2 ) ( n2,5 = n5,2 )

( n5,0 = n0,5 ) ( n5,1 = n1,5 )  ( n5,2 = n2,5 ) ( n5,3 = n3,5 )  ( n5,4 = n4,5 ) ( n5,5 ) 

( n3,0 = n0,3) ( n3,1 = n1,3 ) ( n3,2 = n2,3 ) ( n3,3 ) ( n3,4 = n4,3 ) ( n3,5 = n5,3 )

and a

b is also a line index

( n4,0 = n0,4 ) ( n4,1 = n1,4 ) ( n4,2 = n2,4 ) ( n4,3 = n3,4 ) ( n4,4 ) ( n4,5 = n5,4 )

( n5,0 = n0,5 ) ( n5,1 = n1,5 )  ( n5,2 = n2,5 ) , ( n5,3 = n3,5 )  ( n5,4 = n4,5 ) ( n5,5 ) 

(1.1) ( 7.85 ) (11.74 ) (15.64 ) (19.38 ) (10.6 ) (14.38 ) (17.28 ) ( ? ) ( ? )  (14.38 ) (18.9 ) (?) (?) (?)  .  (17.28 ) ( ? ) (?) (?) (?)  (? ) (?) (?) (?) (?)   (? ) (?) (?) (?) ( ? )  can be graphed as a 2D surface in a 3D space: see the next figure.

Drăgoi; JAMCS, 25(2): 1-32, 2017;; Article no.JAMCS.36895 no.

ME_f(a,b) 20 15 10 5 0

Fig. D-3. 3. The 3D graph of ME c.

f ( a,b)

cell values for a ∈ [ 0,5] and b ∈ [0,5]

The previous matrices generate half half-dome-like like graphs, apparently with no closed “depression” regions, as all elements tend to become greater when: moving on the lines from left to right, moving on the columns from up to down, moving on the diagonals, from sides to the center. The exponents from each column of ME tend to grow almost linearly from up to down (but also f (a ,b)

on diagonals, from left to center-right center and vice versa): see the next figure. IndexCol Col_1 Col_4

ME_f(a,b)

Col_0 Col_2 Col_5

25 20 15 10 5 0 1

Fig. D-4. The almost linear growth of

f ( a,b )

a ∈ [ 0,5]

cell values on lines, columns and diagonals for

and

b∈ [0,5]

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

fx ( a , b ) = fx ( b, a ) (proposed in the inductive variant

Furthermore, the symmetrical function

of VBGC) generates positive integers that are relatively close BUT strictly larger than the values of e.

f ( a, b ) for a ∈ [0,5] and b∈ [ 0,5] , with also a half-dome-like graph.

The function

 (4)   (128 )  ≅  ( 4 096 ) M = fx(a ,b)  5.2 ×10 5 ( )  8  ( 2.7 ×10 )  ( 5.5 ×1011 )

element

(128 )

( 4 096 )

( 5.2 ×10 ) ( 5.4 ×10 ( 5.4 ×10 ) ( 7.6 ×10 (...) ( 7 ×10 ) 5

( 5.2 ×10 ) ( 2.7 ×10 ) ( 5.5 ×10 ) (...) (...)  ) ( 7 ×10 )  (...) (...)  ) (...)  5

(...) (...)

(...) (...) (...)

(...) (...)

larger

than

( 3) ( 40 306)

strictly

 ( 3)  ( 3)   ( 2564) M = f (a,b)  (125 771)   ( 6 204 163)  ( 260 535 479)

fx ( a, b ) has its values in the matrix

its

(...) (...) (...)

correspondent

element

which

each

    

from

f (a,b)

( 2564) (1 765 126)

(125 771) ( 6 204 163) ( 260 535 479)  ( 32 050 272) ( ?) ( ?)   (1 765 126) (161 352 166) ( ?) ( ?) ( ?)  ( 32 050 272) ( ?) ( ?) ( ?) ( ?)   ( ?) ( ?) ( ?) ( ?) ( ?)  ( ?) ( ?) ( ?) ( ?) ( ?) 

Additionally, the function

 2( a +1)(b+1)(a +b+2)   fx2 ( a, b ) =  2[( a +1)(b+1)( a +b+3)/a ]−2a  ( a +1)(b+1)(a +b+2)−( a +b−2)  2

for ( a = b = 0) for ( a = b ) AND ( a > 0 )

has

for ( a ≠ b ) AND ( a > 0) OR ( b > 0 ) 

strictly larger but even more closer values to the values of

f (a ,b)

, BUT predicts distorted inequalities

between some terms (when compared to the inequalities between some elements of

 ( 4)   (128 )  ≅  ( 4 096 ) M = fx2 (a ,b)  5.2 ×105 ( )   ( 2.7 ×108 )  ( 5.5 ×1011 )

( 5.2 ×10 ) ( 2.7 ×10 ) ( 5.5 ×10 ) (...) (...)  ( 2.6 ×10 ) ( 5.4 ×10 ) ( 7 ×10 )  (...) (...)  ( 5.4 ×10 ) (1.8 ×10 ) (...)  (...) (...) (...) (...)  ( 7 ×10 ) (128 )

( 4 096 )

f (a ,b)

(...) (...)

   

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

The first line of

f (a ,b)

(which is identical to its first column), has sufficiently many terms

to create a function that reasonably approximates the elements on this first line/column, such as:

fey ( a ) = 4a = [ 0 4 8 12 16 20 ] ,

with

≅

fy ( a ) = e fey ( a ) = e 4 a = 1 54.6 2981 162754.8 8.8 ×106 which

 (1.1)  ( 3 )

very

(with

larger

values)

(1.1) ( 7.85 ) (11.74 ) (15.64 ) (19.38 ) 

and

to the

the

first

line

485 ×106  , of

f (a ,b)

( 3) ( 2564 ) (125 771) ( 6 204 163) ( 260 535 479)  respectively.

fy ( 6) predicts a value for

f ( 6, 0 ) ≅ fy ( 6 ) ≅ 2.65 × 1010

which is beyond the

verification capabilities of our current software: this hypothesis was also verified with our software and confirmed that

f ( 6,0) is larger than the limit 2 m = 1010 . The exception of

VBGC(6,0) smaller-and-closest to 2 m = 10

ii. iii.

fy ( 7 ) predicts a value for

is 9 997 202 434 = 2 × 4 998 601 217

f ( 7, 0 ) ≅ fy ( 7 ) ≅ 1.45 × 1012 which is far beyond the

verification capabilities of our current software. On the second line/column of ME (1.1)

f (a ,b)

(10.6) (14.38) (?) (?) (?) ,

the <

elements may also grow in a arithmetical progression with an (exponential) step

s≅4

(starting from Ef (1,1) ≅ 10.6 to Ef ( 2,1) ≅ 14.38 ≅ 10.6 + s ), with the exception of a first gap between Ef ( 0,1) ≅ 1.1 and Ef (1,1) ≅ 10.6 , which is correspondent to the gap between

Ef (1,0) ≅ 1.1 and Ef ( 2,0) ≅ 7.85 . As observed, the step s ≅ 4 is conserved on all lines, columns and secondary diagonals, so that the main diagonal probably has a step of

2s ≅ 8 . 1.

The 5th unknown element

Ef ( 4,1) = (? ) from the 2nd line may have a value of

Ef ( 4,1) ≅  ef ( 2,1) + 2 s = 14.38 + 8 ≅ 22.38 as predicted by the same step s ≅ 4 . An ?

Ef ( 4,1) ≅ 22.38

corresponds

hypothetical

f ( 4,1) = e Ef (4,1) ≅ e22.38 ≅ 5 242 162 809 ≅ 5.2 × 109  which is ALSO under the limit 2 m = 1010 and may also be (relatively) verified with our software. However, as ?

f ( 4,1) ≅ 5.2 × 109 is probably very close to the limit 2 m = 1010 , the conjecture VBGC[4,1] may not be testified by a “sufficiently” large gap)

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

Other values of

f ( a, b ) which are predicted to be under the limit 2 m = 1010 are: ?

 f (1, 4 ) = f ( 4,1)  ≅ 5.2×109 ,  f ( 3,2 ) = f ( 2,3)  ≅ ( 5.2 to 8.8 ) ×109 : all the predicted values are very close to the limit 2 m = 10 , so that the conjectures VBGC[a,b] (corresponding to those predicted values) may not be testified by a “sufficiently” large gap. See the next table. 10

Table T-1. The verified values of f ( a , b ) , with a ≥ b ≥ 0 (written as exact positive integers: the shaded cells of the table) and the estimated maximum values of f ( a , b ) using the step s ≅ 4 “rule” (written in exponential format)

f ( a, b ) 0 1 2 3 4 5 6 7

3 3 2,564 125,771 6,204,163 260,535,479 1.4E+10 7.8E+11

3 40,306 1,765,126 32,050,472 5.2E+09 2.9E+11 1.6E+13 8.5E+14

2,564 1,765,126 161,352,166 8.8E+09 4.8E+11 2.6E+13 1.4E+15 7.8E+16

125,771 32,050,472 5.2E+09 2.9E+11 1.6E+13 8.5E+14 4.7E+16 2.5E+18

6,204,163 5.2E+09 2.9E+11 1.6E+13 8.5E+14 4.7E+16 2.5E+18 1.4E+20

260,535,479 2.9E+11 1.6E+13 8.5E+14 4.7E+16 2.5E+18 1.4E+20 7.6E+21

1.4E+10 1.6E+13 8.5E+14 4.7E+16 2.5E+18 1.4E+20 7.6E+21 4.1E+23

7.8E+11 8.5E+14 4.7E+16 2.5E+18 1.4E+20 7.6E+21 4.1E+23 2.3E+25

fy ( a ) = e fey ( a ) = e 4 a predicts so accurately the first line of M

f (a ,b)

, so that this paper also

proposes a secondary inductive (form of) VBGC (siVBGC[a,0]) which states that: “Any/every even positive integer 2m ≥ 2 ⋅ int  fy ( a )  , with

fy ( a ) = e 4a , can be written

as the sum of at least one pair of DISTINCT odd i-primeths a Px > 0 Py , with the positive integers pair

( a , 0 ) , with a > 0

defining the (recursive) orders of the i-primeths pair

( a P , 0P ) AND the distinct positive integers pair x

( x , y ) , with x >

indexes of each of those i-primeths.”. The set of conjectures siVBGC(a,0) can be used to verify much more rapidly (cost/time-

efficiently) ntBGC, by searching using only the subsets

( 4a )

closest to 2m ≥ 2 ⋅ int e i.

y > 1 defining the

down to

a P , starting from a P which is x

(

a P and testing the primality of 2m − a P x 2

)

Interestingly, the differences between consecutive elements on any line or column of  (1.1) (1.1) ( 7.85 ) (11.74 ) (15.64 ) (19.38 )   1.1 10.6 ) (14.38 ) (17.28 ) ( ) ( (?) ( ? )   ≅  ( 7.85 ) (14.38 ) (18.9 ) (?) (?) ( ? )  have a 1st or a 2nd value that ME =  f (a,b)  (11.74 ) (17.28 ) (?) (?) (?) (?)    (15.64 ) (?) (?) (?) (?) (?)    (?) (?) (?) (?) ( ? )   (19.38 )

is slightly above s ≅ 4 , with all the other values (the 2nd /3rd, the 4th etc) being smaller or approximately equal to s ≅ 4 : see the next tables and graphs.

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

Table and figure TF-2a. The differences between consecutive (known) elements from the lines of ME , with a being the index of a column of ME and b being the index of a line of

f (a ,b)

)

f ( a + 1, b ) −

f (1, b ) −

f ( 2, b ) −

f ( 3, b ) −

f ( 4, b ) −

f ( 5, b ) −

f ( a, b )

f ( 0, b )

f (1, b )

f ( 2, b )

f ( 3, b )

f ( 4, b )

b=0 b =1

6.75

3.89

3.9

3.74

9.51

3.78

2.90

b=2

6.53

4.52

2.38

b=3

<4; <3; <2? <4?; <3?; <2?; <1?

5.54

b=4

<5?; <4? <5?; <4?

<5?; <4? <5?; <4? <4?; <3?; <2?; <1?

<5?; <4? <4?; <3?; <2?; <1?

<4; <3; <2? <4; <3; <2? <4?; <3?; <2?; <1?

b=5 10.00 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00

b=0 b=1 b=2 b=3 f(1,b)-f(0,b) f(2,b)-f(1,b) f(3,b)-f(2,b) f(4,b)-f(3,b) f(5,b)-f(4,b)

Table and figure TF-2b. The differences between consecutive (known) elements from the columns of , with a being the index of a column of ME and b being the index of a line of ME ) ME f (a,b)

f ( a , b + 1) − f ( a, b)

f (a ,b)

a=0

a =1

a=2

f ( a,1) − f ( a,0 )

f ( a , 2 ) − f ( a ,1)

6.75

3.78

4.52

f ( a,3) − f ( a,2 )

3.89

2.90

f ( a , 4 ) − f ( a ,3)

3.9

f ( a,5) − f ( a, 4 )

3.74

<4; <3; <2? <4; <3; <2?

<5; <4?; <3? <5; <4?; <3? <4; <3; <2?

9.51

6.53

f (a ,b)

a=3 5.54 <6; <5?; <6; <5?; <6; <5?; <6; <5?;

<4? <4? <4?

a=4 <6?; <5?; <6?; <5?; <6?; <5?; <6?; <5?;

<4? <4? <4?

a =5 <6?; <5?; <4? <6?; <5?; <4? <6?; <5?; <4?

<4?

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10.00 9.00 8.00 7.00 6.00

a=0

5.00

a=1

4.00

a=2

3.00

a=3

2.00 1.00 0.00 f(a,1)-f(a,0) f(a,2)-f(a,1) f(a,3)-f(a,2) f(a,4)-f(a,3) f(a,5)-f(a,4)

6 Conclusions on VBGC 1.

Essentially, VBGC(a,b) is an extension and generalization of BGC as applied on the generalized concept of all subsets of super-primes of any iteration order i, generically named "i-primeths" in this paper. a. b.

VBGC has an inductive variant and an analytical variant, which both apply to any iprimeth subset. Obviously, VBGC contains the attribute “vertical” it its name motivated by the fact that VBGC is a “vertical” (recursive) generalization of the ntBGC on the infinite superset of iprimeths.

VBGC can be considered a "meta-conjecture", as it essentially (and alternatively) states/predicts an infinite number of BGC-like conjectures (“stronger” than BGC) which are generically indexed as VBGC(a,b), each associated with a pair a. b.

( a, b) ∈ N

and a finite positive integer n

( a,b)

∈N

VBGC(0,0) is equivalent to the non-trivial variant of BGC (ntBGC), as defined in the Section III of this article. VBGC(1,0) is a Goldbach-like Conjecture (GLC) stronger and more elegant than ntBGC, as it acts on a limit 2 f (1, 0 ) = 6 identical to ntBGC inferior limit (which is 2 f ( 0, 0 ) = 6 ) BUT the associated G1,0 ( m ) (which counts the number of pairs of possible GIPs for any even integer

m > 3 ) has significantly smaller values than the function G0,0 ( m ) of

ntBGC [which is VBGC(0,0)] c.

VBGC(2,0) is obviously stronger than VBGC(1,0) with G2,0 ( m ) having smaller non-

(

zero values than G1,1 ( m ) for m ∈ f ( 2,0 ) , ∞ d.

)

VBGC(1,1) (anticipated by my discovery of VBGC(1,0) from 2007 and officially registered in 2012 at OSIM) is obviously stronger than VBGC(1,0 ) and is equivalent to BaylessKlyve-Oliveira e Silva Goldbach-like Conjecture (BKOS-GLC) published in Oct. 2013 [49] alias “Conjecture 9.1” (rephrased) (tested by these authors up to 2 m = 10 ): “all even 9

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

(

)

integers 2m >  2 ⋅ 40306 = 2 f (1,1)  can be expressed as the sum of at least one pair



of prime-indexed primes [PIPs] (1-primeths Px and Py )”. This article of Bayless. Klyve and Oliveira (2012, 2013) was based on a previous article by Barnett and Broughan (published in 2009) [50], but BKOS-GLC was an additional result to this 2009 article. Our new software retested and reconfirmed VBGC(1,1) up to 2 m = 10

verifying VBGC(a,b) for much more ( a , b ) positive integer pairs

[10]

and also helped

VBGC is “much stronger” and general than BGC and proposes a much more rapid and efficient (atleast-one Goldbach index partition [GIP])-sieve than the Goldbach-Knjzek-Rivera conjecture (GKRC). a.

The Goldbach Matrices (GMs) (containing all possible GIPs) generated by VBGC has a smaller nof. lines than the GMs of GIPs generated by GKRC. VBGC is a useful optimized sieve to push forward the limit 4·1018 to which BGC was verified to hold [51,52]. All VBGC(a>0,b≥0) (including siVBGC) can be used to produce more rapid algorithms for the experimental verification of ntBGC for very large positive integers. A first experiment would be to re-test BGC up to that limit 2m = 4 × 10 alternatively using siVBGC and to compare the global times of computing. When verifying ntBGC for a very large number N , one can use the aVBGC(a,b) or siVBGC(a,b) with a minimal positive value for the 18

difference  N − f ( a , b )  . For example, in the case of VBGC(1,0), the average number of attempts to find the first pair (x,y) for each integer m, in the interval [3,2m] tends asymptotically to ln

( n ) = ln(n)/2

when searching just the 1-primeths subset in

descending manner, starting from the largest 1-primeth

( 2m − Px ) is a 0- primeth)

Px ≤ 2m − 1 and verifying if

4. When

Ga,b ( f ( a, b ) + 1) →1

a → ∞, b → ∞ and m → ∞ ,

VBGC(a,b)

tend

narrow

≥

progressively

( a2 , b2 ) , with a2 > a1 and b2 > b1 .

for

each

pair

and of

the

“comets”

positive

integers

VBGC is a potential important (unified) conjecture of primes and a very special self-similar

(i−1)

property of the primes as the rarefied set ℘* is self-similar to the more dense set ℘* in respect to the ntBGC. In other words, each of the i-primeths sets behaves as a “summary of” the 0primeths set in respect to the ntBGC: this is a (quasi)fractal-like BGC-related behavior of the infinite number of the i-primeths sets.

Essentially, VBGC conjectures that ntBGC is a common property of all i-primeths sets (for of each any positive integer order i), differing just by the inferior limit f ( a , b ) = n

(a ,b)

VBGC(a,b). The set of values of f ( a , b ) is a set of critical density thresholds/points of each [10] The code-source (written by Mr. George Anescu in Microsoft Studio 2015 - Visual C++ language/environment using parallel processing) that was used to test BKOS-GLC and VBGC up to n=1010 (on a laptop PC with an IntelR CoreTM processor i7-3630 QM CPU at 2.4 GHz with 4 processors (8 hyper-threads), can be found at this URL: https://drive.google.com/open?id=0Bws5l5MW9z0pY0lpbWJRZmNaMGc; the old variant can be found at this URL: http://dragoii.com/test_primes.rar

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

i-primeths set in respect to the superset of VBGC(a,b) conjectures. All Goldbach comets associated with all known VBGC(a,b) are self-similar to each other and to the entire superset of comets. Batchko R.G. has also reported other quasi-fractal/quasi-self-similar structure in the distribution of the prime-indexed primes [53]: Batchko also used a similar general definition for primes with (recursive) prime indexes (PIPs), briefly named in my article as “i-primeths”. Carlo Cattani and Armando Ciancio also reported a quasi-fractal distribution of primes (including i-primeths) similar to a Cantor set (Cantor dust) by mapping primes and iprimeths into a binary image which visualizes the distribution of i-primeths [54]. VBGC may be an intrinsic property of all sets of i-primeths that can also explain OR be explained by this Cantor dust-like distribution of these i-primeths sets.

(i>0)

Obviously, all sets ℘* are subsets of ℘* =℘* and come in an infinite number: this family of subsets is governed/defined by the Prime Number Theorem. There is a potential infinite number of rules/criterions/theorems to extract an infinite number of subsets from ℘ * (grouped in a family of subsets defined by that specific rule/criterion/theorem: like the Dirichlet's theorem on arithmetic progressions for example). It would be an interesting research subfield of BGC to test what are those families (of subsets of primes)

that respect ntBGC and generate functions with finite values similar to f ( a , b ) = n

a ,b .

This potential future research subfield may also help in optimizing the algorithms used in the present for ntBGC verification on large numbers. It is an interesting fact per se that all

compared to

℘* =℘*

(i>0)℘*

subsets have very low densities (when

and N ) , but these low densities are sufficiently… large to

allow the existence of a function f ( a, b ) with finite values for any pair of finites ( a , b ) . A real challenge in the future (concerning VBGC) is to calculate the values of the function

f ( a , b ) = na ,b and test/verify VBGC(a,b) for large positive integers pairs (a,b), including

the pairs ( a, b ) with relatively large ( a − b ) differences.

7 Potential Applications of VBGC: 1.

As the (weak) Ternary Goldbach Conjecture (TGC) is considered a consequence of BGC, VBGC can be used as a model to also formulate a Vertical (generalization) of the Ternary Goldbach Conjecture (VTGC) as an analogous consequence of VBGC, with a corresponding (potential infinite) meta-sequence of conjectures VTGC(a,b,c) with an associated function

f ( a , b, c ) = na ,b,c .

VBGC can be used to optimize the algorithms of finding very large new primes (i-primeths) smaller but closest possible to a chosen (very large) even number q = 2 m : a.

Step 1. One may choose a set order

a℘*

and a conjecture VBGC(a,b) with positive integer

b chosen so that the known n a,b to be smaller but closest possible to q = 2 m . ( )

Step 2. One may then test only the primality of the differences d x = q − from

( a Px ) (starting

x = 1 to larger positive integer indexes, in ascendant order) which have the potential

to be i–primeths of type

bP . y

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

VBGC can offer a rule of symmetric/asymmetric decomposition of Euclidean/non-Euclidean finite/infinite spaces with a finite (positive integer) number of dimensions d = 2n into pairs of spaces, both with a (positive) i-primeth number of dimensions. According to VBGC, an finite regular Euclidian/non-Euclidean 2n-space with volume V (with n>2) can always be

decomposed

permit

   ( aP ) V2n = k ⋅  V a P × V b   = k ⋅  r x × r   ( x )  Py       a.

symmetry/asymmetry b   Py   

as:

  , with k = space volume specific constant 

In this way, VBGC can also be used in M-Theory to simulate decompositions of 2Nbranes (with finite [positive] integer number of dimensions brane and

such

d = 2n ) into pair of a Px -

b P -brane, both branes with a (positive) i-primeth number of dimensions. y

This type of vertical generalization (generating a meta-conjecture) may be the start of a new research sub-field in which other conjectures may be hypothesized to also have vertical generalizations applied on i-primeths. For example, a hypothetical vertical Polignac's conjecture (a “minus” version of BGC: “for any positive even number n, there are infinitely many prime gaps of size n” or “there are infinitely many cases of two consecutive prime numbers with difference n”) may speed up the searching algorithms to find very large primes (smaller but closest to a chosen positive integer superior limit m ).

Acknowledgements I would like to express all my sincere gratitude and appreciation to all my mathematics, physics, chemistry and medicine teachers for their support and fellowship throughout the years, which provided substantial and profound inner motivation for the redaction and completion of this manuscript. I would also like to emphasize my friendship with George Anescu (physicist and mathematician) who helped me verify VBGC up to n=1010 by creating a much more rapid (by also using parallel programming) and robust software in Visual C++ for this purpose, as an alternative to my first (relatively slow) software created in Visual Basic. My special thanks to professor Toma Albu[11] who had the patience to read and listen my weak voice in mathematics. Also my sincere gratitude to professor Şerban-Valentin Strătilă [12] who adviced me on the first special case of VBGC discovered in 2007 and he urged me to look for a more general conjecture based on my first observation.

Competing Interests Author has declared that no competing interests exist.

References [1]

Weisstein EW. Goldbach Conjecture”, web article From MathWorld—A Wolfram Web Resource; 1999-2014. Available: http://mathworld.wolfram.com/GoldbachConjecture.html, http://demonstrations.wolfram.com/GoldbachConjecture/, https://math.dartmouth.edu/~euler/correspondence/letters/OO0765.pdf

[11] The CV of Professor Albu T. is also available online at URL: http://gta.math.unibuc.ro/pages/talbu.html; [12] The CV of Professor Strătilă Ş-V. is also available online at URL: club.infim.ro/public_html/MEMBERS/PAGES/stratila.htm

http://www.humboldt-

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

[2]

Caldwell CK. Goldbach's conjecture (another Prime Pages' Glossary entries); 1999-2015, web article. Available: http://primes.utm.edu/glossary/page.php?sort=goldbachconjecture

[3]

Oliveira e Silva T. Goldbach conjecture verification; 2014, web article. Available: http://sweet.ua.pt/tos/goldbach.html

[4]

Ye JD, Liu C. A study of Goldbach's conjecture and Polignac's conjecture equivalence issues. IACR Cryptology ePrint Archive. 2013;2013. Available: http://eprint.iacr.org/2013/843

[5]

Ye J.D, Liu C. A study of relationship of RSA with Goldbach’s conjecture and its properties; 2014. Available: http://vixra.org/abs/1410.0059; http://vixra.org/pdf/1410.0059v1.pdf

[6]

Liu C. A study of relationship among Goldbach conjecture. Twin Prime and Fibonacci Number; 2015. Available: http://vixra.org/abs/1410.0059; http://vixra.org/pdf/1505.0030v1.pdf

[7]

Liu C, Chang CC, Wu ZP, Ye SL. A study of relationship between rsa public key cryptosystem and Goldbach’s conjecture properties. International Journal of Network Security. 2015;17(4):445-453. Available: http://ijns.jalaxy.com.tw/contents/ijns-v17-n4/ijns-2015-v17-n4-p445-453.pdf

[8]

Sloane NJA, Robert G. Wilson. A007097, Primeth recurrence: a(n+1) = a(n)-th prime (Formerly M0734 [1995])”, web article From The Online Encyclopedia of Integer Sequences (Oeis.org); 1995, 2016-2017. Available: https://oeis.org/A007097

[9]

Fernandez Neil. The exploring primeness project; 1999, Website. Available: http://www.borve.org/primeness/intro.html, http://www.borve.org/primeness/FOP.html; https://oeis.org/A006450/a006450.html, https://oeis.org/wiki/Higher-order_prime_numbers

[10]

Weisstein EW. Prime number”, web article from MathWorld—A wolfram web resource; 1999-2015. Available: http://mathworld.wolfram.com/PrimeNumber.html;

[11]

Wells D. The penguin dictionary of curious and interesting numbers. Middlesex, England: Penguin Books. 1986;31. Available: https://www.scribd.com/doc/190072346/Penguin-Dictionary-of-Curious-and-InterestingNumbers

[12]

Weisstein E.W. Goldbach Partition; 1999-2015. web article Available: http://mathworld.wolfram.com/GoldbachPartition.html); Some JavaScript/ Wolfram Language online calculator of Goldbach partitions can be found. Available: https://plus.maths.org/content/mathematical-mysteries-goldbach-conjecture, http://demonstrations.wolfram.com/GoldbachPartitions, http://wims.unice.fr/wims/wims.cgi?module=tool/number/goldbach.en

[13]

Granville A. Harald Cramér and the distribution of prime numbers” (based on a lecture presented on 24th September 1993 at the Cramér symposium in Stockholm; 1993. Available: https://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

[14]

Granville A. Different approaches to the distribution of primes. Milan Journal of Mathematic. 2009;78:1–25. Available: http://www.dms.umontreal.ca/~andrew/PDF/ItalySurvey.pdf Soundararajan K. The distribution of prime numbers” (ArXiv article); 2006.

[15]

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

Available: http://arxiv.org/abs/math/0606408 [16]

Diamond H.G. Elementary methods in the study of the distribution of prime numbers. Bull. Amer. Math. Soc. (N.S.). 1982;7(3):553-589. Available: http://projecteuclid.org/download/pdf_1/euclid.bams/1183549769

[17]

Liang W, Yan H, Zhi-cheng D. Fractal in the statistics of Goldbach partition. (ArXiv article); 2006. Available: http://arxiv.org/abs/nlin/0601024

[18]

Weisstein EW. Euclid's theorems. Web article from MathWorld—A wolfram web resource; 19992014. Available: http://mathworld.wolfram.com/EuclidsTheorems.html

[19]

Weisstein EW. Prime number theorem. web article from MathWorld—A wolfram web resource; 1999-2014. Available: http://mathworld.wolfram.com/PrimeNumberTheorem.html

[20]

Pakianathan J, Winfree T. Quota complexes, persistent homology and the Goldbach conjecture. 2011;9-10. Available: http://arxiv.org/abs/1104.4324

[21]

Zhang S. Goldbach conjecture and the least prime number in an arithmetic progression. (ArXiv article). 2008;2010:2. Available: http://arxiv.org/abs/0812.4610

[22]

Wikipedia contributors (last update: 2 Aug. 2015). Goldbach's weak conjecture. Wikipedia, The Free Encyclopedia. Available: https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture (As accessed on 13 December 2015)

[23]

Helfgott HA. The ternary Goldbach conjecture is true*; 2013. Available: http://arxiv.org/abs/1312.7748, https://valuevar.wordpress.com/2013/07/02/the-ternary-goldbach-conjecture/, http://aperiodical.com/2013/05/on-equivalent-forms-of-the-weak-goldbach-conjecture/) (*although it still has to go through the formalities of publication, Helfgott's preprint is endorsed and believed to be true by top mathematicians, including the Fields medalist Terence Tao who showed in 2012 that any odd integer is the sum of at most 5 primes, as can be found. Available:https://terrytao.wordpress.com/tag/goldbach-conjecture/, http://arxiv.org/abs/1201.6656)

[24]

Helfgott HA. The ternary Goldbach problem. Snapshots of modern mathematics from Oberwolfach, No. 3/2014; 2014,2015. Available:https://www.mfo.de/math-in-public/snapshots/files/the-ternary-goldbach-problem, http://arxiv.org/abs/1404.2224; http://arxiv.org/abs/1501.05438;

[25]

Platt DJ. Proving Goldbach’s weak conjecture; 2014. Available: http://www.cms.ac.uk/SET/PDF/M-PLATT-751-PSR.pdf

[26]

Lemoine E. L'intermédiare des mathématiciens. 1894;1(1894):179:ibid 3. 1896;151.

[27]

Wikipedia contributors (last update: 25 Nov 2014). “Lemoine's conjecture”, Wikipedia, The Free Encyclopedia. Available: https://en.wikipedia.org/wiki/Lemoine%27s_conjecture (As accessed on 4 Jan 2016): Levy H. On Goldbach's conjecture. Math. Gaz. 1963;47(1963):274.

[28]

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

[29]

Weisstein EW. Levy's conjecture. Web article from MathWorld—A wolfram web resource; 19992014. Available: http://mathworld.wolfram.com/LevysConjecture.html

[30]

Kiltinen J, Young P. Goldbach, Lemoine, and a know/don't know problem. Mathematics Magazine (Mathematical Association of America). 1984;58(4):195–203. Available: http://www.jstor.org/stable/2689513?origin=crossref&seq=1#page_scan_tab_contents, https://www.researchgate.net/publication/265763631_Goldbach_Lemoine_and_a_knowdon%27t_kno w_problem

[31]

Smarandache F. (1999, 2000[republished], 2007[republished]). “Conjectures on partitions of integers as summations of primes. Math Power”, Pima Community College, Tucson, AZ, USA. 1999;5(9):2-4. Available: http://arxiv.org/abs/0707.2098

[32]

Weisstein EW. Twin prime conjecture. Web article from MathWorld—A wolfram web resource; 1999-2014. Available:http://mathworld.wolfram.com/TwinPrimeConjecture.html

[33]

Ikorong GAN. Around the twin primes conjecture and the Goldbach conjecture I. Analele Științifice ale Universității “Al. I. Cuza” din Iași (S.N.), Matematică, Tomul LIII. 2007;f.1. Available:http://www.math.uaic.ro/~annalsmath/pdf-uri anale/F1-2007/Ikorong.pdf

[34]

Ikorong GAN. Playing with the twin primes conjecture and the Goldbach conjecture. Alabama Journal of Mathematics, Spring/Fall. 2008;45-52. Available:http://ajmonline.org/2008/10.pdf

[35]

Gerstein LJ. A reformulation of the Goldbach conjecture. Mathematics Magazine. 1993;66(1):44-45. Available: http://www.ww.ingeniousmathstat.org/sites/default/files/Reformulation-Gerstein20557.pdf

[36]

Polymath DHJ. The "bounded gaps between primes" Polymath project - a retrospective; 2014. Available:http://arxiv.org/abs/1409.8361

[37]

Chen JR. On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao. 1966;11(9):385–386.

[38]

Chen JR. On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Scientia Sinica. 1973;16:157–176.

[39]

Cheng-Dong P, Xia-Xi D, Yuan W. On the representation of every large even integer as a sum of a prime and an almost prime. Scientia Sinica. 1975;XVIII(5):599–610. Available: http://math.scichina.com:8081/sciAe/fileup/PDF/75ya0599.pdf

[40]

Cai YC. Chen's Theorem with Small Primes. Acta Mathematica Sinica. 2002;18(3):597–604. Available:http://actamath.com/Jwk_sxxb_en/EN/abstract/abstract5409.shtml

[41]

Cai YC. On Chen's theorem (II). Journal of Number Theory. 2008;128(5):1336–1357. Available:http://www.sciencedirect.com/science/article/pii/S0022314X07001217

[42]

Rivera C. Conjecture 22. A stronger version of the Goldbach Conjecture (by Mr. Rudolf Knjzek, from Austria)”, web article from Prime Problems & Puzzles; 1999-2001. Available:http://www.primepuzzles.net/conjectures/conj_022.html

[43]

Rivera C. Conjecture 22. A stronger version of the Goldbach Conjecture (by Mr. Rudolf Knjzek, from Austria and narrowed by Rivera C.), web article from Prime Problems & Puzzles; 1999-2001. Available: http://www.primepuzzles.net/conjectures/conj_022.html 29

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

[44]

Woon MSC. On partitions of Goldbach's conjecture; 2000. Available:http://arxiv.org/abs/math/0010027

[45]

Sun ZW. Chapter “Problems on combinatorial properties of primes” (19 pages) in “Number Theory: Plowing and Starring Through High Wave Forms: Proceedings of the 7th China-Japan Seminar” (edited by Kaneko M., Kanemitsu S. and Liu J. ), 2013;2014:169–188. Available:https://books.google.ro/books?id=O9qiCgAAQBAJ, (full book):http://arxiv.org/abs/1402.6641 Sun ZW. Towards the twin prime conjecture. A talk given at: NCTS (Hsinchu, Taiwan, August 6, 2014), Northwest University (Xi’an, October 26, 2014) and at Center for Combinatorics, Nankai University (Tianjin, Nov. 3, 2014); 2014. Available:http://math.nju.edu.cn/~zwsun/TwinPrime.pdf

[46]

[47]

Murthy A. Generalized partitions and new ideas on number theory and smarandache sequences. 2005;91. Available: https://books.google.ro/books?id=dVHzAAAAQBAJ&pg=PA91, (page 181 only):http://www.gallup.unm.edu/~smarandache/SN/ScArt5/ExploringNewIdeas.pdf;

[48]

Seleacu V, Bălăcenoiu I. Smarandache notions. (book), 2000;11:181. Available: https://books.google.ro/books?id=3p7dBAAAQBAJ&pg=PA181

[49]

Bayless J, Klyve D, Oliveira e Silva T. New bounds and computations on prime-indexed primes. 2012;2013:(23 pages article,). Integers: Annual. 2013;13:17. Available:http://www.emis.de/journals/INTEGERS/papers/n43/n43.pdf, https://www.researchgate.net/publication/255711160_New_bounds_and_computations_on_primeindexed_primes; https://books.google.ro/books?id=27LnBQAAQBAJ

[50]

Broughan KA, Ross Barnett A. On the subsequence of primes having prime subscripts. (10 pages), Article 09.2.3 from the Journal of Integer Sequences. 2009;12. Available:https://eudml.org/doc/45537, http://www.emis.de/journals/JIS/VOL12/Broughan/broughan16.pdf, http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Broughan/broughan16.html, http://www.math.waikato.ac.nz/~kab/papers/pp2.pdf

[51]

Oliveira e Silva T. “Goldbach conjecture verification” (web article); 2016. Available: http://sweet.ua.pt/tos/goldbach.html

[52]

Oliveira e Silva T, Herzog S, Pardi S. Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4· 1018”, Article in Mathematics of Computation. 2014;83(288):201. DOI: 10.1090/S0025-5718-2013-02787-1 Available:https://www.researchgate.net/publication/267166169_Empirical_verification_of_the_even_ Goldbach_conjecture_and_computation_of_prime_gaps_up_to_410_18

[53]

Batchko RG. A prime fractal and global quasi-self-similar structure in the distribution of primeindexed primes. ArXiv article; 2014. submitted on 10 May 2014 (v1), last revised 17 May 2014 (this version, v2). Available: http://arxiv.org/abs/1405.2900, https://www.researchgate.net/publication/262263247_A_prime_fractal_and_global_quasi-selfsimilar_structure_in_the_distribution_of_prime-indexed_primes,

DrÄƒgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

[54]

Cattani C, Ciancio A. On the fractal distribution of primes and prime-indexed primes by the binary image analysis. Article in â€?Physica A: Statistical Mechanics and its Applications; 2016. DOI: 10.1016/j.physa.2016.05.013. Available:https://www.researchgate.net/publication/302590480_On_the_fractal_distribution_of_prim es_and_prime-indexed_primes_by_the_binary_image_analysis

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895

Addendum The Short Description of the Software Created and Used to Verify VBGC The software was created in Microsoft Visual C++ and uses parallel programming techniques. At first, it created (and stored on hard-disk) a set of “.bin” files containing all known i-primeths in the double-open

(

)

interval 1,1010 : see the next table. Table E. The files used by the software Set of i-primeths

File storing the set of i-primeths

0-primeths 1-primeths 2-primeths 3-primeths 4-primeths 5-primeths 6-primeths 7-primeths

p1_10000000000.bin p2_10000000000.bin p3_10000000000.bin p4_10000000000.bin p5_10000000000.bin p6_10000000000.bin p7_10000000000.bin p8_10000000000.bin

For every ( a, b ) pair with a

≥ b,

File dimension on hard-disk (non-archived) ~3.55 Gb ~188 Mb ~12 Mb ~900 kb ~86 kb ~11 kb ~2 kb ~1 kb

the soft verified each

(

Number of i-primeths stored in the file … 24,106,415 1,513,371 115,127 10,883 1,323 216 47

)

a P > bP x x from the intersection (less dense)

a℘* ∩ 2,2m ≥ 6 (starting from the a set Px closest to 2m − 1 in descending order): it then verified ( )

(

if the difference 2 m − Px

) is an element in the (more) dense set b℘* by using binary section method.

The soft computed each value of f ( a, b ) (with the additional condition a Px ≠ b Py ⇔ a Px > b Py in at least one Goldbach partition for any m > f ( a, b ) , with a Px + b Py = 2 m ). The

computing

time

for

determining

and

f ( 2,1) = 1 765 126

verifying

and

f ( 2, 2 ) = 161 352 166 was about 30 hours in total. The computing time for determining and verifying f ( 3, 0 ) = 125 771 ,

f ( 4, 0 ) = 6 204 163

and

f ( 5, 0 ) = 260 535 479 was also about 30

hours for each value. The computing time for determining and verifying f ( 3,1 ) = 32 050 472 was a couple of days: no exceptions found between 2 ⋅ f ( 3,1) and 2 m = 10

so that

f ( 3,1) may be a

veritable last exception of VBGC[3,1]. ____________________________________________________________________________________ © 2017 Drăgoi; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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