(VBGC - JAMCS - Original Research Article - 32 pages - 28.10.2017) The "Vertical" Generalization of the Binary Goldbach's Conjecture as Applied on "Iterative" Primes with (Recursive) Prime Indexes (i-primeths) (Journal of Advances in Mathematics and Computer Science [JAMCS] 25(2): 1-32, 2017; Article no.JAMCS.36895; ISSN: 2456-9968)

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 meta-conjecture, 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.

DOI : 10.9734/JAMCS/2017/36895

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

OFFICIAL LINKS OF THIS PUBLISHED (OPEN) PEER-REVIEWED ARTICLE:
http://www.sciencedomain.org/abstract/21625
http://www.sciencedomain.org/issue/3151
http://www.journalrepository.org/media/journals/JAMCS_69/2017/Oct/Andrei2522017JAMCS36895.pdf
http://www.sciencedomain.org/review-history/21625
http://www.sciencedomain.org/metrics/21625

INDEX COPERNICUS:
https://journals.indexcopernicus.com/search/article?articleId=1700660

CROSSREF.ORG meta-data:
https://search.crossref.org/?q=10.9734%2FJAMCS%2F2017%2F36895

OEIS:
https://oeis.org/A282251

VIXRAPEDIA:
https://www.vixrapedia.org/wiki/VBGC

The exceptions of VBGC(2,0) were also approved on OEIS as A282251.
http://oeis.org/A282251
-see also the open review history of A282251 sequence at this URL :
http://oeis.org/history?seq=A282251

The exceptions of VBGC(1,1) were also approved on OEIS as A316460.
http://oeis.org/A316460
-see the full indexed exceptions list, also called "b-file", at this URL:
http://dragoii.com/VBGC(1,1)_ExceptionEvenList_b_file.txt
-see also the open review history of A316460 sequence at this URL :
http://oeis.org/history?seq=A316460

I have mentioned these approved sequences because Mr. N. J. A. Sloane (the founder of OEIS) claims that "OEIS is the equivalent of a top mathematical journal" (see http://oeis.org/history?seq=A316460) (https://www.researchgate.net/profile/N_Sloane)

#DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted_button_id=AQYGGDVDR7KH2

Full text

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)
_____________________________________
*Corresponding author: E-mail: dr.dragoi@yahoo.com;
The “Vertical” Generalization of the Binary Goldbach’s
Conjecture as Applied on “Iterative” Primes with
(Recursive) Prime Indexes (i-primeths)
Andrei-Lucian Drăgoi
1*
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: 21
st
September 2017
Accepted: 19
th
October 2017
Published: 28
th
October 2017
_______________________________________________________________________________
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 meta-
conjecture, 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.
Original Research Article

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895
2
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
0i≥
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, with
ix
P
being the x-th i-primeth, with iteration
order
0i≥
, as noted in this paper and explained later on.
VBGC was discovered in 2007 and perfected until 2017 by using the arrays (
p
S
and
,i p
S
) of Matrices
(M) of Goldbach index-partitions (GIPs) (simple
,
p n
M
and recursive
, ,
i p n
M
, with iteration order
0i≥
, 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
n
P
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
,
p n
M
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
0i≥
) 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 , 1n with n >
can be splitted in at least one Goldbach
partition/pair(GP)corresponding to at least one Goldbach index-partition (GIP) [12].
*
℘

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895
3
Or
(2) Any positive integer
1
n
>
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
p
S
of the Simple Matrix of Goldbach Index-Partitions
(
)
,
p n
M
Definition of
*
℘
and
℘
. We may define the prime subset of
N
as
*
℘ =
( ) ( ) ( )
{
}
1 2 3
2 , 3 , 5 ,..., ,..., ,...
x y
P P P P P P
∞
= = =
, with
, * 0
x y N and x y
∈ < <
, with
(
)
x y
P P
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 2
nd
theorem [18]). The numbering function of primes
(
)
n
P
is a bijection that interconnects
*
℘
with
*
N
so that each element of
*
℘
corresponds to only (just) one
element of
*
N
and vice versa:
(
)
1
1 2
P
↔ =
,
(
)
2
2 3
P
↔ =
, ...,
x
x P
↔
(the x-th prime),
y
y P
↔
(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
stronger than TGC, as TGC is a consequence of BGC), the author of this article also defines 0
1
P
=
(the
unity of all integers, implicitly the unity of all primes) and
℘=
( ) ( ) ( ) ( )
{
}
0 1 2 3
1 , 2 , 3 , 5 ,..., ,..., ,...
x y
P P P P P P P
∞
= = = =
, with , 0
x y N and x y
∈ ≤ <
, although
only
*
℘ =
(
)
{
}
0
1
P=
℘−
shall be used in this paper (as it is used in the mainstream of modern
mathematics).
The 1
st
formulation of BGC. For any even integer
2
n
>
, it will always exist at least one pair of (other
two) integers , *
x y N with x y
∈ ≤
so that
x y
P P n
+ =
, with
(
)
x y
P P
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
(
)
, ,
x y
n P P
: 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
2
n
>
(as BGC claims) may suggest that BGC is profoundly connected to the generic primality (of any
x
P
and
y
P
) and, more specifically, argues that GC is in fact a property of PD in
N
(and a property of
*
℘
as

Drăgoi; JAMCS, 25(2): 1-32, 2017; Article no.JAMCS.36895
4
composed of indexed/numbered elements). The most important property of Primes and PD and is that
or
( )
ln ,
x
P x x forany progressivelylargex≅ ⋅
(which is the alternative [linearithmic] expression of the Prime Number Theorem [19], as if
*℘
is a result
of an apparently random quantized linearithmization of
{ }
* 1N−
so that
( )
ln
n
P n n
→ ⋅
. In
conclusion: For any even integer
2n>
, at least one GIP exists (BGC – 1
st
condensed formulation).
The 2
nd
formulation of BGC using the Matrix of Goldbach index-partitions (M-GIP or M).
[1] Let us consider an infinite string of matrices
{ }
1 2 3
, , ,..., ,...
n
S M M M M M
∞
=
, with each generic
n
M
being composed of lines made by GIPs
(
)
,
x y
, such as:
(
j
is the index of any chosen line of
n
M
,
j
1≥
and
jn
m≤
)
(
n
m
is the total maximum number of j-indexed lines of
n
M
)
(x
n,i
,y
n,i
*N∈
, x
n,i
< x
n,i+1
for
2
n
m≥
,
[ ]
1,
n
i m
∀ ∈
)
[2] Let us also consider the function that counts the lines of any
n
M
, such as:
( )
n
l n m
=
. This function
(that numbers the lines of a GM) is classically named as
( ) ( )
n
r n l n m
= =
(“r” stands for the number
of “rows”).
[3] An empty/null matrix
( )
M
∅
is defined as a matrix with zero rows and/or columns.
Using
S
,
M
,
M
∅
and
( )r n
as previously defined, BGC has two formulations sub-variants:
1.
n
M M
∅
≠
(OR
S
doesn’t contain any
M
∅
) for any even integer
2
n
>
or shortly:
2
n
even integer n M M∅
∀ > ⇔ ≠
(the 2
nd
formulation of BGC – 1
st
sub-variant).
2.
For any even integer
2
n
>
,
( ) 0r n >
or shortly:
2 ( ) 0
even integer n r n
∀ > ⇔ >
(the
2
nd
formulation of BGC – 2
nd
sub-variant).
( ) ( )
ln / ln ,
x x
P x x P x x forx
→ ⋅ ⇔ → →∞
[ ]
, ,
,1 ,1
, ,
, ,
, with , j 1,
n
n j n j
n n
n n
x y n
n j n j
n m n m
x y
x y
M P P n m
x y
 
 
 
 
= + = ∀ ∈
 
 
 
 
 
M M
M M

Dr
ă
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5
The 3
rd
formulation of BGC using the generalization of
S
( )
p
S
and the generalization of
M
(
)
,
p n
M
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:
a)
{ }
2 2,1 2,2 2,3 2 2
, ,
, , ,..., ,...
n
S M M M M M
∞
=
(the generic
2
,n
M
of
2
S
has 2 columns based
on [binary] GIPs
with 2 elements);
b)
{ }
3 3,1 3,2 3,3 3
,
3,
, , ,..., ,...
n
S M M M M M
∞
=
(the generic
3
,n
M
of
3
S
has 3 columns based on
[ternary] GIPs
with 3 elements);
c) …;
d)
{ }
,1 ,2 ,3
, ,
, , ,..., ,...
p p n p
p p p
S M M M M M
∞
=
(the generic
,
p
n
M
of
p
S
has p columns
based on [p-nary] GIPs
with p elements and natural p>3);
e) …,
f)
{
}
,1 ,2 ,3
, ,
, , ,..., ,...
n
S M M M M M
∞ ∞ ∞
∞ ∞ ∞ ∞
=
(the generic
,
M
n
∞
of
S
∞
has potentially
infinite
( )
∞
number of columns based on
nary∞−
GIPs
with a potentially infinite
( )
∞
number of elements)
g) With each generic
,
p
n
M
being composed of
,
p
n
m
lines and p columns made by p-nary GIPs
with p elements, such as:
(
j
is the index of any chosen line of
,p n
M
,
1j≥
and
,p n
j m≤
and
,
p n
m
is the total maximum number of j-indexed lines of
,
p n
M
)
(
k
is the index of any chosen column of
,p n
M
,
k 1≥
and
k p≤
and
p
is the total number of
k
-indexed columns of
,p n
M
)
( ,
,, 1
n j n j
x x
+
≤
for
,
2
p n
m≥
,
[ ]
,
j 1, k 1,
p n
m and p
 
∀ ∈ ∀ ∈
 
)
[2] Let us also consider the function that counts the lines of any
,
p n
M
, such as:
.
[ ]
,1
, ,
,
,,
,
,
,
, ,
,
,
,,
,
,
... ...
... ...
, with ... ... ,
... ...
j 1, 1, ,
n j n p j
n j k
p n p n
p n
n p
nn k
n j n p j
n k j
p n x x x
n m n p m
n k m
p n
x x x
x x x
M P P P n
x x x
m and k p
+
+
+
+
+
+
 
 
 
 
= + + + + =
 
 
 
 
 
 
∀ ∈ ∀ ∈
 
M M M M M
M M M M M
,
*
n k j
x N
+
∈
,
( , ) ( , )
p n
r p n l p n m= =

Dr
ă
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6
Using
p
S
,
,
p n
M
,
M
∅
and
( , )
r p n
as previously defined, BGC has two formulations sub-variants:
1.
2,n
M M
∅
≠
(OR
2
S
doesn’t contain any
M
∅
) for any even integer
2
n
>
or shortly:
2,
2
n
eveninteger n M M
∅
∀ > ⇔ ≠
(the 3
rd
formulation of BGC – 1
st
sub-variant).
2. For any even integer
2
n
>
,
(2, ) 0
r n
>
or shortly:
2 (2, ) 0
eveninteger n r n
∀ > ⇔ >
(the 3
rd
formulation of BGC –2
nd
sub-variant).
3 A Synthesis and A/B Classification of the Main Known Goldbach-like
Conjectures (GLCs) Using the
,
p n
M
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
p
S
and its elements
,
p n
M
other that GC (with
2
n
>
), with possibly other inferior limits
2
a
≥
, with
2
n a
> ≥
).
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
,
p n
M M
∅
≠
for a
chosen p>1 and for any / any odd / any even integer
2
n a
> ≥
(with
a
being any finite natural
established by that A-GLC and
n a
>
) BUT ALSO [2] any other non-trivial(nt) accessory
property/properties of all
(
)
,
p n
M M
∅
≠
of
p
S
. A specific A-GLC is considered authentic if the
other non-trivial accessory property/properties of all
(
)
,
p n
M M
∅
≠
(claimed by that A-GLC)
isn’t/aren’t a consequence of the 1
st
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
,
p n
M M
∅
≠
or just some
,
p n
M M
∅
≠
for a chosen p>1 and for some / some odd / some even integer
2
n a
> ≥
(with
a
being any finite natural established by that B-GLC and
n a
>
), all those
,
p n
M
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
(
)
,
p n
M M
∅
≠
(claimed by that B-GLC for
n a
>
) isn’t/aren’t a consequence of the fact that some
,
p n
M 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).

Dr
ă
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7
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)
3
,
n
M
(based on GIPs with 3
elements) such as:
a. Type A formulation variant as applied to
( *)
not just to
℘ ℘
:
“
3
,
2n
integer n M M
∅
∀ > ⇔ ≠
(with
, ,
0
n j k
x
≥
and , ,n j k
x
P
∈℘
)”
b. Type B (neutral) formulation variant: not supported.
2. “Every even integer
4
n
>
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 non-
trivial BGC (ntBGC) sub-variant that shall be treated in this paper (alias “BGC” or “ntBGC”) is
that: “every even integer
6
n
>
is the sum of at least one pair of distinct odd primes” [20,21]
(which is equivalent to: “every even integer
3
m
>
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:
2 2
c p p p
≠ ≠ +
(with
c
being any composite natural number and
p
being any
prime). ntBGC can be formulated using (binary)
2
,
n
M
(based on GIPs with 2 elements) such as:
a. Type A formulation variant: “
6
eveninteger n
∀ >
,
(
)
2
,n
n
M M M
∅
≠
AND
(
)
2
,
n
n
M M
contains at least one line with both elements (GIPs)≠1 (as
1
2
P
=
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: “
6
eveninteger n
∀ >
, all
(
)
2
,
n
n
M M
that are
non-empty (as
p
S
may also contain empty
(
)
2
,n
n
M M M
∅
= for some specific [but still
unfound]
n
values ) will contain at least one line with both elements (GIPs)≠1 (as
1
2
P
=
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)”.
3. “
5
odd integer n
∀ >
,
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: “
5
odd integer n
∀ >
,
n
is the sum of at
least one triad of [possibly identical] primes”). TGC can be formulated using (ternary)
3
,
n
M
(based on GIPs with 3 elements) such as:

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8
a. Type A formulation variant: “
3
,
5n
odd integer n M M
∅
∀ > ⇔ ≠
”
b. Type B (neutral) formulation variant: not supported.
4. “
17
integer n
∀ >
,
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: “
17
integer n∀ >
,
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)
3
,
n
M
(based on GIPs with 3 elements) such as:
a. Type A formulation variant: “
17
integer n
∀ > ⇒
3
,n
M M
∅
≠
AND
3
,
n
M
contains
at least one line with all 3 elements (GIPs) distinct from each other”
b. Type B (neutral) formulation variant: “
17
integer n
∀ > ⇒
those
3
,
n
M
which are
M
∅
≠
will contain at least one line with all 3 elements (GIPs) distinct from each other”
5. “
5
odd integer n
∀ >
,
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)
3
,
n
M
(based on GIPs with 3 elements) such as:
a. Type A formulation variant: “
5odd integer n
∀ > ⇒
3
,
n
M M
∅
≠
AND
3
,
n
M
contains at least one line with at least 2 identical elements (GIPs)”
b. Type B (neutral) formulation variant: “
5
odd integer n
∀ > ⇒
those
3
,
n
M
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
,
p n
M
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
(
)
, 2
p p
+
“ ) 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

Dr
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9
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
≤
n
0.95
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): “
4
eveninteger n
∀ >
, there is at least one prime number
p
[so that]
/ 2
n p n
< ≤
and
q n p
= −
is also prime [with
n p q= +
implicitly]”. GKC can also be reformulated as: “every
even integer
4n>
is the sum of at least one pair of primes with at least one element in the semi-
open interval ”. GKC can be also formulated using (binary)
2
,n
M
(based on GIPs
with 2 elements) such as:
a.
Type A formulation variant: “
4
eveninteger n
∀ > ⇒
( )
2
,
n
n
M M M
∅
≠
AND
( )
2
,
n
n
M M contains at least one line with at least one element
x
, so that
(
, / 2
x
P n n 
∈
.”
b.
Type B (neutral) formulation variant: “
4
eveninteger n
∀ > ⇒
those
( )
2
,
n
n
M M
which are
M
∅
≠
will contain at least one line with at least one element
x
, so that
(
, / 2
x
P n n

∈

.”
2.
The Goldbach-Knjzek-Rivera conjecture [GKRC] [43] (which is obviously stronger than BGC,
but also stronger than GKC for
64n≥
): “
4
eveninteger n
∀ >
, there is at least one prime
number
p
[so that]
4
n p n
< <
and
q n p
= −
is also prime [with
n p q= +
implicitly]”. GKRC can also be reformulated as: “
4
eveninteger n
∀ >
,
n
is the sum of at
least one pair of primes with one element in the double-open interval
( )
,4n n
”. GKRC can be
formulated using (binary)
2
,
n
M
(based on GIPs with 2 elements) such as:
a.
Type A formulation variant: “
4
eveninteger n
∀ > ⇒
( )
2
,
n
n
M M M∅
≠
AND
( )
2
,
n
n
M M
contains at least one line with one element
x
, so that
(
)
,4
x
P n n
∈
.”
b.
Type B (neutral) formulation variant: “
4
eveninteger n
∀ > ⇒
those
( )
2
,
n
n
M M
which are
M
∅
≠
will contain at least one line with one element
x
, so that
( )
,4
x
P n n∈
.”
3.
Any other GLC that establishes an additional inferior limit
0a>
for
(
)
2,
r n
so that
( )
2, 0r n a≥ >
(like Woon’s GLC [44]) can also be considered stronger that BGC, as BGC only
suggests
( )
2, 0r n >
for any even integer
6n>
(which implies a greater average number of
GIPs per each
n
than the more selective Woon’s GLC does).
(
, / 2
n n



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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
x
P
P
function in which
x
P
is the x-th prime and
x
P
P
is the
x
P
-th prime (which is denoted in the
next cited conjectures as
q
P
which is the q-th prime, with q being also a prime number).
1.
Conjecture 3.1 (Unification of GC and TPC, 29 Jan. 2014). For any integer
2
n
>
there is at
least one triad of primes
( ) ( )
()
2
1 2 1 , 2 , 2
q
q n n q P
+
 
< < − − +
 
 
(Sun’s Conjecture 3.1
[SC3.1 or U-GC-TPC], which is obviously stronger than BGC and was tested up to
8
2 10n= ×
)
2.
Conjecture 3.2 (Super TPC [SPTC], 5 Feb. 2014). For any integer
2n>
there is at least one
triad
( )
(
)
(
)
0 , 2 , 2
n k
P
k
k n P prime P prime
−
 
< < + = + =
 
(Sun’s Conjecture 3.2 [SC3.2
or SPTC], which is obviously stronger than TPC and was tested up to
9
10n=
)
[4,5]
3.
Conjecture 3.3 (28 Jan. 2014). For any integer
2n>
there is at least one pentad
(Sun’s Conjecture 3.3 [SC3.3], which is obviously stronger than TPC as it implies TPC; SC3.3 was
tested up to
7
2 10
n= ×
)
4.
Conjecture 3.7-i (1 Dec. 2013). There are infinite many positive even integers
3n>
which are
associated with a hexad of primes
( ) ( )
( ) ( ) ( ) ( )
1 , 1 , , , 1 , 1
n n n n
n n P n P n nP nP
+ − + − + − 
 
(Sun’s Conjecture 3.7-1 [SC3.7-
i], which is obviously stronger than TPC as it implies TPC;
22 110n=
is the first/smallest value
of
n
predicted by SC3.7-I)
5.
Conjecture 3.12-i (5 Dec. 2013). All positive integers
7
n
>
have at least one associated pair
( )
( )
1 , 2
k
n k
k n P prime
−
 
< − + =
 
(Sun’s Conjecture 3.12-i [SC3.12-i])
6.
Conjecture 3.12-ii (6 Dec. 2013). All positive integers
3n>
have at least one associated pair
( )
(
)
1 , !
n k
k n k P prime
−
 
< − + =
 
(Sun’s Conjecture 3.12-ii [SC3.12-ii])
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
( )
( )
( )
1 , 1 , 1
q r
q r P q P r
 
> = − + − +
 
(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/cgi-
bin/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;
( ) ( ) ( )
(
)
(
)
0 1 , 6 1 , 6 1 , , 2
n k n k
k n k prime k prime P prime P prime
− −
 
< < − − = + = = + =
 

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11
8.
Conjecture 3.21-i (6 Mar. 2014). For any integer
5
n
>
there will always exist at least one triad
( ) ( )
(
)
0 , 2 1 ,
k n
k n k prime P k n prime
⋅
 
< < + = + ⋅ =
 
(Sun’s Conjecture 3.21-i [SC3.21-
i])
9.
Conjecture 3.23-i (1 Feb. 2014). For any integer
13
n
>
there is at least one triad of primes
( ) ( )
(
)
1 , 2 , 1
n q
q n q P q
−
 
< < + + +
 
(the Sun’s Conjecture 3.23-i [SC3.23-i]).
4 The ‘i-primeths’ (
*
i
℘
) Definition
The definition of the generalized “i-primeths” concept
*
i
℘
. 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
1
2
P
=
). 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 (
i
P
containing
i
x
P
elements with
0
i
≥
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 1
P x P x P x
≡ =
o
and
( ) ( )
( )
1
.. ( )
i
i i
nested functionsP
P x P x P P P x
−
 
 
≡ =
 
 
o
:
1. 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
N
using 1 vertical iteration,
but
N
doesn’t contain just primes, as
*
N
℘ ≠
);
a. These iterations numbered by order i are easy to follow when implemented in different
algorithms using a programming language on a computer;
2. The concept of primes can be generalized as “i-primeths”
*
i
℘
, with
*
i
℘
also including
*
℘
as
the special case of 0-primeths
(
)
0
* *
⊂
℘ =℘
*
i
℘
;
3. This definition clearly separates
*
℘
from the ordinary
N
using 0 (not 1) as a starting order (i) for
*
℘
(
)
0
*
℘
and considering
N
as a
(
)
1
*
−
℘
(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”, ”super-
primes”, ” superprimes” or “prime-indexed primes[PIPs]”. URL (OEIS page): http://oeis.org/wiki/Higher-order_prime_numbers

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a.
0
*℘
inevitably “contains”
*N
by its indexes , in the sense that
0
*℘
contains all the generic
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
x
is not necessarily a prime.
b. This slightly different definition of the i-primeths (
*
i
℘
containing generic
ix
P
elements with
0i≥
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 B
ă
l
ă
cenoiu I.
[48].
The elements of the generalized set of i-primeths
*
i
℘
:
(alias 0-primeths)
(alias 1-primeths
[7]
)
,
…
, with
* {1,2}x N∈ −
5 The Meta-conjecture VBGC - The Extension and Generalization of
BGC as Applied on i-primeths
( )
*
i
℘
Meta-conjecture VBGC – main co-statements:
1.
Alternatively defining i-primeths as:
0
[0 ]
xiterations
of P on P
P P x
 
 
= 
 
 
,
( )
1
[1 ]
xiteration
of P on P
P P P x
 
 
= 
 
 
 
,
( )
( )
2
[2 ]
xiterations
of Pon P
P P P P x
 
 
 
= 
 
 
… , with
( )
P x
being the x-th prime in the
set of standard primes (usually denoted as
( )
P x
or
x
P
and equivalent to
0
x
P
alias “0-primeths”)
and the generic
ix
P
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).
2.
The inductive variant of (the meta-conjecture) VBGC (iVBGC) proposed in this paper states
that:
“All even positive integers
( )
,
2 2
fx a b
m
⋅
≥
AND (also)
( )
2
,
2 2
fx a b
m
≥ ⋅
, 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
0
x
P
(
)
(
)
(
)
( )
{
}
0 0 0 0 0
1 1 2 2 3 3
, ,... ,...
* * ,
x x
P P P P P P P P
= = = = = = =
℘ =℘ =
2 3 5
(
)
(
)
(
)
{
}
1 2
1 1 1 1
1 2 2 3
,... ,...
* ,
x
p p x p
P P P P P P P P
= = = =
℘ = = = =
3 5
(
)
(
)
(
)
{
}
1
2
2 2 2 2
5
1 3 2
,... ,...
* ,
P Px
P
p p x p
P P P P P P P P
= =
℘ = = = =
5 11
2
1
1 2
... ... ...
, ,..., ,...
*
P PP x
iP P P
i iterations of P i iterations of P
i itera tions of P
i i i x
P P P
P P P
 
 
 
 
 
 
 
℘ = = = =
( )
( )
( )
( )
[
0
]
...
i
ix
iterations
P P P P P x
≥
 
 
=
 
 
 

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written as the sum of at least one pair of DISTINCT odd i-primeths
x y
a b
P P
>
, with the
positive integers pair
( )
, , 0
a b with a b
≥ ≥
defining the (recursive) orders of each of those
i-primeths pair AND the pair of distinct positive integers
( )
, , 1
x y with x y
> >
defining
the indexes of each of those i-primeths pair, with
( ) ( )
( ) ( )
( ) ( ) ( )
( 1)( 1)( 2)
[( 1)( 1)( 3)/ ]
( 1)( 1)( 2) ( 2)
2 0
, 2 0
2 0 0
a b a b
aab a b a
a b a b a b
for a b
fx a b for a b AND a
for a b AND a OR b
+ + + +
+ + + + −
+ + + + − + −
= =


= = >

≠ > >
 

 

and
( ) ( )
( ) ( )
( ) ( ) ( )
2
( 1)( 1)( 2)
[( 1)( 1)( 3)/ ]
( 1)( 1)(
2
2) ( 2)
2 0
, 2 0
2 0 0
a b a b
a b a b a
a b a b a
a
b
for a b
fx a b for a b AND a
for a b AND a OR b
+ + + +
+ + + + −
+ + + + − + −
= =


= = >

≠ > >
 

 

.”
a.
A secondary inductive (form of) (the meta-conjecture) VBGC (siVBGC[a,0]) proposed in
this paper states that: “All even positive integers
( )
22 int
m fy a
⋅≥
 
 
, with
( )
4
a
fy a e=
, can be written as the sum of at least one pair of DISTINCT odd i-
primeths
0
x y
a
P P
>
, with the positive integers pair
( )
,0 , 0
a with a
>
defining the
(recursive) orders of the i-primeths pair
(
)
0
,
x y
a
P P
AND the distinct positive integers
pair
( )
, , 1