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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

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

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

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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

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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:

Dr

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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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10

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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12

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

≥

=

Dr

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13

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