1. Introduction

Prime numbers are fundamental mathematical objects in

number theory [1, 2, 3, 4, 5, 6]. One of the challenging

questions in number theories yet to be answered is the

Goldbach’s conjecture [7] that queried by Christian Goldbach

in a letter to Leonhard Euler in 1742 [4, 8, 9, 10]. A typical

and informal expression of the Goldbach conjecture may be

stated as follows.

Definition 1. The Goldbach conjecture queries whether

every even integer greater than 2 may be expressed as the sum

of two primes.

Key milestones towards the proof of Goldbach conjecture

in the past 278 years and beyond includ: 1) The Euclid’s

Fundamental Theorem of Arithmetic (FTA) that revealed there

is always a unique prime factorization for any integer [1, 3]; 2)

The Legendre’s Sieve of Eratosthenes (1808) that provided a

foundation for modern sieve theories [2, 11]; 3) The Prime

Number Theorem (PNT)

()

lim 1

/ log

n

nn



→ =

that proved by

Hadamard and de la Vallee Poussin in 1896, independently

[4]; 4) Vinogradov’s theorem (1937) that stated all large odd

numbers may be expressed by

1 2 3

(large)n p p p= + +

with

triple primes [12]; 5) A finding that the number of positive

even integers less than n which are not representable as a sum

of two primes grows slower than

(log )r

n

for any positive r

[13]; 6) There are some integer k such that every sufficiently

large even number is the sum of two primes and the kth

powers of 2 [9]; 7) The refined Linnik’s theorem for k = 8

[10]; 8) The proof that every sufficiently large even number is

the sum of a prime and a number with at most two prime

factors [14]; 9) T. Tao explored obstructions to uniformity of

primes and their arithmetic patterns [15]; 10) Every odd

number ( 7) can be written as the sum of three primes known

as the ternary Goldbach conjecture [16]; 11) Numerical

algorithms for testing the Goldbach conjecture in a certain

scope [17]; 12) Every odd number greater than 1 is the sum of

at most five primes [18]; and 13) Every positive integer can be

written as the sum of a prime number and a square free

number [19]. However, there is no formal proof for the

Goldbach conject yet that may hold in all cases because the

nature of its complexity.

It is revealed in this work that the key to proof Goldbach

conjecture is the missing of a prime decomposition theory for

arbitrary even numbers as a counterpart of Euclid’s FTA on

prime factorization [1]. This work intends to present a formal

proof of the Goldbach conjecture based on the finding of the

set of mirror primes







where the set of primes







o (odd integers) 



except 2 as well as the theorem of

mirror-prime decomposition. A formal model and the

recursive properties of the set of primes



are described in

Section 2. The concept of mirror primes



is introduced in

Section 3 that leads to the proof of the Theorem of Mirror-

Abstract: This work presents a formal proof of Goldbach conjecture based on a novel theory of Mirror-Prime

Decomposition (MPD) for arbitrary even integers. A new concept of mirror primes is introduced

as a set of pairs of primes that are symmetrically adjacent to any pivotal even number on both sides

in finite distance k bounded by 1 ≤ k ≤ (ne/2) − 2. As a counterpart of the Euclidean Fundamental Theorem of

Arithmetic for natural number factorization, the MPD theory enables arbitrary even number decomposition by

mirror primes. MPD paves a way to prove the Goldbach conjecture, i.e., where denoted by the big-R calculus

for representing recursive structures and manipulating recursive functions. An algorithm of Goldbach conjecture

testing is designed for demonstrating the formal proof of the Goldbach theorem. i.e

where

denoted by the big-R calculus for representing recursive structures and manipulating recursive functions. An

algorithm of Goldbach conjecture testing is designed for demonstrating the formal proof of the Goldbach

theorem.

Keywords: Number theory, Goldbach conjecture, proof, mirror primes, mirror prime decomposition, recursive

sequence, numerical algorithm

Received: September 26, 2021. Revised: May 24, 2022. Accepted: June 26, 2022. Published: July 18, 2022.

A Proof of Goldbach Conjecture by Mirror Prime Decomposition

YINGXU WANG

Dept. of Electrical and Software Engineering

Schulich School of Engineering and Hotchkiss Brain Institute, University of Calgary

2500 University Drive NW, Calgary, Alberta, CANADA T2N 1N4

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Prime Decomposition for arbitrary even numbers. These

preparations lead to the proof of the Goldbach conjecture as

formally presented in Section 4 by the universal existence of

mirror primes and the inductive rule of mirror-prime

decomposition. A set of experiments based on an Algorithm of

Goldbach Theorem verification is provided for visualizing the

proven Goldbach conjecture.

The set of prime numbers



is a subset of special odd

integers supplement by 2 in natural numbers



, i.e.,

{2}\{1} .

o

  

Since



is infinite, so is



according

to its countability with respect to



. Therefore,



shares the

generic properties of



o as a necessary condition, but also

obeys special primality properties as their sufficient conditions

as described in this section.

In order to efficiently denote and manipulate infinite sets

and sequences as well as functions operating on them, a

general recursive notation known as the big-R calculus [20] is

introduced. As shown in Section 2.1, a suitable notation may

significantly reduce the complexity of problem modeling and

solving. It may also increase the efficiency in recursive

inferences for hard problems and long-chain reasoning for

mathematical induction and deduction.

2.1 Mathematical Models of Recursive Structures and

Properties of Primes

Definition 2. The big-R calculus is a recursive operator

for neatly modeling finite or infinite sequences of recurrent

structures and manipulating a series of embedded functions

such as:

0 1 2

0

0 1 2

0

1

a) Infinite sequence: ( ) ( , , ,..., , ...)

b) Infinite set: { } { , , ,..., , ...}

c) Infinitely inductive functions: ( )



=



=

−

=

ik

i

ik

i

kk

i

k

q q q q q

n n n n n

ff

R

1 1 0 0

1 1

1 1 0 0

( (... ( )...)),

d) Infinitely deductive functions: ( )

( (... ( )...)),

−

=

−







=



=



kk

d

k

kk

f f f f f

ff

f f f f f

R



(1)

Example 1. The set of even integers



e and the recursive

structures of a series of deductively embedded functions



d

may be formally described by the big-R notation, respectively,

as follows:

0

1 1 1 1 0 0

{ 2 2} {2, 4,6,8,.., 2 2, 2( 1) 2,...}

( ) ( (... ( )...)), constant

e

n

k k n n

d

kn

n n n

f f f f f f f

R



=

−−

=

= + = + + +

= =  =

(2)

Definition 3. The set of natural numbers



is all positive

integers in the scope of [1, ) with a uniform step of

increment that may be denoted by the big-R calculus:

1

{1, 1} {1, 2,3,..., 1, 2, ...}



=

+ = + +

n

n n n

R

(3)

Similarly, the sets of even and odd integers

and , ,

e o e o

=

are denoted, respectively, by:

1

{ 2 } {2, 4, 6,..., 2 , 2( 1), 2( 2), ...}

{ 2 1} {1,3,5,..., 2 1, 2( 1) 1, 2( 2) 1, ...}

e

n

o

n

n n n n

R



=



=

= + +

− = − + − + −

(4)

Definition 4. A prime number p, except 2, is an odd

positive integer

1o

p  

that is not a product of two

smaller integers:

2

( | 0 (mod )), {2}\{1}

n

o

m

p n n m n

R





=

   

(5)

Any prime may be verified based on Definition 4 though

more efficient sieve methods and algorithms exist [21, 22, 24,

25, 26, 27]. A generic method for primality testing may be

formally described as follows.

Definition 5. The primality testing function

()n



determines whether n is prime

{2}\{1}

o

n  

:

2,

0, 0 (mod ) //

()

1, otherwise //

n

mm

n m n

n







=













(6)

where

()n



results in a positive verification iff

0 (mod ) for all 2 .n m m n



   



Otherwise, as a

shortcut, any negative result

0 (mod )nm

will terminate the

testing by returning false.

In classic number theory, the set of prime

numbers



is used to be perceived as a random set.

However, according to Definitions 4 and 5,



may be

rationally perceived as a recursively determinable sequence as

follows.

2. The Big-R Calculus for Manipulating

Recursive Structures and Functions

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Definition 6. The generic pattern of primes



is a

recursive and infinite sequence of monotonously increasing

odd integers (except 2) validated by the primality checker

()n



:

1

1 2 1

31

{}

{ 2, 3, [ ( 2 ) | ( ) 1]},

−



=



−

==

= = = = + = 

i

p

i i i

ik

p

p p p p k p k

R

RR



(7)

Example 2. The following subset of primes may be

recursively derived by Eq. (7) following the first two known

primes

12

2 and 3pp==

in



:

3 2 3

4 3 4

54

45

25 24

( 2) (3 2) 5 {2,3, 5, ...}

( 2) (5 2) 7 {2,3, 5,7, ...}

( 2) (7 2)

( 4) (7 4) 11 {2,3,5,7,11, ...}

...

( 2) (89 2) ,

(89 4)

(89 6)

(89 8) 97

= + = + =   =

= + = + 

→ + = + =   =

= + = + 

→ + 

→ + = 

pp

p

pp

25

26

1

1290000

6

{2,3,5,7,11, ..., 97}

(97 2)

(97 4) 101 {2,3, 5,7,11, ..., 97,101}

...

( 2) (( 1) 2)

{2,3,5,7,11 .

29

, ..., 97,101,

9686303489

8

. .,

52

2996

−



=

=

= + 

→ + =   =

= + = − + 

 = =

nn

ni

n

p

pp

p

R

1290000 1290000

3034895 2 2996863031, 1,

48

95 2

...}

−+



where the largest twin primes, 2996863034895·212900001,

have been found in 2016 [29]. The largest prime number as

known in 2020 is 282,589,933 – 1 which has been revealed by

Patrick Laroche from the group of Great Internet Mersenne

Prime Search (GIMPS) [6].

2.2 Recursive Properties of the Sequence of Primes

The generic pattern of prime numbers described in

Section 2.1 reveals the analytic and distribution properties of

the set of primes



. According to Definition 6, all primes

i

p

are derived from a recursive sequence that

provides a new perspective on the nature of primes and their

manipulations.

Theorem 1 (Recursiveness of the Prime Sequence,

RPS). Primes in



are a recursive sequence where prime pn+1

is derived from and constrained by the preceding ones

1

n

i

p

R

=

based on the following necessary and sufficient conditions:

1

2 1 2

1

2,

(a) The necessary condition

2 ,

(b) The sufficient condition

( ) =

0 (mod )

−−

+



+

= = = 



+

=





= + 













n

nn

n

ni

n i n p

n

mm

n

p p k k

pp

pm

R R R

R

(8)

where n, m, i, k 



, p1 = 2, and p2 = 3

Proof. Theorem 1 holds based on the inherent properties

of primes as a special sequence of particular odd integers,

except 2, according to Definition 6:

1 1 1 2

'

1

''

11

a) Condition (a) is necessary because

, , 2, , , , 2, and 3,

( 2 )

results in either ( ) 1 or some of the potential

would be m



+

+=

++

    = =

+



n

nn

p

nn

k

nn

n k n p p p p p

p p k

pp

R



1

2,

issed. Thus, any eligible prime must be

at one of the positions:

( 2 ) as necessary.

b) Condition (b) is sufficient based on (a) because



+

+=





=

=+

n

p

nn

k

p

n

mm

p

p p k

p

R

11

2,

1

2,

11

2

0 (mod ) = 0 (mod )

eliminates all potential prime divisors among preceding

primes ( ) in , such that each

= ( ) is recu





++

=





+

=



++

=







n

p

n

mm

p

nn

n n n

n

m p m

pp

p p p

R

1

rsively determined by

previously known primes validated by

2 or ( 2 ) 1.



=



   + 



n

p

n

k

pm p k

R





Theorem 1 reveals the nature of primality and the

recursive property of the infinite sequence of primes. It also

indicates that pn+1 would remain indeterminable until the

preceding

1n

p

p+





have been acquired by any inexhaustive

prime sieve method.

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



Based on the preparations in Section 2, a key concept of

mirror primes is introduced in this section to model an

important distribution pattern of primes towards the proof of

Goldbach conjecture.

Definition 7. The mirror primes

/2

e

n

p



with respect to a

pivotal even number ne 



e 



are pairwise primes

symmetrically adjacent to the central en within finite k 



distances:

2

/2 /22

2

/

/ /2

2

1

{ ( )

| ( ) ( )) 1)}

,

22

−+

−

=

=

= − = +

ee

e

nn

e

ne

nn

n

k

nn

p k p k

pp

pR







(9)

where k is called the half interval and the primality validation

function

( ), 1 2

22

ee

nn

kk



   −

as given in Definition 5

eliminates any potential decomposition that is not a pair of

mirror primes.

Example 3. The following sets of mirror primes are

derived according to Definition 7 where the sum of each pair

is always equal to their corresponding pivotal ne:

Based on Definition 7, the entire set of mirror primes



may be recursively derived as given in Definition 8.

Definition 8. The set of mirror primes



is all valid

pairs of mirror primes with respect to each pivotal even

number 4  ne /2 



e in finite distance

12

2

e

n

k  −

:

2

1

/2 /2

/2

/2 4

/4 /

22

( , )

22

[ ]}

)

{ } ( )

{

)(|1(

−+

−

=



=



=



=



− = +



=



=





ee

e

ee

e

ne

k

nn

e

n

e

n

nn

p k p

p

k

p

R

RR





(10)

where all pairs of mirror primes

/2

e

n

p





in the scope

8e

n  

are determined by Definition 7.

It is noteworthy that the set of mirror primes represents

all symmetric pairs of adjacent primes with respect to any

pivotal even number ne/2 in equal distances. Based on the

generic model of mirror primes, the classic twin

primes

e

n

p



may be formally derived as a special subset of



where the half interval

1k

. For instances:

44

66

88

8

10 10

(8 / 2 | 1) {(4 1) | (4 1) 1} {(3,5)}

(12 / 2 | 1) {(6 1) | (6 1) 1} = {(5,7)}

(16 / 2 | 1) {(8 1) | (8 1) 1 (8 1) 0}

= {(7, )} = { }, while {(5,11| 3), (3,13 | 5)}

(20 / 2 |

= = = = =

= = = =

= = = − =  + =

  = = =

=

p p k

p k k

p p k











10

12 12

1) {(10 1) | (9) 0 (11) 1}

= {( ,11)} = { }, while {(7,13 | 3),(3,17 | 7)}

(24 / 2 | 1) {(12 1) | (12 1) 1} = {(11,13)}

...

= = =  =

  = = =

= = = =

p k k

p p k









According to Definition 8, although the number of mirror

primes with respect to different ne is proportional to its value,

i.e.,

( / 2) 2

e

n−

, but at least one pair of mirror primes exists

in finite steps for mirror-prime decomposition. This discovery

22

4

1

4

min

3

5

1

2

1

{ ( ) | ( ) 1)} = { (4 ) | (4 ) 1)}

22

{( ),( , 6) | (4 ) 1} {( )}

( ) = 1 for mirror-prime decomposing = 8 = 3 + 5

{ (5 ) | (5 ) 1}

= {(4,6),

e

ee

k

e

k

n

k

nn

p k k k k

3, 5 2 k 3,5

k p n

p k k

R







−

=

= = =



==

5

min

48

50

1

( ),( ,8) | (5 ) 1} {( )}

( ) = 2 for = 10 = 3 + 7

...

{ (50 ) | (50 ) 1}

= {(49,51), (48,52), ( ),(46,54), (45,55),(44,56),( ,57), (42,58),( ), (40,60),

e

k

3,7 2 k 3,7

k p n

p k k

47,53 43 41,59

R





=

==



==

(39,61), (38,62), ( ,63),(36,64), (35,65),(34,66),(33, ), (32,68),( , 69),(30,70),

( ), (28,72), (27, ), (26,74), (25,75), (24,76),( , 77),(22,78),(21, ), (20,80),

(1 81), (18

37 67 31

29, 71 73 23 79

9, ,82), ( ,83),(16,84), (15,85),(14,86),( ,87),(12,88),( ), (10,90),

(9, 91),(8,92),( ,93),(6,94),( ,95),(4,96),( ),( ,98) | (50 ) 1}

= {( ), ( ), ( ), ( ),( )}

17 13 11,89

7 5 3,97 2 k

47,53 41,59 29, 71 11,89 3,97



=

50

min

( ) = 3 for = 100 = 47 + 53

...

e

k p n





3. Formal Models and Properties of

0LUURU3ULPHV

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of the interesting properties of mirror primes is formalized in

the following theorem.

Theorem 2 (Mirror Prime Decomposition, MPD). Any

even integer

/ 2 4

ee

n

may be decomposed to the sum of

at least a pair of mirror primes

/2 /2 /2

(),

2), 2

(

e e e

n n n ee

pnn

kpp k

 

−+

==− +

adjacent to ne/2

as the pivot within

12

2

e

n

k  −

steps:

2

41

/2 /2

/2 / 2 / 2

/2 /2

/2

, / 2 4 :

( ) bounded

by 1

where = ( )

) (

22

2

,

( , )

22

(

)

{

| (

−+

−



==

    

=





+ − + +

−

= − =



+



ee

e e e

ee

e

ne

nk

ee

nn ee

e

n n n

nn

ee

n

e

np

n

p

nk

k

pp

p

n

kk

n

pn

kk

p

RR









/2

.

)

}

1

+





=



e

n

p



(11)

Proof. Theorem 1 holds according to the principle of

mathematical induction throughout the entire set of mirror

primes





as follows:

2

1

/2 / 2

/2 4

,

/ 2 4 , and ( , :

Applying rules of mathematical induction based on Def. 6

)

( , )

22

[

:

{

( ) ( ) | 1

−+

−−+

=

−+



=

= − = +



=



=









ee

neee

ee

ek

nn

ee

nn

ee

nnn

n k p p

nn

p k p k

pp

R R







( )

(8/2 ) 2

1

8

8 (or /2 4) is true :

{ (4 , 4 ) | (4 ) 1) }

={( ), (2,6) | (4 1) 1)} = {(3, 5)}

The prime d

e

]}

a) The statement f

o

p

r th base c

ecom ositio

e

n

8

as

is

−

=







=

==

− + =

=





=





k

e

en

k k k

n

3,5

R





( )

1

498

min

1000

1 2 2;

b) Ass

ume the sta

3

e

tement

i

tr

m

ue

5

det rmined with n the inimum half interval

( /2)

1,000, . ., /2 500 :

{ (500 , 500 ) | (500 ) 1) }

=

= − =



+



==

−= +=

k

e

ee

n

i e n

k k k

k

n

R



min ;

= {(499, 501), (498, 502), ... ( ), ...

| (500 9) 1)}

= {( ), ...}

1000 491 509 with

= 9 2 ( /2) = 498

=

 = = +

 −

e

491, 509

k

n



( )

499

1

1002

c) Then, the next pair of mirror primes for 1000 2

1002 is also true :

{ (501 , 501 ) | (501 ) 1) }

= {(500, 502), ( ), ... | (501 2) 1)}

= {(

=

=+

=

− + =

=

k

e

k k k

499, 503

4

n

R





min 22

a

00

2

.

), . }

1 2 499 503 with ( / )

= 499

Thus, it has been inductively proven th t 8 ,

;

 = = + 

   



=−

ee

9

k

n

9, 503

nn

/2 /2 /2

/2 /2

at least a pair of mirror primes ,

( , )

that satisfies the ounded in

1 2 steps.

()

22

MPD theorem b

2

−+

− + 

+



=

−

e e e

ee

n n n

ee

nn

e

p p p

nn

n p p

n

kk

k







The Theorem of MPD is a coherent counterpart of

Euclid’s FTA on prime factorization in number theory [1]. It

provides a general theory and methodology for finding all

pairs of mirror primes, including twin primes, on both sides of

any arbitrary even number in the scope of

42

e

n

  

, except

the special case

2

e

n=

where the mirror primes regress to a

pair of reflexive primes

4(2,2), 0pk



==

. The proven

existence of at least a symmetric pair of mirror primes to any

pivotal even numbers according to the MPD theorem paves a

way to formally prove Goldbach conjecture in the following

section.

The classic expression of Goldbach conjecture has been

described in Definition 1. Although in his letter to Euler [7],

Goldbach demonstrated alternative prime compositions for a

few small even integers, he could not go very far perhaps

because of the extreme complexity for deal with both infinite

sets of

and

. More fundamentally, we now understand

that the yet to be proven conjecture was mainly due to the lack

of a formal prime decomposition theory for even numbers

representation as revealed in Theorem 2 supplement to

Euclidean FTA [1] for prime factorization in number theory.

Goldbach conjecture as given in Definition 1 may be

formally described as a hypothesis of general prime

decomposition for even numbers as follows.

Hypothesis 1. Goldbach Conjecture states that any

arbitrary even number

e

n

as equal to or greater than 4 may be

expressed by the sum of two primes

and

ij

p p

:

?, , , 4

e i j i j e e e

n p + p p , p n n   

=

(12)

4. Proof of the Goldbach Conjecture

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On the basis of the MPD theorem, Goldbach conjecture

may be deduced to a general prime decomposition problem for

even numbers. Therefore, the establishment of Theorem 2 has

provided the necessary and sufficient conditions for proving

the Goldbach conjecture based on the mathematical model of

mirror primes



.

Theorem 3 (The Goldbach Theorem). Given any

arbitrary even integer

/ 2 4

ee

n  

, there exist at least

a pair of mirror primes

/2 / 2

(),

ee

nn

pp





−+  

that

satisfy:

2

/2 /2 / /2

/2 4

( [ , ) ]) | (

e e e

e

nn

e

n

nn

n p p p p

R



  

− + − +



=

+

(13)

where

/2 /2

, , .in finite distence 1 2

22 2

−+

= − = +  −

ee

nn

ee e

n

pk

n

kkp

n



Proof. The Goldbach theorem is proven based on the

recursive symmetric property of mirror-prime decomposition

as established in Theorem 2:

2

1

2

/ /2

/2 /2

4:

2

Theorem 2 ensures that

at least a pair of valid mirro

and

{( , )

22

| ( }

pinpoin

r primes is in

l

e

)

tab 1y

b

,

−+

−

=

    

=



−=



+

ee

ne

k

n

e

nn

ee

n

nn

kk

nk

pp

R



 

2

1

/2 /2

/

2/

4

2

2 steps

|{ }| 1,

that satisfies:

[

without exception.

where

2

( , )

( ) = ( ) ( )]

22

−+

−

=



=

−

+−



++







ee

e

ee

ne

k

e

nn

nn ee

e

n

k

pp

n

np nkp

n

R







Example 4. Applying the Goldbach Theorem to the

largest twin primes

1290000

29968630348 195 2

e

n

p



=

discovered in 2016 [29],

1290000

2996863034895 2

e

n=

is

surely decomposable by the pair of known twin primes as a

special case of the mirror primes where k = 1:

1290000

1290

2996863034895 2

(2996863034895 2 1) 1,

2996863034895 2

( 1) ( 1), 1

22

29968630

, and

according to Theorem 3:

=

34895 2

(

− + − +



=

+ + = − + + =

=

e e e e

e

ne

nn

p p p p

n

k

   



000 1290000

2996863034895 2

1) ( 1)

22

− + +

The proven Goldbach conjecture in Theorem 3 may be

numerically explained by an infinitively inductive sequence in



. An algorithm for numerically implementing the Goldbach

theorem is derived for prime decomposition of arbitrary even

numbers. It is formally described using Real-Time Process

Algebra (RTPA) [28] known as a form of Intelligent

Mathematics (IM) [5, 23] for AI programming.

Algorithm 1. The Algorithm of Goldbach Theorem

Verification (AGTV) is designed based on Theorem 3 as a

numerical verification tool for mirror-prime decompositions of

arbitrary even integers. The AGTV algorithm treats the

Goldbach theorem as a recursive function

/2 4

( , ) ee

e

n

nn

f p p

R



−+

−

=

according to Eq. (13), which links the hard problem in number

theory to a deterministic numerical solution. The AGTV

algorithm as a process model (PM) in RTPA, AGTV|PM, is

shown in Fig. 1.

The AGTV algorithm is a computational implementation

of the mathematical models obtained in Theorem 3. The input

(I) of AGTV|PM is the maximum expected prime

decomposition for

max

8

|

, )|( ee

e

ee

nn

n

pp

R

. The output (O) of

AGTV|PM is a set of verified results represented by

max

8

|

)( | | |

ee

e

ee nn

ee

n

n p p

R

. The Hyperstructure (H)

denotes underpinning Structure Models (SMs) to be operated

by the algorithm. AGTV|PM is implemented by a recursive

process in the loop

max

/ 2| 8

|

ee

n

R

(...) after the upper limit for

iteration is validated by the if-then-(else) structure ( ). It

then determines the first or nearest mirror-prime

decomposition for each

| | |

e e pn p

guaranteed by

Theorem 3 within

22

e

n

k −

iterations. Once a validate pair of

mirror-prime decomposition for a given even number

|ee

n

is

found, the algorithm exits (

) and enter the next iteration

until all

max |

ee

n

cases of Goldbach decompositions are

completed.

The time complexity of AGTT|PM is

5

2

max

max max max

1

)(( ( ) )

22

e

e e e

On

n n O n

. The space

requirement for AGTV is constrained by the memory size of

the underpinning computer platform.

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Fig. 1. The algorithm for Goldbach Theorem verification (AGTV)

The AGTV algorithm may be implemented in any

programming language for enabling empirical testing by

readers as illustrated in Experiment 1. A set of numerical

experiments based on AGTV is tested in MATLAB that

provides empirical evidence for demonstrating the nature of

the proven Goldbach theorem.

Fig. 2. Experimental results of Goldbach theorem

for

66 66 66

=



by Algorithm 1

Experiment 1. Applying the AGTV algorithm, a set of

experimental results has been obtained as illustrated in Figures

2 and 3 in the two-dimensional space





. Figure 2

demonstrates the proven decompositions of the first 66

samples in the scope of

4 / 2 66

e

N

. It shows that any even

number Ne can be expressed as the sum of a pair of mirror

primes, Ne  Pm- + Pm+, as predicated by the proven Goldbach

theorem. It is noteworthy that the curve of Ne/2 functions as a

divider for separating the pairs of both sets of symmetric

mirror numbers. The cases where the mirror primes almost

touch the Ne/2 curve indicate those of twin primes (k =1)

decomposition as special cases of mirror primes.

Similarly, the proven decompositions of the set of 100

even numbers in the scope of

4 / 2 100

e

N

is autonomously

generated as illustrated in Figure 3 in a neat form where every

Ne  (Pm-+Pm+)/2 is determined based on the proven Goldbach

theorem. Any large set of experiments may serve as additional

instances to demonstrate the Goldbach theorem in general.

The only constraint for the processing capability of AGTT is

the limit of computer speed and memory space towards

exhaustively decomposing the infinitive sequence of mirror

primes. Therefore, the inductive theorem and mathematical

inferences as proven in Theorems 2 and 3 play more generic

and rigorous roles for manipulating the infinite scope of

mirror-prime decompositions problem beyond any empirical

experiment towards infinitive.

a

mx

/ 2 / 2 / 2 / 2

max

/ 2 4

/ 2 2

/2

4

/

|

/2

|

| [ , |

|PM(< :: >; < :: ( ]>;

2

< :: | | | [( , ( , ) 1 >)

{

)| | |

, )| ]

e e e e

e

ee

ee ee

nn

n n n n

ee

n

nn

n

np

n

p

p p

pp

ppR

R

AGTV I O

H

max

/ 2| 4

|2

2/2

x

|1

/2

ma

|

8

( |B := |B

|

( | 2

|

2

|

ee

e

ee

n

e

nee

o

k

nee

o

e

n

Find F

n

pk

n

p

n

k

R

/2

|

/2

| 2,

( [ | 0 (mod | ) | 0 (mod | )] // Primality test fail

neo

ee

p

nn

oo

mm

p m p m

// Exit

|

/ 2 / 2

// The th+1 find

(" | | | ")

ee

nn

ee

n

Print n p p

)

}

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DOI: 10.37394/23206.2022.21.63

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Fig. 3. Experimental results of Goldbach theorem

for

100 100 100

=



by Algorithm 1

The experiments have provided a visualization of the

Goldbach Theorem, which empirically explain the nature of

Goldbach conjecture. Figures 1 and 2 demonstrate there

always exist at least one pair of mirror primes to decompose

arbitrary even numbers in most of the testing cases according

to Theorem 3. It is found that the expected pair of primes for

satisfying the Goldbach theorem is not arbitrary primes, but

merely those belong to the set of mirror primes. The Goldbach

theorem and MPD theorem proven in this work have found

many interesting applications including a deepened

understanding of the nature and the recursive distribution

patterns of primes, an expected fast recursive algorithm for

primality testing, and a formal proof of the twin prime

conjecture [30].

This work has presented a formal proof of Goldbach

conjecture based on a discovery of mirror primes and their

recursive properties. A theorem of mirror-prime

decomposition for arbitrary even numbers has been

established towards the formal proof of the Goldbach

conjecture. The work has led to a new perception on the

Goldbach theorem as an infinite recursive sequence in





:

/2 / 2 /2

/2 / 2

/2

/2 / 2

2

41

1:

2

,

( , )

22

{ }

s

and 1

|

st

()

= ( )

a i fies )

2

( , )

(

−+

−



==



+



−









= − = +



=





+

   



=−

e e e

ee

e

n n n

nn

e

n

ee

e

n

nn e

e

ne

nk

n

nn

k

p

k

n k N

pp

p

pn

p

n kp

RR











. )( 2+

e

nk

such that there exist at least a pair of mirror primes for any

/ 2 4

e

n

bounded by

1 ( /2) 2

e

kn −

steps. Experiments

using the algorithm of Goldbach theorem testing have

empirically and visually demonstrated the Goldbach theorem

in analytic number theory.

This work is supported by the Intelligent Mathematics

Initiative of the International Institute of Cognitive Informatics

and Cognitive Computing (I2CICI), the IEEE SMC Society

Technical Committee on Brain-Inspired Cognitive Systems

(TC-BCS) and the AutoDefence project of DND, Canada. The

author would like to thank the anonymous reviewers for their

valuable suggestions and comments.

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Volume 21, 2022

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