Elementary proof of the twin prime conjecture, Polygnac conjecture and Goldbach conjecture

7American Journal of Multidisciplinary Research & Development (AJMRD)
Volume XX, Issue XX (June - 2023), PP 22-29
ISSN: 2360-821X
www.ajmrd.com
Research Paper Open Acces

Elementary proof of the twin prime conjecture, Polygnac conjecture
and Goldbach conjecture
Liao Teng
Tianzheng International Mathematical Research Institute, Xiamen, China

Abstract:
In order to strictly prove from the point of view of pure mathematics Goldbach''s 1742 Goldbach
conjecture and Hilbert''s twinned prime conjecture in question 8 of his report to the
International Congress of Mathematicians in 1900, and the French scholar Alfond de Polignac''s
1849 Polignac conjecture, By using Euclid''s principle of infinite primes, equivalent
transformation principle, and the idea of normalization of set element operation, this paper
proves that Goldbach''s conjecture, twin primes conjecture and Polignac conjecture are
completely correct.

Key words:
Twin prime conjecture, Polignac conjecture, Goldbach conjecture, the infinitude of prime numbers,
the principle of equivalent transformations, the idea of normalization of set element operations.


I. Introduction
In a 1742 letter to Euler, Goldbach proposed the following conjecture: any integer greater than 2
can be written as the sum of three prime numbers. But Goldbach himself could not prove it, so he
wrote to ask the famous mathematician Euler to help prove it, but until his death, Euler could not
prove it.The convention "1 is also prime" is no longer used in the mathematical community, but
this paper needs to restore the convention "1 is also prime". The modern statement of the original
conjecture is that any integer greater than 5 can be written as the sum of three prime numbers. (n >
5: When n is even, n=2+(n-2), n-2 is also even and can be decomposed into the sum of two prime
numbers; When n is odd, n=3+(n-3), which is also an even number, can be decomposed into the
sum of two primes.) Euler also proposed an equivalent version in his reply, that any even number
greater than 2 can be written as the sum of two primes. The common conjecture is expressed as
Euler''s version. The statement "Any sufficiently large even number can be represented as the sum
of a number of prime factors not more than a and another number of prime factors not more than
b" is written as "a+b". A common conjecture statement is Euler''s version that any even number
greater than 2 can be written as the sum of two prime numbers, also known as the "strong
Goldbach conjecture" or "Goldbach conjecture about even numbers". From Goldbach''s conjecture
about even numbers, it follows that any odd number greater than 7 can be represented as the sum
Multidisciplinary Journal www.ajmrd.com Page | 22 The proof of the Riemann conjecture
of three odd primes. The latter is called the "weak Goldbach conjecture" or "Goldbach conjecture
about odd numbers". If Goldbach''s conjecture is true about even numbers, then Goldbach''s
conjecture about odd numbers will also be true.Twin primes are pairs of prime numbers that differ
by 2, such as 3 and 5,5 and 7,11 and 13. . This conjecture, formally proposed by Hilbert in
Question 8 of his report to the International Congress of Mathematicians in 1900, can be described
as follows:There are infinitely many prime numbers p such that p + 2 is prime.
Prime pairs (p, p + 2) are called twin primes.In 1849, Alphonse de Polignac made the general
conjecture that for all natural numbers k, there are infinitely many prime pairs (p, p + 2k). The
case of k = 1 is the twin prime conjecture.

II. Reasoning
Before 1900, mathematicians treated 1 as the smallest prime number. This paper will restore this
tradition and treat 1 as the smallest odd prime number, so 1 is the smallest prime number.
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 3 plus (x-1) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... = ( is prime, is prime, ).

If only prime numbers are recorded, and 1 plus x (x ) and 3 plus (x

( ) will always become an infinite number of prime numbers, expressed

by { }( ), which is 3, 7, 11, 13, 17,... ,

If only the primes are taken, it is obvious that 1 plus x ( ) and

3 plus (x-1) ( ) all become the same infinite number of primes 1, 3, 5, 7, 11,

13, 17, ... , with { } ( as the prime, ) and { } ( as the prime numbers, ),

then = ( into primes, as the prime, ). 1 plus (x ) and 1

plus (x ) difference 2, while 3 plus (x-1) (x ) and 3 plus

(x-1) (x ) difference 2, Considering the set of prime numbers { }, { }

and , and differ by 2 or and differ by 2, so 2 can be written as the difference

of infinitely many pairs of prime numbers, This is exactly what the twin prime conjecture
describes. The twin prime conjecture is whether there are infinite pairs of prime numbers
that differ by 2. Using Euclid''s theorem that there are infinite prime numbers, we can easily
prove the twin prime conjecture.
And here we go:
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 5 plus (x-2) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... = ( is prime, is prime, ).

Multidisciplinary Journal www.ajmrd.com Page | 23 The proof of the Riemann conjecture
If only prime numbers are recorded, and 1 plus x ( ) and 5 plus

(x ( ) will always become an infinite number of prime numbers,

expressed by { }( ), which is 5, 7, 11, 13, 17,... ,

If only the primes are taken, it is obvious that 1 plus x ( ) and

5 plus (x-2) ( ) all become the same infinite number of primes 1, 3, 5, 7, 11,

13, 17, ... , with { } ( as the prime, ) and { } ( as the prime numbers, ),

then = ( into primes, as the prime, ). 1 plus (x ) and 1

plus (x ) difference 4, while 5 plus (x-2) (x ) and 5 plus

(x-2) (x ) difference 4, Considering the set of prime numbers { }, { }

and , and differ by 4 or and differ by 4, so 4 can be written as the difference

of infinitely many pairs of prime numbers.
And here we go:
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 7 plus (x-3) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... , = ( is prime, is prime, ).

If only prime numbers are recorded, and 1 plus x ( ) and 7 plus

(x-3)( ) will always become an infinite number of prime numbers,expressed

by { }( ), which is 7, 11, 13, 17,... ,

If only the primes are taken, it is obvious that 1 plus x ( ) and

7 plus (x-3) ( ) all become the same infinite number of primes 1, 3, 5, 7, 11,

13, 17, ... , with { } ( as the prime, ) and { } ( as the prime numbers, ),

then = ( into primes, as the prime, ). 1 plus (x ) and 1

plus (x ) difference 6, while 7 plus (x-3) (x ) and 7 plus

(x-3) (x ) difference 6, Considering the set of prime numbers { }, { }


and , and differ by 6 or and differ by 6, so 6 can be written as the difference

of infinitely many pairs of prime numbers.
And here we go:
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 9 plus (x-4) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... = ( is prime, is prime, ).

If only prime numbers are recorded, and 1 plus x ( ) and 9 plus

(x-4) ( ) will always become an infinite number of prime numbers,

expressed by { }( ), which is (9), 11, 13, 17,...(21) ,23, …,

If only the primes are taken, it is obvious that 1 plus x ( ) and

9 plus (x-4) ( ) all become the same infinite number of primes 1, 3, 5, 7, 11,

13, 17, ... , with { } ( as the prime, ) and { } ( as the prime numbers, ),

Multidisciplinary Journal www.ajmrd.com Page | 24 The proof of the Riemann conjecture
then = ( into primes, as the prime, ). 1 plus (x ) and 1 plus

(x ) difference 8, while 9 plus (x ) and 9 plus

(x ) difference 8, Considering the set of prime numbers { }, { }


and , and differ by 8 or and differ by 8, so 8 can be written as the difference

of infinitely many pairs of prime numbers.
And here we go:
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 11 plus (x-5) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... = ( is prime, is prime, ).

If only prime numbers are recorded, and 1 plus x ( ) and 11 plus

(x-5) ( ) will always become an infinite number of prime numbers,

expressed by { }( ), which is 11, 13, 17,...(21) ,23, …,

If only the primes are taken, it is obvious that 1 plus x ( ) and

11 plus (x-5) ( ) all become the same infinite number of primes 1, 3, 5, 7,

11, 13, 17, ... , with { } ( as the prime, ) and { } ( as the prime numbers, ),

then = ( into primes, as the prime, ). 1 plus (x ) and 1 plus

(x ) difference 10, while 11 plus (x ) and 11 plus

(x ) difference 10, Considering the set of prime numbers { }, { }

and , and differ by 10 or and differ by 10, so10 can be written as the

difference of infinitely many pairs of prime numbers.
And here we go:
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 13 plus (x-6) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... = ( is prime, is prime, ).

If only prime numbers are recorded, and 1 plus x ( ) and 13 plus

(x-6) ( ) will always become an infinite number of prime numbers,

expressed by { }( ), which is 13, 17,19,...(21) ,23, …

If only the primes are taken, it is obvious that 1 plus x ( ) and

13 plus (x-6) ( ) all become the same infinite number of primes 1, 3, 5, 7,

13, 17,19, ... ...(21),23, with { } ( as the prime, ) and { } ( as the prime numbers,

), then = ( into primes, as the prime, ). 1 plus (x )

and 1 plus (x ) difference 12, while 13 plus (x )

and 13 plus (x ) difference 12, Considering the set of prime

numbers { }, { } and , and differ by 12 or and differ by 12, so12 can be

written as the difference of infinitely many pairs of prime numbers.

Multidisciplinary Journal www.ajmrd.com Page | 25 The proof of the Riemann conjecture

And here we go:
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 15 plus (x-7) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... = ( is prime, is prime, ).

If only prime numbers are recorded, and 1 plus x ( ) and 15 plus

(x-7) ( ) will always become an infinite number of prime numbers,

expressed by { }( ), which is (15), 17,19,...(21) ,23, …,

If only the primes are taken, it is obvious that 1 plus x ( ) and

15 plus (x-7) ( ) all become the same infinite number of primes 1, 3, 5, 7,

13, 17,19, ... ,...(21),23,…, with { } ( as the prime, ) and { } ( as the prime

numbers, ), then = ( into primes, as the prime, ). 1 plus (x

) and 1 plus (x ) difference 14, while 15 plus (x

) and 15 plus (x ) difference 14, Considering the set of

prime numbers { }, { } and , and differ by 14or and differ by 14, so 14 can

be written as the difference of infinitely many pairs of prime numbers.
And here we go:
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 15 plus (x-7) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... = ( is prime, is prime, ).

If only prime numbers are recorded, and 1 plus x ( ) and 15 plus

(x-7) ( ) will always become an infinite number of prime numbers,

expressed by { }( ), which is (15), 17,19,...(21) ,23, …,

If only the primes are taken, it is obvious that 1 plus x ( ) and

15 plus (x-7) ( ) all become the same infinite number of primes 1, 3, 5, 7,

13, 17,19, ... ,...(21),23,…, with { } ( as the prime, ) and { } ( as the prime

numbers, ), then = ( into primes, as the prime, ). 1 plus (x

) and 1 plus (x ) difference 14, while 15 plus (x

) and 15 plus (x ) difference 14, Considering the set of

prime numbers { }, { } and , and differ by 14 or and differ by 14, so 14 can

be written as the difference of infinitely many pairs of prime numbers.
And here we go:
if we only remember prime numbers, according to Euclid''s theorem that there are infinitely
many prime numbers, then,1 plus x (x ) will always become an infinite

number of prime numbers, expressed by { }( is a prime, ), is

1,3,5,7,11,13,17,... ,
Multidisciplinary Journal www.ajmrd.com Page | 26 The proof of the Riemann conjecture
If only prime numbers are recorded, 17 plus (x-8) ( ) will always become

an infinite number of prime numbers, expressed by { }( is prime, ),

1,3,5,7,11,13,17,... = ( is prime, is prime, ).

If only prime numbers are recorded, and 1 plus x ( ) and 17 plus

(x-8) ( ) will always become an infinite number of prime numbers,

expressed by { }( ), which is 17,19,...(21) ,23, …,

If only the primes are taken, it is obvious that 1 plus x ( ) and

17 plus (x-8) ( ) all become the same infinite number of primes 1, 3, 5, 7,

13, 17,19, ... ,...(21),23,…, with { } ( as the prime, ) and { } ( as the prime

numbers, ), then = ( into primes, as the prime, ). 1 plus (x

) and 1 plus (x ) difference 16, while 15 plus (x

) and 15 plus (x ) difference 16, Considering the set of

prime numbers { }, { } and , and differ by 16 or and differ by 16, so 16 can

be written as the difference of infinitely many pairs of prime numbers.
…,
And so on,
If only prime numbers are recorded, p (p is any prime number) plus (x-j) (x ,

will always become an infinite number of prime numbers, expressed by { }( is

prime, ), which is 1,3,5,7,11,13,17,... = ( is prime, is prime, ).

1 plus x (x and x ≥j and j ) and p(p is any prime number) plus (x-j) ( x

and x ≥j, j ) ， give , ,( ),…, …, the odd number is placed in

parentheses, and when the odd number is removed, the same infinite number of prime
numbers will be obtained, represented by ( is

prime, ), … … because 1 and (or ) differ by 2k( ), which

is 1 plus x (x ) and 1 plus x (x ) differ by

2k( ), 1 plus (x ) and 1 plus (x )


difference 2k( ), while p plus (x ) and p plus

(x ) difference 2k( ), Considering the set of prime

numbers { }, { } and , and differ by 2k( ) or and differ by 2k( ),

so 2k( ) can be written as the difference of infinitely many pairs of prime numbers.

This is exactly what the Polignac conjecture describes, so the Polignac conjecture holds.
If Polignac''s conjecture is true, Goldbach''s conjecture automatically holds. Because
and are both prime numbers, If , according to the Polignac

conjecture, then = , = , so


= , because represents all even

numbers, so 2k= {0,2,4,6,8,10,... }, and is prime and ≥1, so 2 is twice the number

of all primes, 2 = {2,6,10,14,... , 2p} (p is prime),2k+2 represents all even sets

{0,2,4,6,8,10,...}, the value of each element in {0,2,4,6,8,10,...}is added to the set
{2,6,10,14,... ,2p} (p is prime), the result is still all even, so 2k+ = {0,2,4,6,8,10,... }, so it

can still be expressed as 2k(k is a non-negative integer), then 2k+2 =

{0,2,4,6,8,10,... ,}=2k( ), then + =2k( ).Obviously

Multidisciplinary Journal www.ajmrd.com Page | 27 The proof of the Riemann conjecture
+ =2k( ) is what Goldbach''s conjecture describes. is at least the

sum of a pair of prime numbers, and and can be equal or unequal, so even

numbers greater than zero can be at least written as the sum of a pair of prime
numbers.Since ( is a prime number, ) and ( is a prime number, )

can be expressed as an infinite number of prime numbers, but since the value of the
specific even number is finite, according to = , We can see that

all even numbers can be expressed as the sum of finite pairs of prime numbers,
i.e.Goldbach''s conjecture holds. According to the Polignac conjecture,there are an infinite
number of primes and by 2k ( ).That is


.Also have an infinite number of primes ， are 2k

( ), namely = ， .So

. Then 、 、 form an arithmetic sequence with a tolerance of

2k( ), and there are infinitely many groups, so there are infinitely many groups of

arithmetic sequences made up of prime numbers.

III. Conclusion
The Polignac conjecture, the twin prime conjecture, the Goldbach conjecture are perfectly
valid.

IV. Acknowledgements
Thank you for reading this paper.
V.Contributions
The sole author, poses the research question, demonstrates and proves the question.

VI. Introduction of the author
Name: Teng Liao ([e-mail])
Setting: Tianzheng International Institute of Mathematics and Physics, Xiamen, China
Address: No.237, Gaoqi Airport Road, Huli District, Xiamen City, China
Zip Code: 361001

Reference
[1] 《Problems related to flip graph Equation 》；
[2] Riemann : 《On the Number of Prime Numbers Less than a Given Value 》；
John Derbyshire(America): 《PRIME OBSESSION 》P218,BERHARD RIEMANN
AND THE GREATEST UNSOlVED PROBLEM IN MATHMATICS,Translated by Chen
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Weifeng, Shanghai Science and Technology Education Press,
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