7American Journal of Multidisciplinary Research & Development (AJMRD)
Volume XX, Issue XX (June - 2023), PP 22-27
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 2(x ) will always become an infinite number of prime numbers,
represented by { }( is prime, ), is
1,3,5,7,11,13,17,... ,
3 plus (x-1) 2(x ) will always become an infinite number of prime numbers,
expressed by { }( is prime, ), 1, 3, 5, 7, 11, 13, 17,... ， = ( is prime, is
prime, ).3 plus x 2(x ) always becomes an infinite number of prime
numbers, expressed by { } ( is prime, ) as, 3,5,7,11,13,17,...
And 1 plus (x+1) ( x ) and 3 plus x 2(x ) always become the
same infinite number of prime numbers at the same time, 3, 5, 7, 11, 13, 17,... Is
expressed by { }( is a prime number, ).
If we only remember the prime numbers, obviously (1+2) plus x is
the same as 3 plus x , which is equal to an infinite number of prime
numbers , and the difference between 3 plus x and 1 plus x
(x ) is 2 , so the value of 1 plus x is different by 2
from the value of 3 plus x .1 plus x (x ) equals an
infinite number of primes ( is a prime, ), (1+2) plus x
equals an infinite number of primes ，then the difference between and is 2, so
2 can be written as the difference of infinitely many primes, which 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 only prime numbers are recorded, 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,... ,
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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, ).
If only prime numbers are recorded, 5 plus x (x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ) as
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 5 plus
(x-2) ( ) differ by 4, so 4 can be written as the difference between
infinitely many prime numbers.
Here''s another look:
If only prime numbers are recorded, 1 plus x (x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ), is
1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 7 plus (x-3) ( x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ),
1,3,5,7,11,13,17,... = ( is prime, is prime, ).
1 plus x (x and x≥3) and 7 plus (x-3) (x and x≥3) always become the same
infinite number of prime numbers, 7, 11, 13, 17,... ,is expressed by { }( is a prime
number, ). The difference between 1 and 7 is 6, is also that 1 plus x (x
) and 1 plus x (x and x≥3) difference by 6, so 1 plus x ( ) and 7
plus (x-3) ( ) differ by 6,taking into account the set of prime numbers
{ }, { } and , 6 can be written as the difference of infinitely many pairs of prime
numbers.
Here''s another look:
If only prime numbers are recorded, 1 plus x (x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ), is
1,3,5,7,11,13,17,... ,
If only prime numbers are recorded, 9 plus (x-4) (x ) will always become an
infinite number of prime numbers, expressed by { } ( is prime, ), is
1,3,5,7,11,13,17,... = ( is prime, is prime, ).
1 plus x (x and x≥4) and 9 plus (x-4) (x and x≥4) both get (9), 11, 13, (15),
17,... ,Odd numbers are placed in parentheses, and when odd numbers are removed, the
same infinite number of prime numbers will be obtained,expressed by ( is a prime
number, ), is 11,13,,17,... . Since 1 and 9 differ by 8, is also 1 plus x 2(x
) and 1 plus x 2 (x and x≥4) differ by 8, so 1 plus x ( ) and 9 plus
(x-4) ( ) differ by 8,taking into account the set of prime numbers { }, { }
and , 8 can be written as the difference of infinitely many pairs of prime numbers.
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If only prime numbers are recorded, 1 plus x (x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ), is
1,3,5,7,11,13,17,... ,
0will always become an infinite number of prime numbers, expressed by {Qi}(Qi is
= ( is prime, is prime, ).
1 plus (x and x ≥5) and 11 plus (x-5) (x and x≥5) both 11, 13, (15), 17,...
(21), 23,... , Odd numbers are placed in parentheses, and when odd numbers are
removed, the same infinite number of prime numbers will be obtained,expressed by
( is a prime number, ), is 11,13,17, ... . Since 1 and 11 differ by 10, is also 1 plus
x (x ) and 1 plus x (x and x≥5) differ by 10, so 1 plus x (
) and 11 plus (x-5) ( ) differ by 10,taking into account the set
of prime numbers { }, { } and , 10 can be written as the difference of infinitely many
pairs of prime numbers.
Here''s another look:
If only prime numbers are recorded, 1 plus x (x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ), is
1,3,5,7,11,13,17,... ,
0will always become an infinite number of prime numbers, expressed by QiQi is prime,
i Z+, 1,3,5,7,11,13,17,... ,Pi=Qi(Pi is prime, Qi is prime, i Z+).
1 plus (x or x ≥6) and 13 plus (x-6) (x or x ≥6) both 13 ，(15), 17 ， … ，(21)
23 ， …，Odd numbers are placed in parentheses, and when odd numbers
are removed, the same infinite number of prime numbers will be obtained,expressed by
( is a prime number, ), is ， ， , Since 1 and 13 differ by 12, is also
1 plus x 2 (x ) and 1 plus x (x or x ≥6) differ by 12, taking into account
the set of prime numbers { }, { } and , 12 can be written as the difference of infinitely
many pairs of prime numbers.
Here''s another look:
If only prime numbers are recorded, 1 plus x (x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ), is
1,3,5,7,11,13,17,... ,
0will always become an infinite number of prime numbers, expressed by QiQi is prime,
i Z+, 1,3,5,7,11,13,17,... ,Pi=Qi(Pi is prime, Qi is prime, i Z+).
Multidisciplinary Journal www.ajmrd.com Page | 25 The proof of the Riemann conjecture
1 plus x 2(x or x ≥7) and 15 plus (x-7) (x or x ≥7) both
(15),17, …,(21),23,(25),(27),29, …,Odd numbers are placed in parentheses, and when odd
numbers are removed, the same infinite number of prime numbers will be
obtained,expressed by ( is a prime number, ), is Since 1
and 15 differ by 14, is also 1 plus x (x ) and 1 plus x (x or x ≥7) differ
by 14, taking into account the set of prime numbers { }, { } and , 14 can be written as
the difference of infinitely many pairs of prime numbers.
1,3,5,7,11,13,17, …,
Here''s another look:
If only prime numbers are recorded, 1 plus x (x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ), is
1,3,5,7,11,13,17, …,
0will always become an infinite number of prime numbers, expressed by QiQi is prime,
i Z+ ，1,3,5,7,11,13,17 Pi=Qi(Pi is prime, Qi is prime, i Z+).
1 plus x (x and x≥8) and 17 plus (x-8) (x and x≥8) both
17,19,(21),23,(25),(27),29, …, Odd numbers are placed in parentheses, and when odd
numbers are removed, the same infinite number of prime numbers will be
obtained,expressed by ( is a prime number, ), is Since 1
and 17 differ by 16, is also 1 plus x (x ) and 1 plus x (x and x≥8) differ
by 16, taking into account the set of prime numbers { }, { } and , 16 can be written as
the difference of infinitely many pairs of prime numbers.
…
And so on:
If only prime numbers are recorded, 1 plus x (x ) will always become an
infinite number of prime numbers, expressed by { }( is prime, ), is
1,3,5,7,11,13,17,... ,
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
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2k( ), at the same time considering primes set { 、 , so 2k( ) can
be written as the difference of infinitely many 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. 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 + =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 (1509135693@139.com)
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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
Weifeng, Shanghai Science and Technology Education Press,
China,https://www.doc88.com/p-54887013707687.html;
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