The Apocalyptic Solution to Goldbach's Conjecture

Mojtaba Jalali Koutanaei^*
*Correspondence: Mojtaba Jalali Koutanaei, Department of Geophysics, University of Tehran, Tehran, Iran, Email:

Keywords

Goldbach's conjecture • Number theory • Prime numbers

Introduction

The Goldbach conjecture is a yet unproven conjecture stating that every even integer greater than two is the sum of two prime numbers. The conjecture has been tested up to 400,000,000,000,000. Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. In 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture: Every integer greater than 2 can be written as the sum of three primes. He considered 1 to be a prime number, a convention subsequently abandoned. So today, Goldbach's original conjecture would be written: Every integer greater than 5 can be written as the sum of three primes. Euler, becoming interested in the problem, answered with an equivalent version of the conjecture: Every even number greater than 2 can be written as the sum of two primes, adding that he regarded this as a fully certain theorem, in spite of his being unable to prove it. The Goldbach conjecture, in number theory, is an assertion (here stated in modern terms) that every even counting number greater than 2 is equal to the sum of two prime numbers. Goldbach’s conjecture was published in English Mathematician Edward Waring’s” Meditationes algebraicae”, which also contained Waring’s problem and what was later known as Vinogradov’s theorem. The latter, which states that every sufficiently large odd integer can be expressed as the sum of three primes, was proved in 1937 by the Russian Mathematician Ivan Matveyevich Vinogradov. Further progress on Goldbach’s conjecture occurred in 1973, when the Chinese Mathematician Chen Jing Run proved that every sufficiently large even number is the sum of a prime and a number with at most two prime factors structures in a single function, making it our most important tool in the study of prime numbers [1].

Literature Review

Pseudo-prime number

It is a number that is divisible by prime numbers or other prime numbers. For example, 7 × 5=35

All numbers can be written as 6k, 6k-1, 6k-2, 6k-3, 6k-4, 6k-5, and only 6k-5, 6k-3, and 6k-1 produce pseudo-prime numbers. Among these three, only “6k-5” and “6k-1” generate all prime numbers along with some pseudo-prime numbers [2,3]. Therefore, the formula for twin primes is as follows:

P₁(k)=6k-5, P₂(k)=6k-1, where k=1,2,3,4, etc.

We consider only the numbers (2 and 3) as twin primes and denote them as P (0)=(2,3). The pseudo-prime numbers are twins and the distance between them is equal to 4 (Table 1).

Table 1. The pseudo-prime numbers are twins and the distance between them is equal to 4.

Definition 1: Pseudo Goldbach's conjecture: Every even natural number can be expressed as the sum of two prime pseudo numbers.

Proof of definition 1:

It is possible to combine two prime pseudo numbers in five different ways. In all five scenarios, the output numbers can be produced as “6k”, “6k-2” and “6k-4”.

Mode 1: 6k-2 ≡ (6k₁-5) +3

Mode 2: 6k-4 ≡ 6k1+2 ≡ (6k₁-1) +3

Mode 3: 6k-2 ≡ (6k1-1) + (6k₂-1)

Mode 4: 6k-4 ≡ 6k-10 ≡ (6k₁-5) + (6k₂-5)

Mode 5: 6k≡ 6k-6≡ (6k₁-5) + (6k₂-1)

Definition 2: Pseudo Goldbach's conjecture: Any natural number greater than 5 can be expressed as the sum of three pseudo prime numbers.

Proof of definition 2:

It is possible to combine three pseudo-prime numbers in the following eleven ways, and in the output of all six states, numbers can be produced as 6k, 6k-1, 6k-2,6k-3, 6k-4, 6k-5.

Mode 1: 6k ≡ (6k₁-5) + 3 + 2

Mode 2: 6k-2 ≡ 6k₁+4 ≡ (6k₁-1) + 3 + 2

Mode 3: 6k≡ (6k₁-1) + (6k₂-1) + 2

Mode 4: 6k-5≡ 6k+1≡ (6k₁-1) + (6k₂-1) + 3

Mode 5: 6k-4 ≡ (6k₁-5) + (6k₂-1) + 2

Mode 6: 6k-3 ≡ (6k₁-5) + (6k₂-1) + 3

Mode 7: 6k-2≡ 6k₁-8 ≡ (6k₁-5) + (6k₂-5) + 2

Mode 8: 6k-5≡ 6k₁-11 ≡ (6k₁-5) + (6k₂-5) + (6k₃-1)

Mode 9: 6k-1≡ 6k₁-7 ≡ (6k₁-5) + (6k₂-1) + (6k₃-1)

Mode 10: 6k-3 ≡ 6k₁-15 ≡ (6k₁-5) + (6k₂-5) + (6k₃-5)

Mode 11: 6k-3≡ (6k₁-1) + (6k₂-1) + (6k₃-1)

Prime number

A prime number is a number that is only divisible by one and itself, such as 7, 17.

Definition 3. Goldbach's conjecture: Every even natural number can be expressed as the sum of two prime numbers.

Proof of definition 3:

It is possible to combine two prime numbers in the following five ways. In the output of all five combinations, numbers can be produced as 6k, 6k-2, and 6k-4.

Mode 1: 6k-2 ≡ (6k₁-5) +3

Mode 2: 6k-4 ≡6k₁+2≡ (6k₁-1) +3

Mode 3: 6k-4 ≡ 6k-10≡ (6k₁-5) + (6k₂-5)

Mode 4: 6k-2 ≡ (6k₁-1) + (6k₂-1)

Mode 5: 6k ≡ 6k-6≡ (6k₁-5) + (6k₂-1)

The Bertrand-Chebyshev theorem is one of the theorems concerning prime numbers. This theorem states that for every natural number greater than 3, denoted as n, there exists a prime number, denoted as p, such that: n<p<2n-2.

Bertrand first stated this theorem in 1845 and Chebyshev proved it in 1850.

1) 6k’-4≡ (6k’1-5) + (6k’2-5) , k’=k’1+k’2-1

For every k₁>2 there exists k’₁ which, k₁<k'₁+k'₂-1

It will always be 6k₁-5<6k’₁-5<12k₁-176k₁

6<6k₁-5< 6k’₁-5<12k₁-17<12k₁-14

then 6k₁-6< 6k’1-5 <12k₁-14

On the other hand, according to Bertrand-Chebyshev Theorem, there is at least one prime number in this interval, which is 6k’₁-5.

6k₁-6< 6k’₁-5=p₁<12k₁-14 that: n₁=6k₁-6 (1)

For every k₂>2 there exists k’₂ which, k₂<k'2<2k₂-2

It will always be 6k₂-5<6k’₂ -5<12k₂-17

6k₂-6<6k₂-5<6k’₂ -5<12k₂-17<12k₂-14

then 6k₂-6<6k’₂-5<12k₂-14

On the other hand, according to Bertrand-Chebyshev theorem, there is at least one prime number in this interval, which is 6k’₂-5.

6k₂-6< 6k’2-5= p₂<12k₂-14 that: n₂=6k₂-6 (2)

If we combine relations 1 and 2, we will get:

(6k₁-6)+(6k₂-6)<(6k’₁-5)+(6k’₂-5)=p₁+p₂<(12k₁-14)+(12k₂ -14)

6(k₁+k₂-1)-6<6(k’₁+k’₂-1)-4)=p₁+p₂<12(k₁+k₂-1)-16

(6k-6)<6k’-4=p₁+p₂< (12k-16), that k’=k’₁+k’₂-1 (3)

Then: 6k’- 4=p₁+p₂ (4)

For every even number represented by k’ as 6k’-4, there are k’₁ and k’₂ representing prime numbers 6k’₁-5 and 6k’₂-5, respectively. This means that k’ can be expressed as the sum of k’₁ and k’₂, minus 1.

2) 6k’-2 ≡ (6k’₁-1) + (6k’₂-1), k’=k’₁+k’₂

For every k₁>2 there exists k’₁ which, k₁<k'₁<2k₁-2

It will always be 6k₁-1<6k’₁-1<12k₁-13

6k₁-2<6k₁-1<6k’₁-1<12k₁-13 <12k₁-6

then 6k₁-2<6k’₁-1<12k₁-6

On the other hand, according to Bertrand-Chebyshev theorem, there is at least one prime number in this interval, which is 6k’₁-1.

6k₁-2<6k’₁-1=p₁<12k₁-6 that: n₁=6k₁-2 (5)

For every k₂>2 there exists k’₂ which, k₂<k'₂<2k₂-2 It will always be 6k₂-1<6k'₂-1<12k₂-13

6k₂-2<6k₂-1<6k’₂-1<12k₂-13<12k₂-6 then

6k₂-2<6k’₂-1<12k₂-6

On the other hand, according to Bertrand-Chebyshev theorem, there is at least one prime number in this interval, which is 6k’₂-1.

6k₂-2<6k’₂-1=p₂ <12k₂-6 that: n₂=6k₂-2 (6)

If we combine relations 5 and 6, we will get:

(6k₁-2)+(6k₂-2)<(6k₁-1)+(6k₂-1)=p₁+p₂<(12k₁-6)+(12k₂-6) 6(k₁ +k₂)-4<6(k’₁+k’₂)-2)= p₁+p₂<12(k₁+k₂)-12

(6k-4)<6k’-2 = p₁+p₂ < (12k-12),

that k’=k’₁+k’₂ (7)

Then: 6k’-2=p₁+p₂ (8)

For every even number represented by k’ as 6k’-2, there are k’₁ and k’₂ representing prime numbers 6k’₁-1 and 6k’₂-1, respectively. This means that k’ can be expressed as the sum of k’₁ and k’₂.

3) 6k’≡ 6k’-6≡(6k'₁- 5) + (6k'₂-1) k’=k'₁+k’₂-1

For every k₁>2 there exists k’₁ which, k₁<k'₁<2k₁-2 It will always be 6k₁-5<6k'₁-5<12k₁-17

6k₁-6<6k₁-5<6k’₁-5<12k₁-17<12k₁-14

then 6k₁-6< 6k’₁-5<12k₁-14

On the other hand, according to Bertrand-Chebyshev theorem, there is at least one prime number in this interval, which is 6k’₁-5.

6k₁-6<6k’₁-5= p₁<12k₁-14 that: n₁=6k₁-6 (9)

For every k₂>2 there exists k’₂ which, k₂<k'₂<2k₂-2 It will always be 6k₂-1<6k’₂-1<12k₂-13

6k₂-2<6k₂ -1<6k’₂-1<12k₂-13<12k₂-6 then 6k₂-2<6k’₂-1<12k₂-6

On the other hand, according to Bertrand-Chebyshev theorem, there is at least one prime number in this interval, which is 6k’₂-1.

6k₂-2<6k’₂-1=p₂<12k₂-6 that: n₂=6k₂-2 (10)

If we combine relations 9 and 10, we will get:

(6k₁-6)+(6k₂-2)<(6k₁-5)+(6k₂-1)=p₁+p₂<(12k₁-14)+(12k₂-6)6(k₁ +k₂-1) 2<6(k’₁+k’₂-1)=p₁+p₂<12(k₁+k₂-1)-8

(6k-2)<6k’= p₁+p₂<(12k-8), that k’=k’₁+k’₂-1 (11)

Then: 6k’=p₁+p₂ (12)

For every even number represented by k’ as 6k’, there are k’₁ and k’₂ representing prime numbers 6k’₁-5 and 6k’₂-1, respectively. This means that k’ can be expressed as the sum of k’₁ and k’₂, minus 1.

Discussion

Relations 4, 8 and 12 demonstrate that every even number can be expressed as the sum of the two prime numbers, confirming Goldbach's conjecture.

Definition 4: Goldbach's conjecture: Any natural number greater than 5 can be expressed as the sum of three prime numbers [4-6].

Proof of definition 4:

Assuming the number is even

Every even number can be written as another even number plus 2. According to theorem 3, every even number is the sum of two prime numbers. In general, every even number can be expressed as follows:

N_EVEN=P₁+P₂+2

Assuming the number is odd

Every odd number can be written as the sum of an even number and a prime number. According to theorem 3, every even number is the sum of two prime numbers. In general, every odd number can be expressed as follows:

N_ODD= P₁+P₂+P₃

Conclusions

All numbers can be written as 6k, 6k-1, 6k-2, 6k-3, 6k-4, or 6k-5. Only “6k-5” and “6k-1” generate all prime numbers along with some pseudo-prime numbers. The formula for twin primes is as follows P₁(k)=6k-5, P₂(k)=6k-1 where k=1,2,3,4, etc. The prime numbers are twins and the distance between them is always 4. It is possible to combine two prime numbers in the following 5 ways, resulting in numbers expressed as 6k, 6k-2 and 6k-4. Using the Bertrand- Chebyshev theorem, the existence of two prime numbers within the specified interval was proven. An even number can always be written as the sum of these two confirmed prime numbers, so it corresponds to Goldbach's conjecture.

References

1. Horvat, Tim. "No Country for Old Mathematicians." US- China Econ Rev 5 (2015): 123-132.

   [Google Scholar]

2. Worley, Sara. "Conceivability, possibility and physicalism." Analysis 63 (2003): 15-23.

   [Google Scholar]

3. Avadanei, DragoÈ?. "Narrating The Undefined: A Math Melodrama (Thought) Experiment." J Rom Lit Stud 29 (2022): 146-153.

   [Google Scholar]

4. Chen, Bo. "K ripke's Epistemic Argument Against Descriptivism Revisited." J Chin Philos 40 (2013): 544-562.

   [Crossref] [Google Scholar]

5. Szabo, Zoltan Gendler. "Believing in things." Philos Phenomen Res 66 (2003): 584-611.

   [Crossref] [Google Scholar]

6. Baber, Walter F. "The arts of the natural: Herbert Simon and artificial intelligence." Public Adm Q 12 (1989): 329-347.

   [Google Scholar]