# Solution for Goldbach Conjecture

• hayanisel
In summary, Wilber Valgusbitkevyt provides a solution for Goldbach's Conjecture. He starts by creating another theorem, Victoria Hayanisel Theorem 4, which states that for any positive integer X >= 4, there exists at least one prime number smaller than X that does not divide X. He then proves that all even numbers can be represented as the sum of two odd prime numbers with Arbitrary Modular Arithmetic and Fermat's Infinite Descent Method. Next, he proves that the number of possible X – A such that guarantees X + A to be a prime number is at least 1 for any X >= 4. Finally, he provides a solution for Goldbach's Conjecture by showing that
hayanisel
Can anyone find any stupid mistake in this? Also, can I get some professor names to send solutions for unsolved math problems? I have the solutions for Goldbach Conjecture, Polignac's Conjecture, Hadwiger Conjecture, Ringel-Kotzig Conjecture, Collatz Conjecture, Erdős conjecture on arithmetic progressions, and Erdős–Faber–Lovász conjecture. May I get some professor names that I can send the papers to verify them, mostly to see if there is any stupid mistake.

Title: Solution for Goldbach’s Conjecture

Author: Wilber Valgusbitkevyt

Abstract: For any given even number 2X, there exists prime numbers which can be noted as X – A. If X – A (mod Pq) =/= 2X for all q’s, then X + A is a prime number, and the sum of X – A and X + A is 2X for all even numbers bigger than 4. This is shown with Arbitrary Modular Arithmetic and Fermat’s Infinite Descent Method. Then, it is shown that the number of possible X – A such that guarantees X + A to be a prime number is at least 1 for any X >= 4. In other words, all even numbers can be represented as the sum of two odd prime numbers.

Proof: I am going to start by creating another theorem Victoria Hayanisel Theorem 4: For any positive integer X >= 4, there exists at least one prime number smaller than X that does not divide X. Although this is not used in my proof for Goldbach’s Conjecture, it is still a new theorem with a proof that helps describe the problem.

Suppose that X is the multiple of all prime numbers smaller than X. Then, X – 1 is a prime number. But if X – 1 is included in the set of the prime factors of X, then it cannot be X – 1 (obviously, some positive integer multiplied by X – 1 cannot be X). Hence, there is at least one prime number smaller than X that doesn’t divide X for all X >= 4.

Let’s get back to the proof for Goldbach Conjecture:

Consider all even numbers 2X such that X > = 4. For 2X when X = 2 and X = 3, 4 = 2 + 2 and 6 = 3 + 3 hence the conjecture is true. For 2X when X >= 4, consider Victoria Hayanisel Theorem 4. Consider the following arbitrary modular arithmetic. Consider e + 1 prime numbers less than X + A. Let:

X – A (mod P1) = J1 X (mod P1) = I1 A (mod P1) = I1 – J1

………………………… ………………………… …………………………

X – A (mod X - A) = 0 X (mod X - A) = 0 A (mod X – A) = A

……………………….. ……………………….. …………………………

X – A (mod Pe) = Je X (mod Pe) = Ie A (mod Pe) = Ie - Je

X + A (mod P1) = 2I1 – J1

…………………………

X + A (mod X - A) = 2A

………………………..

X + A (mod Pe) = 2Ie – Je

Now, the question is whether there exists X + A which is a prime number given some X – A which is a prime number, for some A at given X. In other words: Given j1, …, je are not 0, 2Iq – Jq =/= 0. In other words, 2Iq (mod Pq) =/= Jq.

Suppose 2Iq (mod Pq) = Jq. If this were true, 2Iq (mod Pq) = X – A since X – A (mod Pq) = Jq.

Let’s rearrange this equation: X – A (mod Pq) = 2Iq.

But, X (mod Pq) = Iq, which means 2X (mod Pq) = 2Iq.

Hence, X – A (mod Pq) = 2X.

Hence, if there exists X – A such that X – A (mod Pq) =/= 2X for all q, then Goldbach’s Conjecture is true.

The last piece of the puzzle: Is there such X – A for a given X such that X – A (mod Pq) =/= 2X for all q’s?

I am going to create a three-dimensional sequence called Victoria Hayanisel Sequence:

T 1 2 3 4 5 6 7 8 9 10 …
2 1 0 1 0 1 0 1 0 1 0 …
3 1 2 0 1 2 0 1 2 0 1 …
5 1 2 3 4 0 1 2 3 4 0 …
…………………………………………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………………………………………

The first row is just the sequence of positive integers. The first column is the sequence of prime numbers. The other rows are the sequence of the remainders when the first column divides the first row.

Given i prime numbers, there are P1 x P2 x … x Pi number of combinations of the remainders. Given a specific set of remainders, there are (P1 – 1) x (P2 – 1) x … x (Pi – 1) number of different possible combination of remainders.

The last piece of the puzzle was “is there such X – A for a given X such that X – A (mod Pq) =/= 2X for all q’s?” The answer is yes, and there are (P1 – 1) x (P2 – 1) x … x (Pi – 1) number of them. Since 2X is an even number, we are interested in odd prime numbers, and odd prime numbers are always not divisible by 2, we can forget about 2. Hence, let’s consider only from 3. Now, the question is “is one of them a prime number?” The answer is yes. There are at least (P2 – 2) x … x (Pi – 2) of them (to be accurate we would exclude itself, but it is irrelevant for this as it makes no difference because it is still bigger than 1), which is always bigger than 1, because whatever the given set of remainders are, exclude them and 0’s from each rows. so that it would be prime because its remainders are set of non-zeros and each element is different from the given set of remainders.
Hence, given X – A (mod Pq) =/= 2X for all q’s, X + A has to be a prime number, and their sum is 2X for all X >= 4. For 2X when X = 2 and X = 3, 2X can be assigned 2 + 2 and 3 + 3 to it. For the rest, the proof is applicable. There exists at least one such X – A for all X. Hence, Goldbach’s Conjecture is true for all even numbers.

hayanisel said:
Can anyone find any stupid mistake in this? Also, can I get some professor names to send solutions for unsolved math problems? I have the solutions for Goldbach Conjecture, Polignac's Conjecture, Hadwiger Conjecture, Ringel-Kotzig Conjecture, Collatz Conjecture, Erdős conjecture on arithmetic progressions, and Erdős–Faber–Lovász conjecture. May I get some professor names that I can send the papers to verify them, mostly to see if there is any stupid mistake.

Title: Solution for Goldbach’s Conjecture

Author: Wilber Valgusbitkevyt

Abstract: For any given even number 2X, there exists prime numbers which can be noted as X – A. If X – A (mod Pq) =/= 2X for all q’s,

*** What is "Pq"? ***

then X + A is a prime number, and the sum of X – A and X + A is 2X for all even numbers bigger than 4. This is shown with Arbitrary Modular Arithmetic and Fermat’s Infinite Descent Method. Then, it is shown that the number of possible X – A such that guarantees X + A to be a prime number is at least 1 for any X >= 4. In other words, all even numbers can be represented as the sum of two odd prime numbers.

Proof: I am going to start by creating another theorem Victoria Hayanisel Theorem 4: For any positive integer X >= 4, there exists at least one prime number smaller than X that does not divide X. Although this is not used in my proof for Goldbach’s Conjecture, it is still a new theorem with a proof that helps describe the problem.

*** This might seem interesting to you but in fact it's a trivial claim, since $X\geq 4\Longrightarrow (X-1)\nmid X$ , and thus any prime dividing X - 1 doesn't divide X...***

Suppose that X is the multiple of all prime numbers smaller than X. Then, X – 1 is a prime number. But if X – 1 is included in the set of the prime factors of X, then it cannot be X – 1 (obviously, some positive integer multiplied by X – 1 cannot be X). Hence, there is at least one prime number smaller than X that doesn’t divide X for all X >= 4.

Let’s get back to the proof for Goldbach Conjecture:

Consider all even numbers 2X such that X > = 4. For 2X when X = 2 and X = 3, 4 = 2 + 2 and 6 = 3 + 3 hence the conjecture is true. For 2X when X >= 4, consider Victoria Hayanisel Theorem 4. Consider the following arbitrary modular arithmetic. Consider e + 1 prime numbers less than X + A. Let:

X – A (mod P1) = J1 X (mod P1) = I1 A (mod P1) = I1 – J1

………………………… ………………………… …………………………

X – A (mod X - A) = 0 X (mod X - A) = 0 A (mod X – A) = A

……………………….. ……………………….. …………………………

X – A (mod Pe) = Je X (mod Pe) = Ie A (mod Pe) = Ie - Je

X + A (mod P1) = 2I1 – J1

…………………………

X + A (mod X - A) = 2A

………………………..

X + A (mod Pe) = 2Ie – Je

Now, the question is whether there exists X + A which is a prime number given some X – A which is a prime number, for some A at given X. In other words: Given j1, …, je are not 0, 2Iq – Jq =/= 0. In other words, 2Iq (mod Pq) =/= Jq.

Suppose 2Iq (mod Pq) = Jq. If this were true, 2Iq (mod Pq) = X – A since X – A (mod Pq) = Jq.

Let’s rearrange this equation: X – A (mod Pq) = 2Iq.

But, X (mod Pq) = Iq, which means 2X (mod Pq) = 2Iq.

Hence, X – A (mod Pq) = 2X.

Hence, if there exists X – A such that X – A (mod Pq) =/= 2X for all q, then Goldbach’s Conjecture is true.

The last piece of the puzzle: Is there such X – A for a given X such that X – A (mod Pq) =/= 2X for all q’s?

I am going to create a three-dimensional sequence called Victoria Hayanisel Sequence:

T 1 2 3 4 5 6 7 8 9 10 …
2 1 0 1 0 1 0 1 0 1 0 …
3 1 2 0 1 2 0 1 2 0 1 …
5 1 2 3 4 0 1 2 3 4 0 …
…………………………………………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………………………………………

The first row is just the sequence of positive integers. The first column is the sequence of prime numbers. The other rows are the sequence of the remainders when the first column divides the first row.

Given i prime numbers, there are P1 x P2 x … x Pi number of combinations of the remainders. Given a specific set of remainders, there are (P1 – 1) x (P2 – 1) x … x (Pi – 1) number of different possible combination of remainders.

The last piece of the puzzle was “is there such X – A for a given X such that X – A (mod Pq) =/= 2X for all q’s?” The answer is yes, and there are (P1 – 1) x (P2 – 1) x … x (Pi – 1) number of them. Since 2X is an even number, we are interested in odd prime numbers, and odd prime numbers are always not divisible by 2, we can forget about 2. Hence, let’s consider only from 3. Now, the question is “is one of them a prime number?” The answer is yes. There are at least (P2 – 2) x … x (Pi – 2) of them (to be accurate we would exclude itself, but it is irrelevant for this as it makes no difference because it is still bigger than 1), which is always bigger than 1, because whatever the given set of remainders are, exclude them and 0’s from each rows. so that it would be prime because its remainders are set of non-zeros and each element is different from the given set of remainders.
Hence, given X – A (mod Pq) =/= 2X for all q’s, X + A has to be a prime number, and their sum is 2X for all X >= 4. For 2X when X = 2 and X = 3, 2X can be assigned 2 + 2 and 3 + 3 to it. For the rest, the proof is applicable. There exists at least one such X – A for all X. Hence, Goldbach’s Conjecture is true for all even numbers.

It seems pretty obvious that you're not a mathematician and thus I'd advice you to try to get the help of one to whom you can explain

in person and in detail your stuff, and such that she/he will be ready to polish your ideas and your mathematicial writing. As it is now,

your paper above is pretty cumbersome and it's boring and pointless to try to decypher what you meant to write/express.

DonAntonio

Ps Who in the world is Victoria Hayanisel and why in the world do you insist in calling theorems by her name, making your writing even more cumbersome?

Pq refers to prime numbers such as P1, P2, P3, and so on. The "trivial claim" doesn't have much to do with the paper. I suppose I cannot make a theorem out of that one. The paper looks that complicated to need deciphering? All I used was variables with indexes, nothing more. Then there were just modular arithmetics with sequences.

How do I use math symbols, signs, and equations? Is that Latex?

No serious mathematician is going to read anything you have to say if you present your work like this. You have listed some of the most well known unsolved problems of mathematics, and claiming that you have solved any of them is viewed with the same skepticism as if someone were to claim the Earth is flat.

Before you do anything else, I suggest you read this advice article geared toward amateur mathematicians.

Quote from the article: "I've aimed it at people who think they've already solved famous problems"

This does not meet the guidelines of this forum.

## 1. What is the Goldbach Conjecture?

The Goldbach Conjecture is a mathematical hypothesis that states that every even number greater than 2 can be expressed as the sum of two prime numbers.

## 2. Why is the Goldbach Conjecture important?

The Goldbach Conjecture has remained unsolved for over 270 years and has been a topic of fascination for mathematicians and scientists. Its proof would provide a better understanding of the distribution of prime numbers and could potentially lead to new mathematical discoveries.

## 3. Has the Goldbach Conjecture been proven?

No, the Goldbach Conjecture has not been proven. While there is numerical evidence to support the conjecture, a formal proof has not been found.

## 4. What progress has been made towards solving the Goldbach Conjecture?

In the past few decades, there have been significant advances in tackling the Goldbach Conjecture, including the use of computer algorithms and advanced mathematical techniques. However, a complete proof remains elusive.

## 5. What would a solution to the Goldbach Conjecture mean?

A solution to the Goldbach Conjecture would be a monumental achievement in mathematics and would have significant implications for number theory and cryptography. It would also resolve one of the most famous unsolved problems in mathematics.

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