1. ## Goldbach's Conjecture

We know that the strong Goldbach conjecture:

"Every even number $\displaystyle \ge 4$ is the sum of two primes"

is an open question with a price on its head.

What I want to know is has a resticted version of this been proven for
special types of positive even integers, say the squares, cubes, whatever?

RonL

2. Originally Posted by CaptainBlank
What I want to know is has a resticted version of this been proven for
special types of positive even integers, say the squares, cubes, whatever?
Unsolved problem sent as a letter to Euler in 1742.

The numerical results are astonding, for example there are 219,400 such representations for 100,000,000. And in fact it seems as the number of possible representations keeps increasing. And there is nobody who seriously believes that this conjecture is false.

It is know that each even integer is a sum of six or fewer primes.

The first real progess was made by Hardy and Littlewood in 1922, they have shown that for a sufficiently large integer the Conjecture of Goldbach is true! But the sad news is that Hardy and Littlewood used the Generalized Riemann Hypothesis. In 1937 the Russian mathematician Vinogradov was to remove the necessatity of using the generalized Riemann Hypothesis and established that all sufficiently large integers are some of three odd primes. But he was unable to decide how large this number is. In 1956 Borozdkin proved that $\displaystyle 3^{3^{15}}$ is big enough. And in 1989 this number was reduced to $\displaystyle 10^{4300}$. Hence if it can be checked by hand that all of those numbers are true, then the Goldbach Conjecuture is settled. But it is far to difficult for a man to do. And even if we accept computer results (because I do not accept anything that a computer does and many mathematicians agree) it is still computationally difficult.

Note: I have taken the above facts from my Number Theory book. I have changed the wording to avoid copyright material.

3. Originally Posted by ThePerfectHacker
Unsolved problem sent as a letter to Euler in 1742.

The numerical results are astonding, for example there are 219,400 such representations for 100,000,000. And in fact it seems as the number of possible representations keeps increasing. And there is nobody who seriously believes that this conjecture is false.

It is know that each even integer is a sum of six or fewer primes.

The first real progess was made by Hardy and Littlewood in 1922, they have shown that for a sufficiently large integer the Conjecture of Goldbach is true! But the sad news is that Hardy and Littlewood used the Generalized Riemann Hypothesis. In 1937 the Russian mathematician Vinogradov was to remove the necessatity of using the generalized Riemann Hypothesis and established that all sufficiently large integers are some of three odd primes. But he was unable to decide how large this number is. In 1956 Borozdkin proved that $\displaystyle 3^{3^{15}}$ is big enough. And in 1989 this number was reduced to $\displaystyle 10^{4300}$. Hence if it can be checked by hand that all of those numbers are true, then the Goldbach Conjecuture is settled. But it is far to difficult for a man to do. And even if we accept computer results (because I do not accept anything that a computer does and many mathematicians agree) it is still computationally difficult.

Note: I have taken the above facts from my Number Theory book. I have changed the wording to avoid copyright material.
Yea, yea, I know all that, if I could find the information I'm after in an
easily accessible souce I would not be asking

(By the way the text of Goldbach's letter is available here)

RonL

4. Originally Posted by CaptainBlank

(By the way the text of Goldbach's letter is available here)
Look interesting.

So Euler knew German?

Can somebody please translate (by somebody I mean Earboth).

5. Originally Posted by ThePerfectHacker
Look interesting.

So Euler knew German?

Can somebody please translate (by somebody I mean Earboth).

RonL

6. Hard to read, but it seems that Euler is using an infinite series in his approach somewhere.

7. Hi everyone,
any one to help me to know if there are still some reward for someone who prove the goldbach conjecture,
Many thanks.
(PS. I know it was some reward until march 2002...)

8. ## Attempted proof of Goldbach's Conjecture

I have an attempted proof of Goldbach's Conjecture. Does anyone have any advice on how to get it reviewed?

9. Originally Posted by ProfessorB
I have an attempted proof of Goldbach's Conjecture. Does anyone have any advice on how to get it reviewed?
If you indeed have enough knowledge of mathematics to solve this problem then you would not not be asking where you can get it reviewed.

10. Thank you for your kind help. Your obvious superior knowledge has produced in me a hitherto unequaled humility. It goes without saying, of course, that you could not possibly be wrong. That condition is reserved for the merely human.

11. Originally Posted by ProfessorB
I have an attempted proof of Goldbach's Conjecture. Does anyone have any advice on how to get it reviewed?
Write it up as a paper and submit it to a refereed mathematical journal. Of course they will not even look at it if it does not conform to accepted convention for maths papers. Even then they may well not look at it if you do not have an identifiable mathematical background (journals receive so may erroneous proofs of GC and FLT that they often do not even look at such papers unless the author has credentials and or background).

Alternatively you could post it on ArXiv.org where you may get some feedback.

CB

12. Thank you for responding. ArXive.org is, however, not available (under construction). I have posted it to a non-mathematical forum but don't have a lot of optimism for response.

13. Originally Posted by ProfessorB
Thank you for your kind help. Your obvious superior knowledge has produced in me a hitherto unequaled humility. It goes without saying, of course, that you could not possibly be wrong. That condition is reserved for the merely human.
You should take note of what ImPerfectHacker says for a number of reasons, one of which is that mathematical background will be taken into account by anyone qualified to read your proof before they bother.

CB

14. Originally Posted by ProfessorB
Thank you for responding. ArXive.org is, however, not available (under construction). I have posted it to a non-mathematical forum but don't have a lot of optimism for response.