We know that the strong Goldbach conjecture:
"Every even numberis 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
Unsolved problem sent as a letter to Euler in 1742.Originally Posted by CaptainBlank
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 thatis big enough. And in 1989 this number was reduced to
. 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 anUnsolved 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
Note: I have taken the above facts from my Number Theory book. I have changed the wording to avoid copyright material.
easily accessible souce I would not be asking
(By the way the text of Goldbach's letter is available here)
RonL
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
Again, I appreciate your advice. I realize the improbability of my claim - ImPerfectHacker just has a rather boorish way of communicating.
I may indeed be wrong, but I am not without some justification in believing I have, if not having solved GC, then at least having made some progress. I am also aware that history has provided us with some examples of progress being made in mathematics by those without formal education in the field - Ramanujan being the most striking example.
Once again, thank you.
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