Academic: Read up on it! (Score 1)

It would be nice if someone here actually showed some academic initiative and looked up Golbach's problem in standard references to comment on the work that as already been done in the field.

I happen to have a copy of the Encyclopedic Dictionary of Mathematics (Mathematical Society of Japan, translated, 1980 by MIT Press) here in my den.

Looking up Goldbach I find:

"Goldbach's problem is found in letters (1742) he exchanged with L. Euler. In them he stated that every positive integer can be expressed as the sum of primes. More precisely, he conjectured that any even integer not smaller than 6 can be expressed as the sum of two odd primes and that any odd integer not smaller than 9 can be expressed as the sum of three odd primes.

I.M. Vinogradov (1937) proved that every sufficiently large odd integer can be expressed as the sum of three primes."

The article then goes on to summarize the gist of this proof which involves a summation of (exp( - 2 * pi * i * ( a/q ) * N)) and a convergent series S(N)= sum(q=1, inf, A(q,N)) (which is "absolutely convergent and is equal to" [a copy of product expressions than can't be typeset here]).

It doesn't mention the magnitude of "sufficiently large" (read articles at Wolfram Research as referenced by the /. "related links" box near this article for more on that). Anyway --- Vinogradov solved Goldbach's conjecture for "sufficiently large odd integers."

Apparently "a finite or infinite sum of exponential functions such as this is called a trigonometric sum."

The article then discusses the case of even integers: "I.G. van der Corput, T. Estermann, and N. G. Cudakov proved simultaneously (1938) that almost all even integers (i.e., except a set of density 0) can be expressed as the sum of two primes." [emphasis mine].

All of this is in section 5.C of EDM.

The article goes on to comment on further work by Cudakov and Linnick, their introduction of "function-theoretic methods" and "the density theorem concerning the zeros of L-series" (which is greek to me).

They conclude with reference to the work of Zulauf (1952) which "continued along the same lines" as suggested by G.H. Hardy and J.E. Littlewood (1919).

I'm sure that some of the more academic participants here at /. might have access to a more comprehensive library than the bookshelf in my den. I'm not a mathematician, my education of the subject pretty much ended with high school Calculus, one audited class in combinatorics at Reed College and a few forays into Mathematica (tech support for my father, who is the arm chair mathematician and physicist of the family).

Personally I'd recommend starting with a thorough review of the literature to see what approaches have already failed. When you understand why those approaches failed then you can apply those tests to any methods you propose as a solution.

For example, if one could show that all even numbers less than some number N are the sum of pairs of primes less then N' and you could show that N and N' are related --- that N+x results in N'+y --- you might have a inductive proof in there somewhere.

If you win the $1M (or was that in pounds?) then feel free to donate a few grand to the FSF in my name.

I suspect that a proof of Goldbach's conjecture might entail a more fundamental formula or series that could generate the set of all primes. That would be a much more important finding since it would shatter the mathematical security of many cryptographic algorithms --- particularly the RSA PK (public key) system. (There's billions of $ at stake in that case).