Did David Hilbert consider these two problems insoluble? (Because the Goldbach-Euler binary problem is most likely unsolvable due to Gödel's theory). Undecidability here implies the impossibility of either proving or disproving. I will be glad to opinions of experts.
No, he didn’t. And many people fail to realize that it’s not enough to invoke Gödel’s theorem like some sort of incantation: What’s necessary is to prove that a specific proposition is undecidable. Gödel’s theorem doesn’t state which propositions are undecidable, it, simply, implies that such propositions exist. Proving that a particular proposition is undecidable is of comparable difficulty to that of proving it true or false, if, in fact, it isn’t.
It might be useful to study these lecture notes: https://www.dpmms.cam.ac.uk/~wtg10/3primes.dvi , by Timothy Gowers, regarding Vinogradov's proof of the ``three primes theorem''; and this paper, Article Every odd number greater than 1 is the sum of at most five primes
by Terence Tao, in order to better understand what the issues are.
Of course, the fact that these problems are unsolvable still needs to be proven if this is true. However, David Hilbert combined the two problems mentioned above into the eighth problem long before the sensational works of Gödel and Vinogradov.
Goldbach conjecture is equivalent, let me guess, to the conjecture saying that every prime is paired with another via some power of 2.
Hardy and Littlewood showed in 1923 that if some generalization of the Riemann hypothesis is true, the Goldbach hypothesis is valid for all sufficiently large odd numbers. However, this was years after Hilbert's report. The circular method was also already created after Hilbert's speech at the Second International Congress of Mathematicians.
The problem with Hilbert is that he resorts to generalizations that do not often lead to a solution of the problem. As I have proved after 10 years of work on the Riemann Hypothesis, the assumption is wrong. The other unsolved problem is solved by simple reasoning. And the two exist in my site.
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