It is written on Wikipedia:

"During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdős (1949). While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, later proofs introduced various simplifications through the use of Tauberian theorems but remained difficult to digest. A short proof was discovered in 1980 by American mathematician Donald J. Newman. Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis."

I must admit that I did not go through those proofs but it is to be expected that, up to a certain point, proofs that are more simpler will appear, althugh it could be that their length will increase, proportional to their simplicity, but that also does not need to be the case generally.

My thoughts on Goldbach are mainly of the form:

"Prime numbers become less frequent as we advance trough a set of natural numbers so that gives an aspect of less chance that bigger even numbers will be sum of two primes but as even numbers get bigger and bigger then there is more and more of primes that occur before some even number so the chance that an even number will be sum of two primes increases, and this latter chance seems to dominate since an even number seems to don´t care much how will prime numbers be distributed after him, he enjoys the fact that the number of possible sums that constitute him increase as he increases."

Of course , this ain´t nothing but philosophy, but I was thinking to what would an approach of the form "Suppose that there exists even number that cannot be written as a sum of two primes." lead to?

Also, do you think that we will need complex analysis and various integrals and logarithmic functions in order to give (first) proof of Goldbach?

Can some combination of enormous number of more or less elementary results from number theory lead to a resolution of Goldbach or you think that probably a first proof will use heavy machinery?

What are your thoughts on this? (Of course that I believe that Goldbach is provable.)

  • $\begingroup$ We can hope of course, but I think it's impossible. $\endgroup$ – Michael Rozenberg Oct 3 '17 at 14:36
  • $\begingroup$ @MichaelRozenberg But, why? $\endgroup$ – user480281 Oct 3 '17 at 14:55
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    $\begingroup$ Do you know the Vinogradov's theorem on the weak Goldbach conjecture ? We can hope elementary ways to show Goldbach is almost always true in some strong sense, but solving the remaining "almost" part can't be elementary. Also there are no simple proofs of the prime number theorem, all of them use non-trivial properties of $\zeta(s)$. $\endgroup$ – reuns Oct 3 '17 at 15:08
  • $\begingroup$ Goldbach's conjecture apparently is unbelievably difficult. My personal guess is that it will not be solved in the near future. I am quite convinced that a proof (if Goldbach is provable at all, which is far from certain) will need "big guns" of mathematics, like Andrew Wiles' proof. The conjecture is very famous, much effort was made by the best mathematicians and not the slightest idea how it COULD be proven has been detected. So, an "obvious" proof is extremely unlikely to exist. $\endgroup$ – Peter Jul 5 '18 at 13:28

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