Why Goldbach's conjecture is still non-solved and is difficult to prove? What makes the mathematicians fail when trying to prove it?
closed as off-topic by Peter, Namaste, Brian Borchers, Ethan Bolker, Matthew Leingang Apr 2 '18 at 17:24
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Goldbach’s Conjecture is only difficult to prove right now, with our [limited] mathematical toolbox. One day, when a proof is obtained — and I have no doubt at all that, barring the unexpected end of the human species, the Goldbach Conjecture will eventually be proved — we will be able to know precisely why it was so difficult until that moment.
My intuition says, it’s because we do not truly understand the relationship between addition, multiplication, and exponentiation. Once we have a better grasp of their interactions, many problems that once seemed nearly or totally intractable (e.g., Landau’s problems, Fermat’s Last Theorem, Ore’s Harmonic Divisor Conjecture, Waring’s Problem, etc.) will all fall relatively easily.
According to Donald Knuth (All questions answered) maybe is a "random truth":
Goldbach’s conjecture is just, sort of, true because it can’t be false. There are so many ways to represent an even number as the sum of two odd numbers, that as the numbers grow the number of representations grows bigger and bigger. Take a 101010-digit even number, and imagine how many ways there are to write that as the sum of two odd integers. For an n-bit odd number, the chances are proportional to 1/n that it’s prime. How are all of those pairs of odd numbers going to be nonprime? It just can’t happen. But it doesn’t follow that you’ll find a proof, because the definition of primality is multiplicative, while Goldbach’s conjecture pertains to an additive property. So it might very well be that the conjecture happens to be true, but there is no rigorous way to prove it.
Interesting discussion in https://mathoverflow.net/questions/27755/knuths-intuition-that-goldbach-might-be-unprovable.