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Why Goldbach's conjecture is still non-solved and is difficult to prove? What makes the mathematicians fail when trying to prove it?

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    $\begingroup$ The best way to answer your question is this: Try proving it. $\endgroup$ – Shaun Ault Nov 1 '13 at 12:05
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    $\begingroup$ Difficulty of proof can only be talked about in retrospect. By observing that we do not have a proof yet even after many attempts, we may arrive at the firm opinion that a proof is difficult. Then again, recent results are very close to the goal (but with nontrivial methods) $\endgroup$ – Hagen von Eitzen Nov 1 '13 at 12:13
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    $\begingroup$ .. but for an intuitive explanation: Priems were made for multiplying, not adding. $\endgroup$ – Hagen von Eitzen Nov 1 '13 at 12:14
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    $\begingroup$ Tou can see here that this is also the case. There are proofs for questions closely related to the Goldbach's conjecture. $\endgroup$ – Mateus Sampaio Nov 1 '13 at 12:36
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    $\begingroup$ Recently the Goldbach's conjecture has been proved to be true. The paper about it has been accepted to publish in one of the reputed journal. One of my friend has a copy of the paper. $\endgroup$ – D. N. Nov 4 '13 at 10:04
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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.

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  • $\begingroup$ "because we do not truly understand the relationship between addition, multiplication, and exponentiation" - can you elaborate on that? $\endgroup$ – shuhalo Mar 13 '17 at 22:55
  • $\begingroup$ @shuhalo: I don't know if I can, really… If I could explain it to any satisfactory degree, I could probably already prove some of those “intractable problems“. ;) $2+2=2*2=2^2$, and that [triple-]relationship doesn't hold for any other number. $a^n+b^n=c^n$ has an infinite number of integer solutions for n=1 and n=2, and then suddenly none when n > 2. This isn't just "Strong Law of Small Numbers" stuff (though some will claim it is) — there are deep truths we simply haven't uncovered yet. $\endgroup$ – Kieren MacMillan Mar 14 '17 at 18:24
  • $\begingroup$ I am not so sure that Goldbach's conjecure will ever be proven, even if it is $(1)$ true (which is believed by virtually all mathematicians, but we cannot be sure. If we could , a proof would be obsolete) and $(2)$ provable, if it is true (Goedel's incompleteness-theorem might apply to Goldbach's conjecture). Fermat's last problem was finally solved, but that does not mean that it is suddenly an easy problem. It is still a very deep theorem that required extrem powerful tools to solve it. $\endgroup$ – Peter Apr 1 '18 at 10:22
  • $\begingroup$ Concerning the problem that primes are "for multiplications". The problem whether an odd perfect number exists is unsolved as well, and I do not see where addition-properties play any role in this problem. So, the problem with the additions is apparently not the only problem with Goldbach, but I admit it makes it more difficult. $\endgroup$ – Peter Apr 1 '18 at 10:24
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    $\begingroup$ @Peter: When Terjanian, in 1977 [!!], developed his direct proof of FLT I in the case of even exponents, most mathematicians were very surprised. (cf. Ribenboim: “The proof will require only elementary considerations, so it is surprising that it was not found beforehand.”). Chein similarly shocked people when he massively simplified the proof of Catalan’s Conjecture in the case $p=2$. More recently, the limit on the twin prime gap was massively reduced, taking the entire math world by surprise. Notice a pattern? ;) $\endgroup$ – Kieren MacMillan Apr 1 '18 at 20:59
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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.

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