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I just read about the Goldbach Conjecture and it got me thinking about probabilities. Supposing that prime numbers are somewhat randomly distributed) then if we calculate the odds of a given even number being the sum of two primes, the odds are much higher the larger the prime number. The exact formula is a bit beyond me, but suppose we were looking at a situation where the odds of there NOT being two primes that sum to the even number are cut in half each time the number increases by two. So if we want to know if all the even numbers from 2 to n are sums of primes we would use (1-(1/2)(3/4)(7/8)..(((n^2)-1)/(n^2))) the limit of which appears to be in the neighborhood of a constant .711212 according to excel.

And I remember learning that there must be theorems that are unprovable.

Is it possible that Goldbach's Conjecture might fall into a class of problems that true but unprovable NOT because the theorem is self-contradictory like the "this theorem is false", but instead is unprovable because there just isn't a good reason for it to be true? That is, the theorem could have been true or false, and had a fixed probability of being one or the other, but unless the roll of the dice hits right and comes out false (and we can do a proof by contradiction), we'll never be able to prove it? We would be able to make the probability smaller by finding more cases (that is, if we show the first 100000 even numbers are sums of primes then we can recalculate the odds that a larger number will be not be a sum of primes and the new odds will be smaller than the old odds), but we'll never be able to eliminate the odds completely.

I'm sure there must be someone who has looked into this before. What is the attitude of the math community toward this point of view? Is it seen as useless and ignored because it can't be used to prove anything? Is it considered heretical to suggest something could be true without having a reason?

Readin
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  • Prime numers are not randomly distributed though, see answer of this question: http://math.stackexchange.com/questions/421353/are-primes-randomly-distributed – Sak Feb 29 '16 at 04:51
  • I find some parts of your question being more or less of the form "what will be in the future?", which is rather hard to answer. – barak manos Feb 29 '16 at 04:52
  • It is possible that the Goldbach conjecture is not provable in say first-order Peano arithmetic, or ZFC. However, results close in spirit to the Goldbach conjecture have been proved. For example, every large enough odd number is the sum of $3$ primes. – André Nicolas Feb 29 '16 at 05:07
  • Incidentally, $1-\frac12\times \frac34 \times \frac56 \times \cdots \times \frac{2n-1}{2n}= 1- \frac{(2n)!}{4^n (n!)^2} \to 1$ as $n \to \infty$ and $1-\frac1{\sqrt{\pi n}}$ is a good approximation for large $n$. On a separate point, $2$ is not the sum of two primes. – Henry Feb 29 '16 at 08:07
  • I haven't forgotten about this question; I'm trying to figure out a better way to ask it. – Readin Mar 19 '16 at 04:55
  • It is true that any random set stops being random as soon as you generate it. If I want to say something about 5 rolls of the dice before I roll them it is entirely different from saying something about those rolls after I've rolled them. Some sets, for example the even numbers, we are able to say a great deal about without actually calculating all of them. I can tell you how many even numbers there are between 0 and 2 billion without generating an total list of them. I can't say the same thing about primes, nor are we sure that we ever will be able to. (cont.) – Readin Mar 19 '16 at 04:58
  • At this point, the best we can do is give a probability of how many primes there are between 0 and a very large n, unless we've gone to the trouble of generating all those primes. (cont.) – Readin Mar 19 '16 at 05:00
  • There is no possibility of ever creating a proof that lists all the primes, because there are an infinite number of primes. – Readin Mar 19 '16 at 05:02
  • If Goldbach's conjecture is false, then we can prove it so by finding a counterexample. If the conjecture is true, then we have to prove that there cannot be a counterexample because of some property of primes. But what if there is no such property of primes that says it but instead it just so happens that no matter how many you look at you just never hit one that is a counterexample? What role would probability play in that? – Readin Mar 19 '16 at 05:04
  • If I get it correctly, you mean that Goldbach Conjecture could be "true" not because of some specific prime related property, but just because there are so many of them. Even if that's the case that doesn't make it an unprovable statement. Think of this scenario: [1] find a "less dense" set of odd numbers that have the "Goldbach property" [2] demostrate that if an infinite set of odd numbers has the Goldbach property, all the denser odd numbers set have the same property too. This way you could "demostrate" Goldbach conjecture without talking about primality. – Insac Nov 20 '16 at 14:47

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