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In particular, I realize that with each number we check there is slightly less of chance of finding a violation, but instead I am asking if larger numbers are somehow 'padded' with more solutions? And by going higher up in bigger numbers does the chances drop way down due to the nature of them being large? Does it seem more likely that if a violation was found, it would have been early in the series, like say, below the number 20? It seems that after a googolplex for example, the odds would be essentially zero. Or, is it still the same chance as before?

Juel Herbranson
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    The expected number of solution grows with the magnitude of the given even number $n$ , hence counterexamples in fact get more and more unlikely. Nevertheless , we cannot rule out a huge counterexample beyond , say , googolplex. On the other hand, we won't be able to show that there is a representation for , say , googolplex because we cannot check numbers in that range for primality in practice. – Peter Jan 25 '23 at 08:55
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    In fact, I do not think that any mathematician seriously doubts about the truth of Goldbach's conjecture, the heuristical evidence is overwhelming. Even , if the smallest counterexample would have , say , $50$ digits , we could not detect this in practice. – Peter Jan 25 '23 at 09:01
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    It's really quite difficult to give a satisfying answer to this question, because as far as I know there is no formal theory of how to assign probabilities to the truth or falsehood of mathematical statements. We can really only make heuristic arguments here. You may be interested in the probabilistic heuristic in favor of Godlbach's conjecture which is discussed on Wikipedia here: https://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Heuristic_justification – Qiaochu Yuan Jan 25 '23 at 09:08
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    @Peter ?? The heuristic predicts that Goldbach is true for $n$ large enough not for all $n$. – reuns Jan 25 '23 at 11:14
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    @reuns If you have data of a careful analysis how "likely" a counterexample is considering the current search limit and if this analysis indicates a reasonable possibility for a counterexample, then prove me wrong. – Peter Jan 25 '23 at 11:40
  • @Peter I explained why you are wrong – reuns Jan 25 '23 at 13:01
  • No, you didn't. Clear , a heuristical evidence is never a proof, but you did not explain why a single (small) counterexample has a reasonably high chance to exist. If the expected number of counterexamples is near $0$ , the chance that Goldbach is true is very high. – Peter Jan 25 '23 at 13:04
  • Let $E(x)$ be the number of even integers $\le x$ that are not expressible as sum of two primes. Then it is known that $E(x)<x^{1-\delta}$ for some $\delta>0$. – TravorLZH Jan 25 '23 at 16:20
  • @Peter, if you are available would you consider this other entry? I'm not confident that it is well stated, and I've tried to clarify in the comments. It is relevant and touches on one of your comments here.https://math.stackexchange.com/q/4625781/730354 – Juel Herbranson Jan 26 '23 at 02:22

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