Timeline for Can anything deep be said uniformly about conjectures like Goldbach's?
Current License: CC BY-SA 4.0
13 events
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| Mar 13, 2019 at 3:33 | comment | added | user21820 | Ah I see okay then perhaps we don't disagree after all. Sure, if we consider the heuristic as always an approximant, then we can indeed use it to make 'good guesses' about the truth value of various conjectures. But I don't get why you said "Hardy-Littlewood conjectures" in the plural, since the first one contradicts the second. In fact, the second one is of the exact same sort as Goldbach-like conjectures in the sense that it makes universal local claims about primes (one claim for each $y$). Whereas the first one makes only global (asymptotic) claims. | |
| Mar 13, 2019 at 3:09 | comment | added | H A Helfgott | Well, I didn't mean tweaking the model, but rather recognizing it as a first approximation. The Cramér model itself is a little crude - the Hardy-Littlewood conjectures are really the right statement; Maier's theorem is an example of a non-obvious situation where the difference matters. | |
| Mar 13, 2019 at 2:26 | comment | added | user21820 | I do agree that a heuristically unlikely counter-example would be a strong argument for not relying on the heuristic. What I don't agree is your implicit suggestion that we might be able to refine the heuristic or identify which situations the heuristic is applicable to, apart from what can actually be proven. I consider primes as completely non-random, and so there is no reason to expect to be able to refine a wrong heuristic without it being like whack-a-mole. On the other hand, I believe that Goldbach-like facts are pure 'luck', and so there is not even any reason to attempt refinement. | |
| Mar 12, 2019 at 22:49 | comment | added | H A Helfgott | A heuristic is precisely that: it cannot be proved or disproved. What it tells you is that, in a broad range of situations, primes should behave as if they were random, and that the effect becomes (typically) stronger, much stronger, for larger numbers - to the point that a single very large counterexample would be a strong argument for narrowing the kind of situations to which the heuristic is applicable (or else for refining the model). | |
| Mar 12, 2019 at 8:01 | comment | added | user21820 | This is different from things like Merten's conjecture, where the probabilistic heuristic is used not for single (small) cases but for asymptotic behaviour, and so we can actually say something if the heuristic predicts something wrongly. (I did upvote your post, but as explained I still think your point about "revising models on finding large counter-examples" is unjustifiable.) | |
| Mar 12, 2019 at 7:58 | comment | added | user21820 | Specifically, we already know absolutely that the primes are not random, so the presence of facts about natural numbers that are unlikely if the primes were randomly distributed according to the probabilistic heuristic tells us absolutely nothing that we didn't already know, and hence I still don't see why you think we can possibly revise the heuristic in a mathematically justified manner. | |
| Mar 12, 2019 at 7:56 | comment | added | user21820 | Hmm I think you're still missing the point. If you actually have an unbiased source of random bits, you can compute the probability that $n$ bits from that source are all zero, getting $1/2^n$. And so if you have an unknown source and get $20$ zero bits in a row, you are justified in saying that it's likely not an unbiased source of random bits. However, applying that kind of argument here yields nothing! All we can conclude from existence of large counter-examples to Goldbach is that the primes were not generated by a random source with a PNT-based distribution, which... we already know! | |
| Mar 12, 2019 at 4:58 | comment | added | H A Helfgott | On small numbers: Goldbach's conjecture used to say that every positive even number is the sum of two primes. Then $1$ got demoted in the 19th century and is no longer a prime, so now the conjecture says that every even number $\geq 4$ is the sum of two primes. | |
| Mar 12, 2019 at 4:55 | comment | added | H A Helfgott | Well, the probability (according to a given model) that $n$ be a counterexample generally decreases as $n$ increases. In this case, if our models are correct, it is very unlikely that there are any counterexamples of size $n\geq 10^{20}$ (say), and so the appearance of a single counterexample of size about $10^30$ would be strong evidence that the model needs work. | |
| Mar 10, 2019 at 10:07 | comment | added | user21820 | Anyway thanks for the remark about Merten's conjecture, though as you say it isn't an example since the probabilistic models agreed with its eventual failure. | |
| Mar 10, 2019 at 10:00 | comment | added | user21820 | Also, I'm not sure I understand your point that if there is a single counter-example to Goldbach at around $10^{30}$, we would be well-advised to see whether we need to revise our models. The whole point of the probabilistic model is that it is based on a assumption that holds for large intervals but not for what we apply it to. So I'm not sure how one could possibly revise it when failures arise by 'random luck'. More precisely, how would we tell if it isn't 'random luck'? Until we see too many conjectures have 'unlikely' large counter-examples, I would believe it is 'random luck'. | |
| Mar 10, 2019 at 9:54 | comment | added | user21820 | Thanks for your answer! So do you agree with user36212 and I that the truth value of such conjectures for small numbers is more or less 'random' (where "small" here depends on the specific conjecture, say less than one billion for Goldbach)? | |
| Mar 9, 2019 at 16:52 | history | answered | H A Helfgott | CC BY-SA 4.0 |