Timeline for Can anything deep be said uniformly about conjectures like Goldbach's?
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| Mar 7, 2019 at 16:48 | comment | added | user21820 | I'll think about that. I believe it should be possible, just like the answer to the generalized Collatz conjecture is uncomputable, but of course it would have to be simple enough to be convincing that one isn't 'cheating' somehow. In the best case, one might hope for an MRDP-based theorem where the required polynomial is actually simpler when we can include primality restrictions. After all, encoding sequences is often done using prime factorization. | |
| Mar 6, 2019 at 18:35 | comment | added | Timothy Chow | @user21820 : That is an interesting angle. Perhaps you can formulate a precise conjecture (or question) along these lines; e.g., define a precise family of "Goldbach-like conjectures" such that, heuristically, each member has finitely many counterexamples, and conjecture that the sequence of counterexamples is uncomputable. | |
| Mar 6, 2019 at 17:35 | comment | added | user21820 | Hmm... does the MRDP theorem make a compelling enough case (to you) that some kind of diophantine equations generally have ad-hoc solution sets? Do you think that Goldbach-like conjectures (i.e. involving primality restrictions) do not suffer from this general phenomenon, or that the ones mathematicians have been historically interested in are somehow special? | |
| Mar 5, 2019 at 18:06 | comment | added | Timothy Chow | @user21820 : I think we're getting into unanswerable philosophical territory. The answer to your question depends on what one deems "interesting," and is highly contingent on the current state of human knowledge rather than on anything intrinsic or objective. Perhaps an extraterrestrial who is much smarter than we are and who has a very different sense of mathematical taste would see lots of deep connections, yet when confronted with the same facts, we might find the connections uninteresting or unintelligible. | |
| Mar 5, 2019 at 15:15 | comment | added | user21820 | Yes, if there are connections between phenomena that don't have the same 'surface presentation', then I would consider the underlying reasons 'deep'. Do you believe that in general Goldbach-like conjectures have such kind of connections to other areas of mathematics, or do you believe (as I do) that they are usually like 'random program' kind of phenomena? I don't doubt that a few may have deep connections, but I think the vast majority don't (and user36212 seems to agree). In contrast to this. | |
| Mar 5, 2019 at 14:51 | comment | added | Timothy Chow | @user21820 : Thank your for your clarification. I don't understand what you mean by "deep" when it comes to small cases. The sum of the first $k$ squares equals $n^2$ only when $(k,n) = (1,1)$ or $(24,70)$. Is this fact "deep"? It is related to properties of the Leech lattice. Similarly, sporadic counterexamples to other asymptotic number-theoretic statements could have connections to sporadic phenomena in other areas of mathematics. Would that be "deep"? | |
| Mar 5, 2019 at 6:38 | comment | added | user21820 | Yes, as I commented on my question, I did see this discrepancy with Maier's theorem. But I do not consider that it is a true discrepancy, because prime gaps are not as 'random' as Goldbach-like sums; there is a 'correlation' between a prime and the gap after it. So I do consider Maier's theorem as deep, but it isn't of the same kind as Goldbach-like conjectures. The other conjecture you mention does qualify as a uniform reason for asymptotic behaviour, but doesn't answer my question about the small cases. | |
| Mar 5, 2019 at 1:16 | history | answered | Timothy Chow | CC BY-SA 4.0 |