Timeline for Examples in which probabilistic heuristic reasoning fails
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2018 at 0:21 | comment | added | Craig Feinstein | That argument has been in the literature for quite some time. It is a good heuristic probabilistic argument. If ever someone proves the Collatz Conjecture false, then that would be an answer to my question. | |
| Aug 23, 2018 at 21:04 | comment | added | David G. Stork | Tao describes a probabilistic argument that the orbits of the Collatz conjecture should decrease in magnitude, and thus ultimately pass through $1$. It is of course conceivable that the Collatz conjecture is false (and possibly the probabilistic argument correct), but all simulation studies suggest that the conjecture is true. Admittedly, this is no proof, but it suggests that there may be a hidden flaw in the statistical argument. (See: terrytao.wordpress.com/tag/collatz-conjecture ) | |
| Aug 22, 2018 at 16:17 | comment | added | Craig Feinstein | @Wojowu, I am now reading the paper by Ian Richards that I linked to above, and it appears that it addresses the issue that Greg Martin raised. Anyway, the paper is a good read. | |
| Aug 22, 2018 at 7:58 | comment | added | Wolfgang | @CraigFeinstein Funny coincidence (serendipity??), as I've asked that question less than a week ago!! I was convinced that you'd seen it and wanted to narrow it down to your specific field of interest. Anyway, no need to delete it, and that would kill the specific discussion going on in here. | |
| Aug 22, 2018 at 6:50 | comment | added | Wojowu | @GregMartin Do you have a reference? I have not heard of that refined heuristic. | |
| Aug 22, 2018 at 1:04 | comment | added | Craig Feinstein | @Wolfgang, I was not aware of that link. It seems that my question is only about probabilistic arguments, while that question includes all types of heuristics. I am not sure what to do. Should I delete this question? | |
| Aug 22, 2018 at 1:01 | comment | added | Craig Feinstein | Actually, negative numbers are not allowed to be prime in that particular conjecture. See the link. | |
| Aug 21, 2018 at 22:38 | comment | added | Greg Martin | When examined carefully, the heuristic actually says that density of primes increases as the absolute value of numbers get larger. (Here negative numbers are allowed to be prime.) Therefore the interval of length $y$ with the largest number of primes should be the interval whose elements are smallest in absolute value — that is, the interval $(-\frac y2,\frac y2]$. | |
| Aug 21, 2018 at 20:22 | comment | added | Wolfgang | related: Are there examples of conjectures supported by heuristic arguments that have been finally disproved? | |
| Aug 21, 2018 at 19:52 | comment | added | Craig Feinstein | @GregMartin, I don't see how the original probabilistic heuristic is flawed, even if in retrospect it is. After all, the density of primes decreases as the numbers get larger. Even the possibility that it is flawed is shocking. | |
| Aug 21, 2018 at 16:47 | comment | added | Sam Hopkins | "Cohen–Lenstra heuristic and roots of unity" by Gunter Malle: "We report on computational results indicating that the well-known Cohen–Lenstra–Martinet heuristic for class groups of number fields may fail in many situations. In particular, the underlying assumption that the frequency of groups is governed essentially by the reciprocal of the order of their automorphism groups, does not seem to be valid in those cases. The phenomenon is related to the presence of roots of unity in the base field or in intermediate fields." (sciencedirect.com/science/article/pii/S0022314X08000504) | |
| Aug 21, 2018 at 16:46 | comment | added | Greg Martin | In (2), I would argue that the original probabilistic heuristic was flawed; the corrected heuristic (proposed after it was shown that (1) and (2) are inconsistent) predicts that $\pi(x+y) \le \pi(x) + 2\pi(\frac y2)$, which I think is believed to be true. | |
| Aug 21, 2018 at 16:21 | comment | added | Sylvain JULIEN | Maier's theorem shows the Cramer probabilistic model can lead to false assumptions. | |
| Aug 21, 2018 at 16:18 | history | asked | Craig Feinstein | CC BY-SA 4.0 |